Analytic Methods for Simulated Light Transport

ANALYTIC METHODS FOR

SIMULATED LIGHT TRANSPORT

James Richard Arvo

Yale University

This thesis presents new mathematical and computational to ols for the simula

tion of light transp ort in realistic image synthesis New algorithms are presented for

exact computation of direct il lumination eects related to light emission shadow

ing and rstorder scattering from surfaces New theoretical results are presented

for the analysis of global il lumination algorithms which account for all interreec

tions of light among surfaces of an environment

First a closedform expression is derived for the irradiance Jacobian which is

the derivative of a vector eld representing radiant energy ux The expression

holds for diuse p olygonal scenes and correctly accounts for shadowing or partial

o cclusion Three applications of the irradiance Jacobian are demonstrated lo cat

ing lo cal irradiance extrema direct computation of isolux contours and surface

mesh generation

Next the concept of irradiance is generalized to tensors of arbitrary order A

recurrence relation for irradiance tensors is derived that extends a widely used

formula published by Lamb ert in Several formulas with applications in com

puter graphics are derived from this recurrence relation and are indep endently

veried using a new Monte Carlo metho d for sampling spherical triangles The

formulas extend the range of nondiuse eects that can be computed in closed

form to include illumination from directional area light sources and reections from

and transmissions through glossy surfaces

Finally new analysis for global illumination is presented which includes both

direct illumination and indirect illumination due to multiple interreections of

light A novel op erator equation is prop osed that claries existing deterministic

algorithms for simulating global illumination and facilitates error analysis Ba

sic prop erties of the op erators and solutions are identied which are not evident

from previous formulations A taxonomy of errors that arise in simulating global

illumination is presented these include p erturbations of the b oundary data dis

cretization error and computational error A priori b ounds are derived for each

category using prop erties of the prop osed op erators

ANALYTIC METHODS FOR

SIMULATED LIGHT TRANSPORT

A Dissertation

Presented to the Faculty of the Graduate Scho ol

Yale University

in Candidacy for the Degree of

Do ctor of Philosophy

James Richard Arvo

December

James Richard Arvo

Dedicated to the memory of my parents

Mathilda Marie Arvo and Helmer Matthew Arvo

Acknowledgements

There are many p eople I wish to thank First and foremost I oer my sincere

thanks to my advisor Martin Schultz for his unhesitating supp ort Without his

encouragement and his steadfast condence in me this thesis would never have

been completed I am also deeply indebted to Ken Torrance who served as an

external committee memb er Ken has been a wonderful inuence and a guiding

light I am grateful to him for his kindness his constant willingness to listen and

and the encouragement he oered when it was most needed Because of to help

Ken I have learned to trust my own instincts

I also wish to thank my other internal committee memb ers Ken Yip and

Vladimir Rhoklin who were a pleasure to interact with and my external commit

tee memb ers Al Barr Pat Hanrahan and John Hughes who generously oered

their time and exp ertise Thanks also to Michael Fischer for kindly helping me to

nd my way to completion

I am grateful to Don Greenb erg for the opp ortunity to be part of the Pro

of the research for gram of Computer Graphics at Cornell University where most

this dissertation was conducted Sp ecial thanks to Erin Shaw Steve Westin Pete

Shirley and John Hughes for carefully reading various p ortions of this work and

oering detailed comments Many thanks to my coauthors Julie Dorsey Dave

Kirk Kevin Novins David Salesin Francois Sillion Brian Smits Ken Torrance

and Steve Westin from whom I have learned so much over the years and to Pete

Shirley Dani Lischinski Bill Gropp and Jim Ferwerda for enumerable stimulating vi

vii

discussions Thanks also to Ben Trumb ore and Alb ert Dicruttalo for mo deling and

software supp ort to Dan Kartch for all the help with do cument preparation to

Jonathan CorsonRikert Ellen French Linda Stephens and Judy Smith for admin

istrative supp ort and to Hurf Sheldon for manyyears of cheerful and professional

systems supp ort From my days at Ap ollo Computer Id like to thank Al Lop ez

Fabio Pettinati Ken Severson and Terry Lindgren for all their encouragement

Many fellow students and assorted friends have also help ed and inspired me along

the way including Lenny Pitt Mukesh Prasad Michael Monks Ken Musgrave

Andrew Glassner Mimi and No el Mateo and Susan Vonderheide

The p erson to whom I am most indebted is my wife Erin whose love and

understanding are truly without b ound It is dicult to imagine how I could have

completed this work without her supp ort at every step along the way

This researchwas conducted at the Program of Computer Graphics supp orted

by the NSFARPA Science and Technology Center for Computer Graphics and

Scientic Visualization ASC and p erformed on workstations generously

provided by the HewlettPackard Corp oration

Much of the material in this thesis was presented at ACM SIGGRAPH under

the titles The Irradiance Jacobian for Partially Occluded Polyhedral Sources

and A Framework for the Analysis of Error in Global Illumination Algorithms

and at SIGGRAPH under the titles Applications of Irradiance Tensors to the

Simulation of NonLamb ertian Phenomena and Stratied Sampling of Spherical

Triangles The copyright for this material is held bytheACM and is included in

this thesis by p ermission of the ACM

Table of Contents

Acknow ledgments vi

List of Figures xi

Nomenclature xiii

Intro duction

LightTransp ort and Image Synthesis

Physical Principles

Computational Asp ects

Historical Bac kground

Thesis Overview

Summary of Original Contributions

Particles and Radiometry

Phase Space Measure

Particle Measures

Particle Axioms

Extension to a Measure

Phase Space Density

Phase Space Flux

Radiance

Irradiance

Derivatives of Irradiance

Intro duction

Lamb erts Formula

The Irradiance Jacobian

Vertex Jacobians

Intrinsic Vertices

Apparent Vertices

Polygon DepthClipping

Prop erties of the Irradiance Jacobian

Applications of the Irradiance Jacobian viii

Finding Lo cal Extrema

Direct Computation of Isolux Contours

Solving the Isolux Dierential Equation

Examples of Isolux Contours

Irradiance Tensors

NonLamb ertian Phenomena

Preliminaries

Three Related Quantities

Polynomials over the Sphere

Elements of Dierential Geometry

Generalizing Irradiance

Denition of Irradiance Tensors

The Form Dierential Equation

Extending Lamb erts Formula

The Radiation Pressure Tensor

A General Recurrence Relation

Angular Moments

Axial Moments

DoubleAxis Moments

Exact Evaluation of Moments

Solid Angle

Boundary Integrals

Algorithms for Ecient Evaluation

Normalization

Optimizations

Applications to Image Synthesis

Directional Luminaires

Glossy Reection

Glossy Transmission

Further Generalizations

Spherical Luminaires

Spatially Varying Luminaires

Comparison with Monte Carlo

Sampling by Rejection

Stratied Sampling

An Algorithm for Sampling Spherical Triangles

Implementation and Results

Verication of Assorted Formulas

Op erators for Global Illumination

Classical Formulation

Linear Op erator Formulation

Normed Linear Spaces

Function Norms

Op erator Norms

Norms of Sp ecial Op erators K G and M

Related Op erators

Adjoint Op erators

Op erators for Diuse Environments

The Rendering Equation

Derivation from the Classical Formulation

MeasureTheoretic Denition of Transp ort Intensity

Error Analysis for Global Illumination

Pro jection Metho ds

ATaxonomy of Errors

Perturb ed Boundary Data

Discretization Error

Computational Errors

Error Bounds

Bounding Errors Due to Perturb ed Boundary Data

Bounding Discretization Error

Bounding Computational Errors

The Combined Eect of Errors

Conclusions

Summary

Future W ork

A Additional Pro ofs and Derivations

A Pro of of theorem

Pro of of theorem A

i j k

A Pro of that x y z d is closed

A Pro of of lemma

A Pro of of theorem

A Reducing to Known Sp ecial Functions

A The L and L Norms of K

A Pro of of theorem

Bibliography

List of Figures

Phase space density remains constant along straight lines

Phase space ux is the number of particles crossing the surface dA

There is no distinction between incoming and outgoing rays

The surface P and its spherical projection P

The ve ctor irradiance at point r

The irradiance at point r due to source P

The view from r of the two types of intrinsic vertices

The view from r of the two types of apparent vertices

A dierential change in the position r

The vertex Jacobian J u with respect to two skew lines

Source P is partial ly occluded by two blockers

The vertex Jacobian does not exist at the interse ction of three edges

Projecting the gradient onto a surface denes a D vector eld

A family of isolux contours for three unoccluded sources

Isolux contours on a planar receiver due to a rectangular source

Fil led isolux contours corresponding to the previous gures

Isomeshes generated from families of isolux contours

A stained glass window reected in a Phonglike glossy surface

Reected radiance is determined by the scattering distribution

The three direction cosines x y and z

Acosine lobe dened by a thorder monomial

Ahypothetical bidirectional reectance distribution function

The geometrical relation between the forms d and dA

For any region A S the unit vector n is normal to the boundary

The outward normal to the boundary curve

The integral of u over a great arc

Crosssections of the weighting functions

The solid angle of a spherical triangle

The vectors used to parametrize the arc dened by a polygon edge

Normalizing the weighting function of a doubleaxis moment

Computing the irradiance at the point r

Examples of directional luminaires

Asimple BRDF dened by an axial moment xi

xii

Analytical ly computed glossy reection of a convex polygon

Analytical ly computed glossy reection of a stained glass window

Afrosted glass simulation demonstrating glossy transmission

Apolynomial over the plane

The function plotted as a function of

The irradiance at the origin due to a triangular luminaire

The vertex C is determined by specifying the area of the subtriangle

The point P is chosen using a coordinate system

Aspherical triangle sampled randomly with the new algorithm

The signed maximum deviation of the Monte Carlo estimate

The exact element T and a Monte Carlo estimate

Uniform sampling of a doubleaxis moment of order

Stratied sampling of a doubleaxis moment of order

Uniform sampling of a doubleaxis moment of order

Stratied sampling of a doubleaxis moment of order

The local coordinates of a bidirectional reectance function

The actions of G and K

The net ux at the point r

Operators connecting the spaces of radiance functions

b b

The operators K and G

The three points and four angles used in dening transport quantities

The form dened on the surface M is the pul lback

The conceptual stages for computing an approximate solution

A Clausens integral over the domain

xiii

Nomenclature

Symbol Page Meaning

A A subset of S

A The algebra of p olynomials over S

c Sp eed of lightinavacuum

Ca b c The corner matrix

d The form corresp onding to solid angle

E Particle measure

f r u Radiance function IR S IR wattsm sr

f r u Emission function IR S IR wattsm sr

f r r Transp ort intensity IR IR IR wattsm

f r r Transp ort emittance IR IR IR wattsm

F A B The radiant power leaving A that reaches B directly

g r r The geometry term m

G The eld radiance op erator

H Isomorphism taking surface radiance onto eld radiance

h Plancks constant

I The multiindex i i

matrix I The identity op erator or

J F The Jacobian matrix of F at r

k r u u Bidirectional reectance distribution function sr

k r r r Dimensionless scattering function

K The lo cal reection op erator

L The Leb esgue function spaces

m Leb esgue measure over M

M A collection of smo oth manifolds in IR

M The op erator I KG

nr u Phase space density IR S IR m sr

Total phase space density IR IR m nr

nr Surface normal function MS

P A surface in IR usually a p olygon

xiv

Symbol Page Meaning

IP Fivedimensional phase space IR S

P A algebra of subsets of IP

pr u Ray casting function MS M

qr r Twop oint direction function MMS

Qp The cross pro duct matrix p

r The Euclidean lenght of the vector r

r A p ointinIR or in M dep ending on context

IR The real numb ers

S The unit sphere in IR

T A In tegral of u over A S

n q

T A w Integral of u u w over A S

ur Mono chromatic energy density IR IR joulesm

u A p ointinS ie a direction usually r jj r jj

W Avector normal to the k th face of a p olygonal cone

X A space of radiance functions

X A nitedimensional subspace of X

A Monte Carlo estimator for T

A Monte Carlo estimator for

Kronecker delta function

The unit normal to the k th face of a p olygonal cone

LeviCivita p ermutation symbol

ij k

i j k

i j k Integral of the monomial x y z over S

The angle subtended by the k th edge of a p olygon

Number of times i o ccurs in the multiindex I

Cosineweighted measure over S

r u Distance to the manifold M from r along u

Arandomvariable distributed over a subset of S

An nthorder tensor related to i j k

S Pro jection of the set S IR onto the unit sphere

Bidirectional reectance distribution function sr r

Symbol Page Meaning

Phase space measure

Leb esgue measure on the sphere S

nthorder axial moment

Doubleaxis moment

x y Atwoparameter sp ecial function

r Vector irradiance IR IR wattsm

r Irradiance M IR wattsm

r u Phase space ux IR S IR m sr

X x The characteristic function of the set A

r Radiation pressure tensor IR IR joulesm

Abound on directionalhemispherical reectivity

r The incoming hemisphere of S at r M

r The outgoing hemisphere of S at r M

jjjj L norm for radiance functions

hi Exp ected value

h j i Inner pro duct

j Normalized orthogonal comp onent ofavector

Within a subscript indicates partial derivatives

Equal by denition

Absolute continuity partial order on measures

for dierential forms Exterior pro duct

Double factorial xvi

Chapter

Intro duction

One of the central problems of computer graphics is the creation of physically

accurate synthetic images from complete scene descriptions The computation of

such an image involves the simulation of light transport the largescale interac

tion of light with matter Within computer graphics the problem of determining

the app earance of an environmentbysimulating the transp ort of light within it is

known as global il lumination While direct simulations of this typ e are easier in

remain many many resp ects than the inverse problems of computer vision there

unsolved problems The diculties arise from complex geometries and surface re

ections encountered in real scenes and from the mutual illumination of ob jects

in a scene byinterreected light Fortunately the predominant eects are time in

variant and o ccur at scales much larger than the wavelength of visible light which

p ermits simplied mo dels of light transp ort to be applied with little sacrice in

delity As a result the physical principles that underlie global illumination come

primarily from geometrical optics radiometry and radiative transfer

The current trend in image synthesis researchtoward increasing physical accu

racy stems from a desire to make images that are not only realistic but are also

predictive Prediction is clearly a requirement for applications such as architectural

and automotive design One strategy for reliably predicting the app earance of a

hyp othetical scene from its physical description is to rst simulate the physics of

light transp ort then approximate the visual stimulus of viewing the scene by map

ping the result to a display device Adhering to physical principles also makes the

pro cess of realistic image generation intuitive It is far easier to control a simulation

using familiar physical concepts than through arcane parameters

The eectiveness of a synthetic image hinges on more than correct physics

Other factors include characteristics of the display device such as nonlinearities and

limited dynamic range the physiology of the eye and even higherlevel cognitive

asp ects of p erception Nevertheless it is the physical and computational asp ects of

the problem that currently dominate the eld Consequently global illumination is

largely the study of algorithms for the simulation of visible light transp ort which

includes the pro cesses of light emission propagation scattering and absorption

The cen tral contributions of this thesis are an improved theoretical founda

tion for the study of these pro cesses and a collection of new computational to ols

for their simulation In particular metho ds of functional analysis are employed to

quantify the pro cess of light transp ort whichallows for analysis of error and e

cient algorithms are develop ed for computing various asp ects of illumination and

reection in simplied settings

Light Transp ort and Image Synthesis

This section summarizes the underlying physical principles and computational pro

cedures of light transp ort and justies their application to the problem of global

illumination Some historical context is also provided b oth to clarify the origin of

the fundamental ideas and also to emphasize connections with other elds

Physical Principles

Image synthesis involves the simulation of visual phenomena those that are ob

e or some instrument capable of discerning light intensity and servable by the ey

frequency This level of description is sometimes referred to as phenomenological

as it fo cuses on phenomena corresp onding to the p ercepts of brightness and color

There are manylevels of physical description that can predict and explain the

phenomenology of light Geometrical optics adequately describ es several largescale

b ehaviors of light such as linear propagation reection and refraction but do es

not incorp orate radiometric concepts that quantify light Physical optics is

based on Maxwells equations for electromagnetic radiation which subsumes geo

metrical optics and includes additional phenomena such as disp ersion interference

and diraction these eects can dominate at scales near or b elow the wavelength

of light This level of description is also quantitative However physical optics

is overly detailed at large scales when the radiation eld is incoherent which is

true of macroscopic environments illuminated by common lighting instruments or

luminaires

A third level of description is known as the transport level in the context of

electromagnetic radiation the study of transp ort pro cesses is known as radiative

transfer Radiative transfer combines principles of geometrical optics and

thermo dynamics to characterize the ow of radiant energy at scales large compared

with its wavelength and during time intervals large compared to its frequency

tral to the theory of radiative transfer are the principles of radiometrythe Cen

measurementoflight This level of description is compatible with the phenomenol

ogy of light without b eing overly detailed whichmakes it appropriate to the

task of global illumination and image synthesis

Radiative transfer do es not explain all phenomena at the level of electromag

netism or quantum mechanics yet it may incorp orate information from more de

tailed physical descriptions such as these More complete theories that op erate

at microscopic scales can be used to predict rstorder eects such as lo cal scat

tering and absorption which enter into simulations at larger scales as b oundary

reection mo del of He et al employs or initial conditions For instance the

physical optics to characterize reection from rough surfaces the mo del can then

be incorp orated into a global illumination algorithm op erating at the transp ort

level Another avenue by whichphysical optics can enter into the simulation

is through physical measurement For instance the reectance prop erties asso ci

ated with b oundaries surfaces may be obtained from samples of real materials

using a gonioreectometer

Global illumination can also b e placed within the much broader context of trans

port theory a eld encompassing all macroscopic phenomena that result

from the interaction of innitesimal particles within a medium The macroscopic

b ehaviors of photons neutrons and gas molecules are all within its purview The

unifying concepts of transp ort theory help to clarify the nature of global illumina

tion and its relation to other physical problems For instance radiative transfer

neutron transp ort and gas dynamics all emphasize volume interactions collisions

involving the threedimensional medium through which the particles migrate while

the b oundary conditions are of secondary imp ortance In contrast global illumi

nation generally neglects volume scattering altogether but incorp orates b oundary

conditions that are far more complex in terms of geometry and surface scattering

distributions This distinction gives the solution metho ds for global illumination

a unique character Nevertheless there are p oints of contact with other problems

for example when a participating medium is included in global illumination the

governing equation is virtually identical to that of neutron migration

Computational Asp ects

techniques in use to day are physicallybased yet none ac Many image synthesis

count for the entire rep ertoire of optical eects that are observable at large scales

The failure is generally not in the physical mo del but rather in the computational

metho ds Nearly all the limitations result from the diculty of faithfully represent

ing the distributions of light scattering from real materials and in accurately simu

lating the longrange eects of scattered light To deal with this complexity global

illumination algorithms frequently incorp orate idealized reection mo dels such as

Lamb ertian ideal diuse sp ecular mirrorlike or a combination of the two

Both of these extremes are easy to represent and corresp ond to wellundersto o d

computational pro cedures

Ray tracing was intro duced by App el and signicantly extended by Whit

ted to incorp orate many of the principles of geometric optics however the

metho d neglects all multiple scatterings of light except those from mirrorlike sur

faces The radiosity metho d rst applied to image synthesis by Goral et al

accounts for multiple diuse interreections using techniques from thermal

engineering yet it do es not accommo date mirrored surfaces Metho ds combin

ing ray tracing and radiosity typically neglect more complex mo des of reection

such as glossy surfaces Other techniques accommo date more complex sur

face reections but either suer from statistical errors or fail to

handle extremely glossy surfaces in addition none account for light scattering by

participating media such as smoke or fog Of the metho ds that can account for par

ticipating media none can accommo date surfaces with lo cally complex geometry

or reectance functions

In studying the computational asp ects of global illumination it is often conve

nien t to partition illumination into two comp onents direct and indirect By direct

illumination we mean the pro cesses of light emission from luminaires area light

sources propagation through space and subsequent scattering at a second sur

face By indirect illumination we mean light that undergo es multiple scatterings

Both of these asp ects are global in that wellseparated ob jects of a scene can inu

ence one anothers app earance either by blo cking light casting shadows in the

case of direct lighting or by scattering light reecting it in the case of indirect

lighting

The direct and indirect comp onents of illumination oer dierent computational

challenges Techniques for direct illumination include mo deling the distribution of

light emitted from luminaires computing features of irradiance such as

shadow b oundaries and gradients and mo deling surface reections

Handling these eects is a prerequisite to simulating indirect illumination

Indirect illumination is imp ortant to consider b ecause a signicant p ortion

of the illumination in a ro om for example may come from multiply scattered

light Both nite element metho ds and Monte Carlo metho ds

have b een applied to the simulation of global illumination which includes

b oth direct and indirect illumination

The most signicant computational asp ects of the niteelement approaches for

global illumination are discretizing computing element interactions and

solving a linear system Discretization includes surface mesh generation which is

one of the more challenging problems of global illumination Thus far the most

eective meshing schemes have b een based on adaptive renement such as the

hierarchical scheme of Hanrahan et al a priori lo cation of derivative discon

tinuities as prop osed by Heckb ert and Lischinski et al or a combination

of the two as prop osed by Lischinski et al

Computing element interactions is p erhaps the most costly asp ect of global

volves a illumination This is a signicant computational challenge b ecause it in

potentially large number of multidimensional integrals over irregular domains

Each multidimensional integral represents the transfer of light among discrete

elements and corresp onds to a coupling term of the nite element matrix The

number of interactions can be large b ecause they are nonlo cal any element may

potentially interact with any other element The regions are irregular b ecause of

o cclusion and lo cal geometric complexityof surfaces The integrals are minimally

involve as many as six dimensions for nondiuse sur twodimensional but may

faces moreover the integrands may be discontinuous due to changing visibility

Owing to these complexities the problem is frequently approximated using Monte

Carlo integration There are virtually no to ols available for solving these problems

analytically when nondiuse reection is involved

Finallyevery global illumination problem involves the solution of a linear sys

tem at some level Because the nite element matrices can be large and dense it

is imp ortant to exploit some structure of the matrix to sp eed its construction and

solution For example b oth blo ck matrices and wavelet representations

have been used for this purp ose

Historical Background

Global illumination draws from many elds two of which are of particular rele

vance radiative transfer and illumination engineering The origins and underlying

physical mo del of global illumination can b e traced directly to the theory of radia

tive transfer while the aims and metho dologies of global illumination are to day

most closely aligned with those of illumination engineering

The birth of radiative transfer theory is generally attributed to the astrophysi

cal work of Schuster and Schwarzschild near the turn of the century The

theory was initially develop ed for the study of stellar atmospheres Among the rst

problems encountered were the inverse problems of inferring physical prop erties of

a stellar atmosphere from the light it emitted To solve the inverse problems it

was rst necessary to obtain a governing equation for the direct pro cess of radia

tive transfer through a hot atmosphere The resulting governing equation known

to day as the equation of transferisanintegrodierential equation that describ es

eraged interaction of light with matter and accounts for the largescale timeav

emission absorption and scattering The emphasis in astrophysics as in most en

gineering applications is on mo deling interactions with participating media With

the appropriate b oundary conditions however the equation of transfer can also

account for arbitrary surface reections which is the asp ect that is emphasized in global illumination

The concepts of radiative transfer have subsequently found application in areas

such as illumination engineering thermal engineering hydrologic

optics agriculture remote sensing computer vision

and computer graphics The equation of transfer was also the starting point

for the development of neutron transp ort theory in which the diusion of neutrons

through matter is governed by essentially the same principles as the diusion of

photons through a participating medium This accounts for the many

similarities b etween solution metho ds for global illumination and simulated neutron

migration particularly among Monte Carlo metho ds

Thesis Overview

The contributions of this thesis are b oth theoretical and practical The theoretical

asp ects of the work are an attempt to strengthen the mathematical foundation of

realistic image synthesis Although computer graphics has drawn from a number

of related elds with more develop ed foundations many of the features unique to

this eld are not yet well understo o d For instance numerous techniques havebeen

adopted from nite element analysis yet many fundamental questions concerning

the nature of the solutions in the context of global illumination have not been

addressed New formalisms that b egin to rectify this are presented

The practical asp ects of the work provide a number of useful computational

for image synthesis The fo cus is on new closedform solutions for a vari to ols

ety of subproblems relating to direct illumination in particular those involving

nondiuse emission and scattering Previously the few to ols that existed for

computing illumination from area light sources were limited to diuse luminaires

New expressions are derived here that accommo date luminaires with directionally

varying brightness these expressions p ermit us to handle a much larger class of

luminaires analytically

Chapter intro duces the fundamental building blo cks of radiometry that are

used throughout the thesis The concepts are develop ed in an untraditional manner

using the formalism of measure theory The approach is inspired by the axiomatic

foundation of radiative transfer theory put forth by R W Preisendorfer This

approach illustrates the origin of the most basic principles that underlie global

illumination

Chapter intro duces a new to ol that is derived directly from a widelyused

result attributed to Lambert The to ol is a closedform expression for the derivative

of the irradiance a measure of incident energy due to diuse p olygonal light

sources The interesting asp ect of the problem is in correctly handling o cclusions

The chapter describ es the complication intro duced by partial o cclusion provides

a complete solution for diuse p olygonal luminaires and demonstrates several

applications

Chapter intro duces the most powerful to ols of the thesis A tensor gener

alization of irradiance is prop osed that leads to a number of new computational

metho ds involving nondiuse surfaces The expressions derived in this chapter

allow a number of fundamental computations to be done in closedform for the

rst time The new expressions and the algorithms for their ecient evaluation

are indep endently veried in chapter by comparison with Monte Carlo estimates

Chapter intro duces new theoretical to ols for the study of global illumination

The fo cus is on a new way to express the governing equation for global illumination

in terms of linear op erators with very convenient prop erties These prop erties lend

themselves to standard metho ds of error analysis which are describ ed in chapter

sources of error in solving the new op erator This nal chapter also discusses the

equation and relates these to existing global illumination algorithms

Throughout the thesis the emphasis is on physical rather than p erceptual as

pects of light transp ort This bias is reected in the consistent use of radiometric

rather than photometric units whichtake the resp onse of the human visual system

into account Thus the admittedly dicult issues of display and p erception are

largely ignored Moreover this work do es not address participating media trans

parent surfaces timedep endent or transient solutions or probabilistic solution

metho ds other than those used for indep endentverication The characteristics of

the problems considered here maybe summarized as follows

Mono chromatic radiometric quantities radiance reectivities emission

A linear mo del of radiative transfer

Direct radiative exchange among opaque surfaces

Steadystate solutions

Deterministic b oundary element formulations

Many applications meet these restrictions b ecause environments that we wish to

simulate frequently come from everyday exp erience where these assumptions tend

to be valid For instance at habitable temp eratures reemission of absorb ed light

by most materials is eectively zero at visible wavelengths which p ermits us to

simulate dierent wavelengths indep endently and also leads to a linear mo del of

light Atmospheric eects over tens of meters are insignicant under normal cir

cumstances so we may assume that surfaces exchange energy directly with no

attenuation Furthermore b ecause changes in our surroundings take place slowly

with resp ect to the sp eed of light transient solutions are of little interest Also

since common sources of ligh t are incoherent eects related to phase such as in

terference are usually masked While diraction and interference can b e observed

in diraction gratings as well as in natural ob jects such as buttery wings and thin

lms these do not dominate architectural settings and other scenes that we

commonly wish to simulate

Finally this work fo cuses on deterministic algorithms primarily to limit the

scop e but also b ecause deterministic metho ds are vital to Monte Carlo Frequently

both the accuracy and eciency of Monte Carlo metho ds can be vastly improved

by rst solving a nearby problem deterministically and estimating the dierence

sto chastically Thus it is hop ed that fundamental advances made for deterministic

algorithms will also b e useful in the context of Monte Carlo which usually has much

broader applicability

Summary of Original Contributions

This section summarizes the original contributions of the thesis which are orga

nized into three ma jor categories I new derivations of basic physical quantities

such as radiance II new metho ds for direct illumination involving b oth diuse

and nondiuse surfaces and I I I new to ols for the analysis of global illumination

Parts I and I I I are primarily of theoretical interest while part I I presents a number

of practical algorithms

I Derivations of Fundamental Physical Quantities Chapter

The most fundamental quantity of radiative transfer known as radiance is

derived from macroscopic prop erties of abstract particles using formalisms from

measure theory The new approach claries some of the assumptions hidden within

classical denitions and demonstrates deep connections with other physical quanti

ties Wellknown prop erties of radiance such as constancy along rays in free space

are shown to hold using the new formalism

Chapters and II Metho ds for Direct Illumination

Derivatives of Irradiance A closedform expression is derived for the Irra

diance Jacobian the derivative of vector irradiance that holds in the presence of

o cclusions The new expression makes p ossible a number of computational tech

niques for diuse p olygonal scenes such as generating isolux contours and nding

lo cal irradiance extrema A new meshing scheme based on isolux contours is also demonstrated

Irradiance Tensors A natural generalization of irradiance is presented that

emb o dies highorder moments of the radiation eld The new concept is a useful

to ol for deriving formulas involving moments of radiance distribution functions

Extending Lamb erts Formula Irradiance tensors are shown to satisfy a

recurrence relation that is a natural generalization of a fundamental formula for

irradiance derived by Lamb ert in the th century The new formula extends

Lamberts formula to nondiuse phenomena

Moment Metho ds for NonLamb ertian Phenomena Closedform solu

tions for moments of irradiance from p olygonal luminaires are presented along

with ecient algorithms for their evaluation The expressions are derived using

irradiance tensors The new algorithms are demonstrated by simulating three dif

ferent nondiuse phenomena each computable in closedform for the rst time

glossy reection glossy transmission and directional area luminaires

Results on Nonp olygonal and Inhomogeneous Luminaires The prob

lem of computing irradiance from spatially varying luminaires is reduced to that

of integrating rational p olynomials over the sphere A generalization of irradiance

tensors is intro duced to handle rational moments It is shown that these integrals

generally cannot be evaluated in terms of elementary functions even in p olygonal

environments

Stratied Sampling of Solid Angle A direct Monte Carlo sampling al

gorithm for spherical triangles is derived The metho d allows stratied sampling

of the solid angle subtended by a p olygon The algorithm is used to construct

a lowvariance estimator for irradiance tensors and related expressions providing

indep endent validation of the closedform expressions

III Analysis of Global Illumination Chapters and

Linear Op erator Formulation A new op erator formulation of the well

known governing equation for global illumination is presented and is shown to

be equivalent to the rendering equation The new formulation cleanly separates

notions of geometry and reection allowing these asp ects to be studied indep en

dently Several related op erators such as adjoints are easily analyzed using the

new op erators

Theoretical Bounds on Op erators and Radiance Functions Bounds

are computed for the new op erators using principles of thermo dynamics and basic

to ols of functional analysis Based on this analysis it is shown that the pro cess of

global illumination is closed with resp ect to all L spaces

Taxonomy of Errors with a priori Bounds Using the prop osed linear

op erators and standard metho ds of analysis a priori b ounds are computed for three

categories of error perturbed b oundary data discretization and computation

Chapter

Particles and Radiometry

In this chapter we dene anumb er of fundamental radiometric quantities that is

concepts p ertaining to the measurement of lightthat are essential to the study of

radiative transfer Rather than enumerating standard denitions we shall instead

deduce the imp ortant concepts starting from observable behaviors of particle

based phenomena that we take as axioms This approach will exp ose some of the

assumptions that are hidden within the classical denitions

The simulation of any real pro cess or phenomenon involves simplifying assump

tions that aect b oth the physical mo del and its mathematical representation

the problem usually Physical assumptions are intro duced to limit the scop e of

by ignoring or altering known physical laws This always involves a compromise

between what is desired and what can be eectively simulated Mathematical

assumptions include those in which structure is imp osed that is not necessarily

present in reality For instance a problem may be embedded in a richer math

ematical framework so that wellundersto o d metho ds of analysis can be applied

Assumptions of b oth typ es are made in dening radiometric quantities

The most common and far reaching mathematical assumption in radiometry is

be faithfully represented by continuous analogues The that light and matter can

utility of this assumption is that it allows us to express new quantities in terms

of limiting pro cesses that can only be approximated by actual exp eriments By

emb edding the phenomena we wish to study in a continuum wemay employ all the

usual machinery of mathematical analysis such as measure theory and dierential

geometry In the physical world of course the corresp onding limiting pro cesses

cease to make sense below a certain scale Consequently the mathematical

abstractions generally extend beyond the phenomena they mo del

We shall begin with macroscopic timeaveraged prop erties of light that are

theoretically observable byaneye or a lab oratory instrument and deduce the basic

concepts of photometry and radiometry that is quantitative asp ects of lightthat

are based on a continuum We shall adopt the common simplifying assumption that

lightmay b e adequately mo deled as a o w of noninteracting neutral particles

From elementary principles that op erate at the macroscopic scale we then construct

functions that corresp ond to radiance and related quantities which will b e essential

building blo cks in the chapters that follow

Much of the largescale behavior of particles can be understo o d in terms of

abstract particles with minimal semantics that is particles with only those fea

tures common to photons neutrons and other neutral particles This observation

typies the p ointofview taken in transport theory which is the study of abstract

particles and their interaction with matter Transp ort theory applies classical

notions of physics at the level of discrete particles to predict largescale statistical

b ehaviors The emphasis on statistical explanations dierentiates transp ort theory

from other classical theories such as electromagnetism Essential to the theory are

fundamental simplifying assumptions ab out the particles For example it several

is assumed that the particles are so small and numerous that their statistical

distribution can b e treated as a continuum andatany point in time a particle

is completely characterized by its p osition velo city and internal states such as

p olarization frequency charge or spin

These assumptions lead naturally to the concept of phase space an abstraction

used in representing the spatial and directional distribution and internal states of

a collection of particles We shall employ a continuous phase space whose p oints

sp ecify p ossible particle states A distribution of particles can then b e represented

as a density function dened over phase space The aim of transp ort theory is to

determine such a density function commonly describing a distribution of particles

at equilibrium based on individual particle behaviors as well as the geometrical

and physical prop erties of the medium through which the particles travel

The setting of abstract particles is sucient to deduce a primitive version of

radiance as well as other radiometric quantities using only measuretheoretic con

cepts and elementary facts ab out the b eha vior of groups of particles Each distinct

quantity corresp onds to a dierent form of channeling of the particles with radi

ance b eing the most fundamental for radiative transfer To deduce the prop erties of

radiance which will b e needed in subsequent chapters wemust ultimately imp ose

further semantics on the abstract particles that will distinguish them as photons

Phase Space Measure

Phase space is an abstraction used in many elds to represent congurations of

discrete particles Frequently the term connotes a N dimensional Euclidean space

with eachpoint enco ding the p osition and velo cityofN distinct particles A single

pointofsuch a phase space completely sp ecies the conguration of all N particles

and the timeevolution of the particles denes a spacecurve

er dimensions suces For many physical problems a phase space of far few

which has b oth conceptual and computational advantages For example when the

particles cannot inuence one another their aggregate b ehavior may b e determined

by characterizing the behavior of a single particle Each phase space dimension

then represents one physical degree of freedom of a particle with the space as a

whole representing all p ossible particle states We shall assume that the particles

are indep endent which implies that particles mayinteract with their surroundings

but not with each other This assumption is valid to very high accuracy for rareed

gases neutrons and photons when interference is ignored

We shall further assume that particlematter interaction or scattering consists

of collisions that are either p erfectly elastic and p erfectly inelastic That is a

particle either retains its original internal state after scattering such as energy or

wavelength or is completely absorb ed This precludes pro cesses bywhich particles

migrate from one energy band to another for photons this may happ en when a

particle is absorb ed and reemitted as in blackb o dy radiation phosphorescence

and uorescence

In transp ort theory this simplication is known as the onespeed assumption

since sp eed is prop ortional to energy for most particles and energy is assumed to

remain constant despite multiple collisions When applied to photons however

the term implies constantwavelength rather than sp eed Hence this simplication

is also known as the gray assumption in the context of radiative transfer

Finallywe assume that the particles p ossess no internal states other than en

ergy or wavelength which is xed for all particles Thus we assume mono chromatic

radiation and ignore phenomena such as p olarization Under these assumptions

each particle has only ve degrees of freedom three for p osition and two for direc

tion The corresp onding dimensional phase space is

IP IR S

in IR where IR is Euclidean space and S is the unit sphere

The abstraction of a distribution of particles in phase space requires some ad

ditional structure for the purp ose of analysis In order to quantify collections of

particles and dene various transformations it is necessary to endow the space

with a measure We therefore intro duce the concept of phase space measure con

sisting of the triple IP P where is a p ositive set function or measure dened

on the elements of a set P which is a algebra of subsets of IP That is P is

subsets that contains IP and is closed under complementation and a collection of

countable unions The elements of P are called the measurable sets

We shall construct the phase space measure IP P from the natural Leb esgue

measures on IR and S which corresp ond to standard notions of volume and

surface area we shall denote these measures by v and resp ectively We then dene

to b e the pro duct measure v and form the algebra P from the two comp onent

algebras More formallywe dene P be the completion of the cartesian pro duct

of the two algebras which is again a algebra p Completion simply

extends the collection of measurable sets to include those formed by a cartesian

pro duct of an unmeasurable set with a set of measure zero

Leb esgue measure is the appropriate abstraction to adopt here since it is in

variant under rigid transformation and p ermits the consistent denition of a family

of useful function spaces However the use of such formalisms requires that the

domain of be restricted to P the algebra of subsets of IP This restriction is

necessary in Euclidean spaces of three or more dimensions where measures that are

dened on all subsets cannot p ossess certain essential prop erties For example such

measures cannot be invariantunder rigid motions a prop erty we require of space

and the particles that travel through it The BanachTarski paradox is a more

colorful example of why such a restriction on the domain is necessary This famous

result states that a measure dened on all subsets of Euclidean space must either

assign the same measure to all v olumes or allow unacceptable consequences

for example one unit sphere could be decomp osed into a nite number of pieces

and reassembled into two identical spheres of the same volume

Particle Measures

Phase space measure applies only to the space on which the particles are dened

and not the distributions themselves To quantify the distributions and relate

them to phase space measure we require several additional prop erties that apply

to collections of noninteracting particles We shall identify four such prop erties

whichwe take to b e axiomatic that is we shall not provide explanations in terms

of more basic phenomena These axioms closely parallel macroscopic prop erties of

real particles although the abstractions transcend their physical counterparts in

several resp ects The axioms lead naturally to the concept of a particle measure

from which we may deduce basic prop erties of neutral particle migration that

underlie radiative transfer and other particlebased phenomena

Particle Axioms

Phenomena such as radiation that arise from microscopic particles can exhibit

largescale features only through the correlated motion of collections or ensembles

of particles We may therefore express prop erties of particles that are relevant to

largescale transp ort in terms of these ensembles We shall consider four such prop

erties that are motivated by physical intuition and corresp ond to real phenomena

that can be measured at the macroscopic scale at least in principle From

these prop erties wemay deduce the existence of a density function over phase space

that exactly represents the conguration of particles

We b egin by assuming the existence of a eld of noninteracting onesp eed

particles in space Every region of space then has a particle content a number

indicating how many particles exist within that region This number may be

which the particles move By further partitioned according to the directions in

asso ciating values to subsets of space and direction we are dening a realvalued

set function over phase space whichwe shall denote by E We assert that E ob eys

the following physically plausible axioms

A E

A A B E A B E AE B

A A E A

Here A and B are subsets of IP and in the case of the last two axioms A and B

must also be measurable with resp ect to Axiom A states that E is a p ositive

set function dened on all subsets of the phase space IP to every subset there

corresp onds a nonnegative value indicating its particle content

Axiom A states that the set function E is additive By rep eated application

of this axiom E is clearly additiveover any nite collection of disjoint sets When

applied to photons this prop erty illustrates a div ergence of the phenomenological

formulation from physical optics The electromagnetic description of light includes

eects that can violate the additivity prop erty for instance distinct volumes in

phase space may interact via interference Thus in the context of photon trans

port axiom A restricts us to incoherent sources of light as interference can arise

whenever the illumination has some degree of coherence

Axioms A and A establish the only connection between the set function E

and the phase space through which the particles migrate Sp ecically axiom A

phase space has a nite particle content and states that every nite volume of

axiom A states that a region of phase space with zero volume cannot contain

a meaningful number of particles The latter axiom has imp ortant implications

as it simultaneously disallows two physically implausible situations p oint sources

and p erfectly collimated thin beams Because is a pro duct measure axiom A

implies that E D when v D which is true at isolated p oints or when

whichistrueofany p erfectly collimated b eam These prop erties greatly

inuence the nature of radiometric calculations since any meaningful transfer of

energy will entail b oth spatial and directional integration

Extension to a Measure

Axioms AA express macroscopic prop erties of particles by means of a set func

tion E that is similar to a measure With slight alterations to two of the axioms E

can b e made a genuine measure Doing so will allow us to study radiance functions

as elements of the standard L or Lebesgue function spaces which are dened in

terms of the Leb esgue integral Toward this end we weaken A by restricting its

domain and strengthen A to apply to innite sums over mutually disjoint sets

Sp ecicallywe replace axioms A and A with

A E P

A E A E A for mutually disjoint fA g P

i i i

i i

Axiom A only requires E to be dened on the algebra P while axiom A

states that E is countably additive In summary axioms A A A and A

taken together state that the function E is a p ositive countably additive set function

dened on a algebra By denition then E is a positive measure

With this interpretation of E axiom A now has additional signicance given

that E and are b oth measures over the same algebra axiom A states that the

particle measure is absolutely continuous with resp ect to the phase space measure

Absolute continuity denes an order relation on the set of measures over the same

m algebra which is usually denoted by p The notation m

simply means that m E whenever m E

Note that E IP need not be nite since an innite number of particles may

exist within an innite volume of phase space without violating basic physical

principles However a particle measure will always p ossess the weaker prop ertyof

b eing nite that is it is nite on each element of some countable partition of

the domain This follows immediately from axiom A and the fact that the

nite phase space measure is

Given the abstraction of indep endent particles in a phase space IP equipp ed

with a measure axioms A A A and A imply that the concept of particle

content coincides with the concept of a p ositive measure We summarize this p oint

which is largely a matter of denitions in the following theorem

Theorem Existence of Particle Measures Given a conguration of non

interacting onespeed particles in an abstract phase space IP with measure if the

particle content function E satises axioms A A A and A then E denes

a positive nite measure over IP with E

We call the threetuple IP P E the particle measure space and E the particle

measur e The existence of measures analogous to E were taken as axiomatic by

Preisendorfer here we have identied the physical and mathematical

assumptions from which they may b e deduced

Phase Space Density

Tocharacterize the distribution of particles in a continuous phase space we require

a function known as phase space density dened over the space Wenowshowthat

such a density function emerges naturally from the concept of particle measure

the central observations parallel the measuretheoretic development of radiative

transfer due to Preisendorfer

y A P is given by a nite measure By theorem the particle content of an

that is absolutely continuous with resp ect to This characterization of particle

content is sucient to deduce the existence of phase space density the nal step

is provided by the RadonNiko dym theorem p p

Theorem RadonNikodym Let B B m be a measure space over the set

algebra B If m B and let be a nite positive measure dened on the

then there exists a nonnegative mintegrable function p such that

E pdm

for al l E B with mE Moreover the function p is unique to within a set

of mmeasure zero

The function p in the ab ove theorem is commonly referred to as a density

function While any nonnegative function p can b e used to dene anew measure

by equation the RadonNikodym theorem shows that under rather mild

conditions the converse also holds in particular the density function p can be

recovered almost uniquely from the two measures m and provided that m

Theorems and together imply the following theorem

Theorem Existence of Phase Space Density For every conguration of

noninteracting onespeed particles in an abstract phase space IP with measure

there exists a measurable function n IP IR which is unique to within a set

of measure zero satisfying

E A n d

where E A denotes the particle content and A IP is a measurable subset

The function n whose existence is guaranteed by the ab ove theorem is called the

density By virtue of equation this function is also referred to phase space

as the RadonNikodym derivative of E with resp ect to since it exhibits algebraic

prop erties analogous to a standard derivative To emphasize the similarity

with dierentiation it is typically denoted by

which also suggests dimensional relationships Since E A is dimensionless n must

have the dimensions of inverse phase space volume or m sr We shall treat n

as a function of two variables and write nr u where r IR and u S

Phase space density makes no mention of material attributes such as mass

or energy these concepts do not enter until we assign physical meaning to the

phase space density makes it quite universal in particles This abstract nature of

one form or another it underlies virtually all particle transp ort problems including

radiative transfer neutron migration and gas dynamics When further semantics

original solid angle

Et(S) Et(S)

S r S

reduced solid angle

(a) (b)

Figure Phase space density remains constant along straight lines in empty space

but total phase space density decreases with the timeevolution of a region of phase space

such as energy content are added to the particles the meaning is inherited by the

resulting density function and is also reected in its physical units

Part of the semantics of a particle is its b ehavior with time Wehave assumed

neutral noninteracting particles which implies that the particles travel in straight

lines except when interacting with matter This behavior is consistent with the

notions of geometrical optics Another timerelated semantic prop erty is that par

ticles travel at a xed sp eed which is true of mono chromatic photons within a

uniform medium These prop erties give rise to a crucial prop erty of phase space

density that is inherited by concepts such as radiance

Theorem Invariance of Phase Space Density In steadystate the phase

space density of onespeedparticles is constant along straight lines in empty space

Pro of We consider the timeevolution of a collection of neutral noninteracting

onesp eed particles in a region of phase space Let S be an arbitrary subset of IP

and dene the timeevolution op erator E by

E S fr tv u ur u S g

where v is the constant sp eed of the particles This function denes a new set by

expanding the spatial comp onent of S in the direction of travel of its particles

leaving the set of directions unchanged Thus any particle whose p osition and

direction of travel are within the set S at time t will also be in the set E S

at time t t Similarly by conceptually reversing the direction of travel of the

particles we see that any particle in E S at time t must have been in the set S

at time t t Note that particles arriving from other directions may exist within

the spatial extent of E S at time t but these do not contribute to the content

of E S It follows that E S contains exactly those particles previously existing

t t

in S See Figure a In steady state this equality holds at all times Thus

E S E E S

for all t such that the particles of S encounter no matter Consequently

Z Z Z

d nr tv u u d nr u d nr u

S S E S

for all S P and tprovided that the evolution of S into E S takes place in

empty space When these conditions are met equation yields

nr u nr tv u u d

which implies that nr unr tv u u for almost all r IR and u S such

that the p oints r and r tu are mutually visible that is separated by empty

space

The classical demonstration of the ab oveinvariance principle pro ceeds by equat

ing the radiant energy passing consecutively though two dierential surfaces or

p ortals Here wehave reached the same conclusion by considering the time

evolution of arbitrary ensembles of particles within a phase space

This invariance principle has imp ortant implications For example under the

given assumptions the directional derivative of phase space density must be zero

in every direction and at all p oints in empty space This is equivalent to

rnr u u

where the gradient is with resp ect to the spatial variable As we shall see this fact

also causes derivatives of related quantities to vanish

The invariance of phase space density along lines is not shared by other closely

related quantities For instance consider the quantity known as total phase space

density which is dened by

nr u d u n r

where is the canonical measure on the sphere p Given a b ounded set

S P the total phase space density must everywhere decrease in the set E S

when t is suciently large This follows from that fact that nr is b ounded ab ove

by the pro duct of the solid angle subtended by S at r and the maximum phase

space density attained within S See Figure b A number of quantities related

to n are intro duced in chapter where we investigate moments of radiance

Phase Space Flux

It is frequently more convenient to characterize the density of particles by their

develop this idea we shall now rate of ow across a real or imaginary surface To

pro ceed using traditional heuristic arguments based on innitesimal quantities

dA u

dω

Figure Phase space ux is the number of particles crossing the surface dA perpen

dicular to u per unit area per unit solid angle per unit time

Consider the particles that pass through a dierential area dA in time dt with

directions of travel within a dierential solid angle d ab out the surface normal

All of the particles are contained within the volume dA ds where ds vdt as

shown in Figure Assuming that r is a point within this volume the particle

content of the volume is given by

nr u dA ds d

But the particles can also b e counted using the rate at which they cross the surface

dA That is the particle content of the volume dA ds is also given b y

vnr u dA d dt

since ds vdt This metho d of counting the particles which substitutes a tem

p oral dimension for a spatial dimension motivates the notion of phase space ux

denoted which is dened by

r u vnr u

m sr s

The concept of phase space ux can be used to express virtually any quantity

relating to the macroscopic distribution of particles

Radiance

Radiance is the physical counterpart to the physiological concept of brightness or

intensity and is the most ubiquitous concept in radiometry We denote the radiance

at the point r and in the direction u by f r u At the macroscopic scale this

function completely sp ecies the distribution of incoherentmonochromatic radiant

energy in the medium consequently all radiometric quantities may be dened in

terms of radiance

To dene radiance from phase space ux we need only intro duce the concept of

energy per particle given by h where h is Plancks constant and is frequency

(a) (b) (c)

Figure There is no distinction between incoming and outgoing rays at points in

space At surfaces the radiance distribution is natural ly partitioned into surfaceradiance

outgoing and eld radiance incoming

We then have

f r u h r u

ch nr u

where c is the sp eed of light Radiance therefore has the units of joulesm sr sec

or equivalentlywattsm sr

is phase Radiance is dened at all points in space and in all directions as

space ux However it is convenient to dene several restrictions of the radiance

function We shall adopt the terminology of Preisendorfer in naming these

restrictions When the p oint r IR is xed the function of direction f r is

called the radiance distribution function at the point r this function is dened at

all p oints in space as well as on surfaces Note that a radiance distribution function

is distinct from a radiant intensity distribution which is a mathematical construct

with the units of wattssr that characterizes a point light source We shall not

make use of the latter concept here

When r is constrained to lie on a surface r M the surface tangent plane

naturally partitions the radiance distribution function at eachpointinto two com

ponents corresp onding to incoming and outgoing radiation We dene surface

radiance to b e the radiance function restricted to a given surface M and to direc

Π(P)

Figure The surface P and its spherical projection P

tions pointing away from the surface by denition it is zero elsewhere Similarly

we dene eld radiance to be the other half of the radiance function restricted to

M that is the comp onent with directions pointing toward the surface See Fig

ure The directions that are exactly tangent to the surface are to b e ignored in

b oth cases as they comprise a set of measure zero are physically meaningless

Irradiance

Additional radiometric quantities can b e dened in terms of radiance by means of

arious domains To dene irradiance and related quanti weighted integrals over v

ties we require a convenient notation for solid angle Let P IR be a surface

and consider its pro jection onto the unit sphere centered at the p oint x asshown

in Figure Formallywe dene its spherical projection P by

P f u S x tu P for some t g

The solid angle subtended by the surface P can b e dened in terms of this op erator

is the surface area of any If is Leb esgue measure on the sphere then A

measurable subset A S The solid angle subtended by the surface P with

resp ect to the point x IR is then P the surface area of its spherical

pro jection Since we mayalways select a co ordinate system in which the center of

pro jection x is at the origin we shall omit the point x without loss of generality

which will simplify notation

The quantity known as irradiance is a density function dened at surfaces it

corresp onds to the radiantpower p er unit area reaching the surface at eachpoint

which is the most basic characterization of the illumination reaching a surface We

denote irradiance by r where r M By denition

watts

r f r u cos du

where is the hemisphere of incoming directions with resp ect to the surface at

the point r and f r u denotes a mono chromatic eld radiance function The

angle within the integral is the incident angle of the vector u with resp ect to the

surface at r Thus cos ju nrj where nr is the unit normal to the surface

at r The presence of the cos accounts for the fact that an incident pencil of

radiation is spread ov er larger areas near grazing thereby decreasing the density

of its radiation This fact is sometimes called the cosine law

Given the role of the surface normal in the ab ove denition it is convenient to

intro duce a new vectorvalued function r dened by

watts

u f r u d u r

The new function is called the vectorirradiance at the p oint r other common

names for the same quantity are light vector and net integrated ux This

vector quantity do es not dep end on surface orientation and is dened at all p oints

in IR It corresp onds to the timeaveraged Poynting vector of the electromagnetic

eld The resulting vector eld over IR is known as the light eld It

follows from equations and that

rn r r

when only eld radiance is considered that is when the vector irradiance do es

light reected or emitted from the surface Furthermore from equa not include

tion it follows that

rr u rf r u d u

A function with zero divergence is said to b e solenoidal

Both radiant exitance and radiosity have denitions similar to that of irradiance

and share the same units that is wattsm The dierence is that the domain of

integration is the outgoing hemisphere rather than the incident hemisphere The

semantic dierence between these two quantities is that radiant exitance refers to

emitted light while radiosity includes b oth emitted and reected light

Chapter

Derivatives of Irradiance

A p erennial problem of computer graphics is the accurate representation of light

leaving a surface In its full generality the problem entails mo deling b oth the lo cal

reection phenomena and the distribution of lightreaching the surface Frequently

the problem is simplied by assuming p olyhedral environments and Lamb ertian

diuse emitters and reectors With these simplications the remaining chal

lenges are to accurately mo del the illumination and shadows resulting from area

light sources and o ccluders and to simulate interreections among surfaces

Many asp ects of surface illumination have b een studied in order to accurately

reected light In previous work Heckb ert mo del features of the incident or

and Lischinski et al identied derivative discontinuities in irradiance and used

them to construct eective surface meshes Nishita and Nakamae lo

cated penumbrae due to o ccluders in p olyhedral environments while Teller

p erformed an analogous computation for a sequence of p ortals to lo cate antip enum

brae Ward and Heckb ert and Vedel estimated irradiance gradients by

Monte Carlo ray tracing and used them to improve the interp olation of irradiance

functions Vedel also computed the gradient analytically in uno ccluded cases

Salesin et al and Bastos et al employed gradients to construct higher

for irradiance functions Drettakis and Fiume estimated order interp olants

gradients as well as isolux contours from a collection of discrete samples and used

them to guide subsequent sampling placing more samples where the curvature of

the isolux contours was large for example Drettakis and Fiume and Stewart

and Ghali prop osed metho ds for incrementally computing visible p ortions of

a luminaire to increase the eciency of many of the ab ove metho ds

This chapter intro duces a new computational to ol that applies to diuse p oly

hedral environments and is useful in any application requiring derivatives of irra

diance The central contribution is a closedform expression for the derivative of

vector irradiance as dened in chapter which is termed the irradiance Jacobian

The new expression prop erly accounts for o cclusion and subsumes the irradiance

gradientas a sp ecial case

Intro duction

The irradiance at a p oint on a surface due to a p olyhedral luminaire of uniform

brightness can be computed using a wellknown formula due to Lamb ert In this

chapter we derive the corresp onding closedform expression for the irradiance Jaco

bian the derivative of the vector representation of irradiance Although the result

is elementary for uno ccluded luminaires within penumbrae the irradiance Jaco

bian must incorp orate more information ab out blo ckers than is required for the

computation of either irradiance or vector irradiance The expression presented

b er of p olyhedral blo ckers and requires only a minor exten here holds for anynum

sion of standard p olygon clipping to evaluate To illustrate its use three related

applications are briey describ ed direct computation of isolux contours nding

lo cal irradiance extrema and isomeshing Isolux contours are curves of constant

irradiance across a surface that can b e followed using a predictorcorrector metho d

based on the irradiance Jacobian Similarly lo cal extrema can be found using a

gradient descent metho d Finally isomeshing is a new approach to surface mesh

generation that incorp orates families of isolux contours

In Section we derive the irradiance Jacobian for p olygonal luminaires of

uniform brightness starting from a closedform expression for the vector irradiance

In previous work the same closedform expression was used in scalar form by Nishita

and Nakamae to accurately simulate p olyhedral sources and by Baum et

al for the computation of form factors which relate to energy transfers among

surface patches Section intro duces a metho d for characterizing changes in the

apparent shap e of a source due to dierential changes in the receiving p oint which

is the key to handling o cclusions In Section basic prop erties of the irradiance

Jacobian are discussed including existence and the connection with gradients

To illustrate the p otential uses of the irradiance Jacobian Section describ es

several computations that employ irradiance gradients We describ e a metho d for

direct computation of isolux contours which are curves of constant irradiance on

a surface Each contour is expressed as the solution of an ordinary dierential

equation which is solved numerically using a predictorcorrector metho d The

resulting contours can then be used as the basis of a meshing algorithm Finding

lo cal extrema is a related computation that can be p erformed using a descent

metho d

Lamb erts Formula

For sources of uniform brightness can be expressed analytically for a number

of simple geometries including spheres and innite strips Polygonal sources

are the fo are another imp ortant class with known closedform expressions and

cus of this chapter Supp ose P is a simple planar p olygon in IR with vertices

v v v If P is a diuse source with constant radiant exitance M wattsm

then the light eld due to P is given by

r r r

i i

where are the angles subtended by the n edges as seen from the vantage

point r or equivalently the arclengths of the edges pro jected onto the unit sphere

vk P unit sphere

vk+1 Θk uk

uk+1 r

Γk

Figure The vector irradiance at point r due to a diuse polygonal luminaire P can

be expressed in closed form The contribution due to edge k is the product of the angle

and the unit vector

k k

ab out r The vectors are unit normals of the p olygonal cone with cross

section P and ap ex at ras shown in Figure For any k n the functions

and can b e written

k k

v r v r

k k

r cos

jj v r jj jj v r jj

k k

and

v r v r

k k

jj v r v r jj

k k

where jj jj is the Euclidean norm and v v Equation most commonly

app ears in scalar form which includes the dot pro duct with the surface normal

With M the corresp onding expression r n r is the form factor

between a dierential patchatr and the p olygonal patch P Equivalent expressions

have b een indep endently discovered byYamauti Fok and Sparrow

However as noted bySchroder and Hanrahan the scalar expression app eared

much earlier in Lamb erts treatise on optics published in Henceforth

equation shall b e referred to as Lamb erts formula

The light eld dened by Gershun and describ ed in chapter is a true

vector eld That is the vector irradiance due to multiple sources may be ob

tained by summing the vector irradiance due to each source in isolation Thus

p olyhedral sources can be handled by applying equation to each face and

summing the resulting irradiance vectors Alternatively when the faces have equal

brightness equation can be applied to the outer contour of the p olyhedron

as seen from the p oint r as describ ed by Nishita and Nakamae Partially

o ccluded sources can be handled similarly by summing the contributions of all

the visible p ortions Determining the visible p ortions of the sources in p olyhedral

environments is analogous to clipping p olygons for hidden surface removal

The closedform expression for vector irradiance in equation provides an

eective means of computing related expressions such as derivatives In the re

for derivatives of the mainder of the chapter we derive closedform expressions

irradiance and vector irradiance due to p olyhedral sources in the presence of oc

cluders and describ e several applications

The Irradiance Jacobian

If the function F IR IR is dierentiable then its derivative DF is represented

by a Jacobian matrix We shall denote the Jacobian matrix of F at r by

J F That is

F r

J F DF r

In this section we derive the Jacobian matrix of the vector irradiance which we

shall call the irradiance Jacobian when the illumination is due to a diuse p olyg

onal luminaire The obvious approach to obtaining an expression for J in this

case is to simply dierentiate equation with resp ect to the p oint r this is a

polygonal source

blocker B

A B blocker A

receiver r

Figure The irradiance at point r due to source P is the same with either blocker

but the slopes of the irradiance curves are dierent

straightforward exercise that can b e p erformed by a symb olic dierentiation pack

age for example The resulting expression will b e valid for uno ccluded p olygonal

luminaires but not for partially o ccluded luminaires

To see why the irradiance Jacobian is more dicult to compute when blo ckers

are present consider the arrangement in Figure First observe that the irra

diance at the point r can be computed by applying equation to the visible

p ortion of the source which is the same in the presence of either blo cker A or

blo cker B Although the common expression can be dierentiated the resulting

derivative do es not corresp ond to the irradiance Jacobian To see this observe

that the two blo ckers pro duce irradiance functions with dierent slop es at r the

blo cker closest to the receiving plane pro duces the sharp er shadow which implies

a larger derivative Consequently the irradiance Jacobians must also dier in the

two cases to account for blo cker p osition

To derive an expression that applies within penumbrae we express r in

terms of vertex vectors which corresp ond to vertices of the spherical pro jection of

v1 v1 P v2 r P r P v2 B B vertex vector

(a)(b) (c)

Figure a The view from r of the two types of intrinsic vertices b The vertex

vector for the unoccluded source vertex v c The vertex vector for the blocker vertex

v whose projection fal ls within the interior of P

the p olygon as depicted in Figure Vertex vectors may p ointtoward vertices of

two distinct typ es intrinsic and apparent An intrinsic vertex exists on either the

source or the blo cker as shown in Figure An apparentvertex results when the

edge of a blo cker as seen from r crosses the edge of the source or another blo cker

es of the vertex as shown in Figure We shall express J in terms of derivativ

vectors Since a vertex vector is a mapping from points in IR to unit vectors its

derivativeisa matrix whichwe call the vertex Jacobian The vertex Jacobians

account for the geometric details of the vertices which yields a relatively simple

closedform expression for J even in the presence of o ccluding ob jects

Let v v v b e the vertices of P the source P after clipping away p ortions

that are o ccluded with resp ect to the point r Without loss of generality w e may

assume that P is a single p olygon if it is not we simply iterate over the pieces

The vertex vectors u ru ru r are dened by

v r

u r

jj v r jj

To simplify notation we also let w rw r denote the cross pro ducts

w r u r u r

k k k

Henceforth we assume that u and w are functions of p osition and omit the

k k

B B 1 1 B1 B r r v1 2 v 1 v 2 B v2 2 P P

(a) (b) (c)

Figure a The view from r of the two types of apparent vertices b The vertex

vector for v results from a blocker edge and a source edge c The vertex vector for v

results from two blocker edges

explicit dep endence on r Expressing and in terms of w we have

k k k

sin jj w jj

k k

and

jj w jj

Note that equation is equivalent to equation only for acute angles

that is only when u u Nevertheless b ecause the new expression for

k k

simplies some of the intermediate expressions that app ear below it will be

retained for most of the derivation The restriction to acute angles will eventually

be removed however so that the nal result will apply in all cases

To compute J in terms of the vertex Jacobians J u J u w e rst

consider the k th term of the summation in equation Dierentiating wehave

J r J

k k k k k k

where r is the outer pro duct of the vector and the gradient r We

k k k k

now compute r and J For brevity we shall denote the vertex vectors u

k k k

and u by a and b resp ectively and the cross pro duct a b by w Then the

gradientof with resp ect to r is

r r sin jj w jj

J w

jj w jj

w w

w J w

a b jj w jj

Similarly dierentiating with resp ect to r we have

J D

jj w jj

J w ww

J w

jj w jj

ww J w

w w jj w jj

From Equations we obtain an expression for J in terms of J w

k k

and the vertex vectors a and b

T T

w ww w J w

J sin jj w jj I

k k

T T

jj w jj a b w w jj w jj

If the factor of sin jj w jj is now replaced by the angle b etween a and b or cos a b

then the expression will hold for all angles removing the caveat noted earlier The

ab ove expression may be written compactly as

J Ea b J a b

k k

where the function E is the edge matrix dened by

T T T

ww ww cos a b

I Ea b

T T T

a b w w jj w jj w w

In equation we have retained w as an abbreviation for a b Because the

edge matrix contains no derivatives it can be computed directly from the vertex

vectors a and b To simplify the Jacobian of a bwe dene another matrixvalued

function Q by

p p

z y

p p

z x

p p

y x

Then for any pair of vectors p and q we have p q Qpq Writing the cross

pro duct as a matrix multiplication leads to a convenient expression for the Jacobian

matrix of F G where F and G are vector elds in IR Thus

J F G QF J G QGJ F

Applying the ab ove identity to equation summing over all edges of the

clipp ed source p olygon P and scaling by M we arrive at an expression for

the irradiance Jacobian due to the visible p ortion of p olygonal source P

Eu u Qu J u Qu J u J

i i i i i i

This expression can be simplied somewhat further by collecting the factors of

each J u into a single matrix We therefore dene the corner matrix C to b e the

matrixvalued function

Ca b c Ea bQa Eb cQ c

Then the nal expression for the irradiance Jacobian can b e written as a sum over

y corner matrices all the vertex Jacobians transformed b

J Cu u u J u

i i i i

where wehave made the natural identications u u and u u Note that

m m

each corner matrix C dep ends only on the vertex vectors and not their derivatives

All information ab out changes in the visible p ortion of the luminaire due to changes

in the p osition r is emb o died in the vertex Jacobians J u J u which we

now examine in detail

Vertex Jacobians

To apply equation we require the vertex Jacobians whichwenow construct

for b oth uno ccluded and partially o ccluded p olygonal sources First observe that

each vertex vector ur is a smo oth function of r almost everywhere that is ur

is dierentiable at all r IR except where two or more edges of distinct p olygons

app ear to coincide as describ ed in section Dierentiability follows from the

smo othness of the Euclidean norm and the fact that the apparentpointofintersec

tion of two skew lines varies quadratically in r along each of the lines From

this it is evident that the vertex Jacobian exists whenever the real or apparent

intersection of two edges exists and is unique

vertex

du dr r change in change in

vertex vector position

Figure Adierential change in the position r results in a change in the unit vertex

vector u The locus of vectors du forms a disk or more general ly an el lipse in the plane

orthogonal to u

When the vertex Jacobian exists it can be constructed by determining its

action on each of three linearly indep endent vectors that is by determining the

instantaneous change in the vertex vector u as a result of moving r Dierential

changes in u are orthogonal to u and collectively dene a disk or in the case

The rates of change that are of partial o cclusion an ellipse See Figure

easiest obtain are those along the ma jor and minor axes of the ellipse which are

the eigenvectors of the vertex Jacobian We rst treat intrinsic vertices and then

generalize to the more dicult case of apparent vertices

Intrinsic Vertices

Supp ose that u is the vertex vector asso ciated with an uno ccluded source vertex

as shown in Figure b In this case the vertex Jacobian J uiseasy to compute

since it dep ends solely on the distance b etween r and the vertex which we denote

by Moving r in the direction of the vertex leaves u unchanged while motion

p erp endicular to u causes an opp osing change in u The changes in u are inversely

prop ortional to the distance This behavior completely determines the vertex

Jacobian Thus we ha ve

J u I uu

where the matrix I uu is a pro jection onto the tangent plane of S at the p oint

u which houses all dierential motions of the unit vector u The same reasoning

applies to vertex vectors dened byablocker vertex as in Figure c In this case

is the distance along u to the blo cker vertex

Apparent Vertices

Within penumbrae apparent vertices may be formed by the apparent crossing of

noncoplanar edges The two distinct cases are depicted in Figure Let u be

the vertex vector asso ciated with such a vertex where the determining edges are

segments of skew lines L and L Let s and t be vectors parallel to L and L

resp ectively as depicted in Figure As in the case of intrinsic vertices moving

r toward the apparent vertex leaves u unchanged so J uu To account for

other motions we dene the vectors s and t by

s I uu s

t I uu t

which are pro jections of s and t onto the plane orthogonal to u Now consider

the change in u as r moves parallel to s as shown in Figure a In this case

the apparent vertex moves along L while remaining xed on L Therefore the

change in u is parallel to s but opp osite in direction to the change in r If is

the distance to L along uwe have

J u s

Evidently s is an eigenvector of J u with asso ciated eigenvalue A similar

argument holds when r moves along t as shown in Figure b Here the apparent

vertex moves along L while remaining xed at L If is the distance to L along

uwe have

J u t

which provides the third eigenvector and corresp onding eigenvalue Collecting

these relationships into a matrix equation we have

h i

s t

J u s t u

t s

It follows immediately that whenever the lines L and L are distinct and non

colinear as viewed from the point r then

J u A A

h i

where A s t u Note that equation reduces to equation when

Equation therefore suces for all vertex vectors but the sp ecial

s t

eciency case for intrinsic vertices can b e used for

Polygon DepthClipping

To compute the irradiance or vector irradiance at a point r it suces to clip all

sources against all blo ckers as seen from r and apply equation to the resulting

vertex lists This op eration is also sucient to compute the corner matrices and

L2 L2

ttL1 L1 ss

B2 B2 s t uu B1 rrB1

(a) (b)

Figure The vertex Jacobian J u with respect to two skew lines L and L is found

by determining how the vertex vector u changes as r moves paral lel to a the vector s

and b the vector t

the vertex Jacobians at uno ccluded source vertices However the vertex Jacobians

for the cases illustrated in Figures c b and c all require information ab out

the blo ckers that is missing from traditionallyclipp ed p olygons Sp ecically the

distances to the blo cker edges that dene each vertex are needed to form the

matrices in Equations and

Thus additional depth information must be retained along with the clipp ed

p olygons for use in computing vertex Jacobians Here we prop ose a simple mecha

nism called depth clippingbywhich the required information app ears as additional

vertices The idea is to construct the clipp ed p olygon using segments of source and

blo cker edges and joining them by segments called invisible edges which cannot b e

seen from the p oint r See Figure The resulting nonplanar contour is identical

to that of the traditionallyclipp ed p olygon when viewed from r Each invisible

edge denes a vertex Jacobian of the form in equation the end p oints of

such an edge enco de the two distances from r while the adjacent edges provide the

two directions Each vertex that is not adjacent to an invisible edge pro duces a

vertex Jacobian of the form in equation

source invisible edge blockers

(a) (b) (c)

Figure a Source P is partial ly occluded by two blockers as seen from r b The

vector irradiance at r due to P can be computed using the simplyclipped polygon c

The irradiance Jacobian at r requires the depthclipped polygon

The depthclipp ed p olygon and the radiant exitance M completely sp ecify the

irradiance Jacobian Most p olygon clipping algorithms can b e extended to generate

this representation using the plane equation of each blo cker The depthclipp ed

p olygon also illustrates the geometric information required for the computation of

irradiance Jacobians

Prop erties of the Irradiance Jacobian

In this section we list some of the basic prop erties of the irradiance Jacobian

b eginning with existence By denition the Jacobian J exists wherever

is dierentiable which requires the existence of each directional derivative at r

Because we consider only area sources the variation of is continuous along any

line except when a blo cker is in contact with the receiving surface Instantaneous

o cclusion causes discontinuous changes in the vector irradiance In the absence

of contact o cclusions the variation of is not only continuous but dierentiable

everywhere except along lines where edges app ear to coincide that is p oints at

which a source or blo cker edge app ears to align with another blo cker edge

For instance when both blo ckers are present simultaneously in Figure the

irradiance curve coincides with curve B to the left of r and with curve A to the

right Therefore the irradiance at r has a discontinuity in the rst derivative Only

contact o cclusion and edgeedge alignments cause the Jacobian to be undened

other typ es of events cause higherorder discontinuities in the vector irradiance

but are rstorder smo oth

B2 P B3

(a)(b) (c)

Figure a The vertex Jacobian does not exist at the intersection of three edges A

smal l change can produce b a single apparent vertex or c two apparent vertices

From equation it would app ear that the irradiance Jacobian do es not

exist if any one of the vertex Jacobians fails to exist this is not always so A

vertex Jacobian may be undened b ecause the vertex lies at the intersection of

three edges as shown in Figure a In cases such as this a minute change in

r can lead to several p ossible congurations with dierent vertex Jacobians See

Figures b and c However the uno ccluded area of the source still changes

smo othly despite such a diculty atasinglevertex To ensure that equation

is valid wherever is dierentiable we simply restrict the edges that are used in

computing the vertex Jacobians to those that actually b ound the clipp ed source

Thus in Figure blo cker B is ignored until it makes its presence known bythe

addition of anew edge as in Figure c

prop erty of the Jacobian matrix is its connection with direc The most basic

tional derivatives For any S the directional derivative of at r in the

direction is

D r J

Although directional derivatives of may be approximated to second order with

central dierences using the irradiance Jacobian has several advantages First all

directional derivatives of r are easily obtained from the irradiance Jacobian at

r which requires a single global clipping op eration That is sources need only

be clipp ed against blo ckers once per p osition r when computing the irradiance

Jacobian according to equation In contrast nite dierence approxima

tions require a minimum of two clipping op erations to compute a single directional

derivative More imp ortantly directions of maximal change follow immediately

from the Jacobian but require multiple nite dierences to approximate

A nal prop erty which we build up on in the next section is the connection

with the rate of change of surface irradiance Dierentiating equation with

resp ect to p osition we have

T T

r J nn J

which asso ciates the irradiance gradient with the irradiance Jacobian Note that

J n is the curvature tensor of the surface at eachpoint r M For planar surfaces

J n so equation reduces to

r n J

which is the form we shall use to compute isolux contours on p olygonal receivers

When evaluating equation several optimizations are p ossible by distributing

the vector multiplication across the terms of equation which changes the

summation of matrices intoasummationofrow vectors

Applications of the Irradiance Jacobian

Thus far we have derived a closedform expression for the irradiance Jacobian

and describ ed the geometrical computations required to evaluate it By means of

equation the irradiance Jacobian can b e computed analytically at anypoint

r within a diuse p olyhedral environment illuminated by diuse luminaires The

most dicult asp ect of the computation is in determining the visible p ortions of

the luminaires although this is also a requirement for computing irradiance The

steps are summarized as follows

Matrix IrradianceJacobian Pointr

Matrix J

for each source P with radiant exitance M

b egin

P P depthclipp ed against all blo ckers as seen from r

for each i J vertex Jacobian for the ith vertex of P

P for each i E edge matrix for the ith edge of

for each i C corner matrix using E and E

i i i

sum of all C J J J

i i

end

return J

Here the inner lo ops all refer to the vertices as seen from r pairs of vertices

asso ciated with invisible edges are counted as one Gradients can then b e computed

using equation or equation The pro cedure ab ove is a generalpurp ose

to ol with many applications several of which are describ ed in the remainder of this

section

Finding Lo cal Extrema

The rst application we examine is that of lo cating irradiance extrema on surfaces

which can be used in computing bounds on the transfer of energy between sur

faces Given the availabilityof gradients the most straightforward approach

to lo cating a p ointof maximal irradiance is with an ascent metho d of the form

T T i i i

I nr n rr r r r

where r is a given starting p oint and the factor is determined by a line search

that insures progress is made toward the extremum For example the line search

may simply halve until an increase in irradiance is achieved The extremum has i

been found when no further progress can be made Minima are found similarly

The principle drawbacks of this metho d are that it nds only lo cal extrema and

convergence can be very slow when the irradiance function is at In the absence

of a global metho d for lo cating all extrema seed p oints near each of the relevant

extrema must b e supplied

Direct Computation of Isolux Contours

Curves of constant irradiance over surfaces are known as isolux contours

Within computer graphics isolux con tours have been used for depicting irradi

ance distributions volume rendering simplifying shading computa

tions and identifying isolux areaswhich are regions of approximately constant

illumination In computer vision isolux contours have been used to p erform

automatic image segmentation Because of their utility in visualizing the il

lumination of surfaces illumination engineers had develop ed metho ds for plotting

isolux contours by hand for one or two uno ccluded sources long b efore the advent

of the computer In this section we show how isolux contours can be

computed directly using the irradiance Jacobian

Every isolux contour on a surface M can be represented by a function of a

single variable r M that satises

rs r

for all s To compute such a curve we construct a rstorder ordinary dier

ential equation ODE to which it is a solution and solve the ODE numerically

The direction of most rapid increase in r at a point r M is given by the

gradient r r which generally do es not lie in the tangent plane of the surface

The pro jection of the gradientonto the surface is a tangentvector that is orthogonal

to the isolux curve passing through its origin See Figure a If the pro jected

gradient is rotated by degrees we obtain a direction in which the irradiance

remains constant to rst order The rotated gradients dene a vector eld whose

(a) (b)

Figure a Projecting the gradient onto a surface denes a D vector eld everywhere

orthogonal to the level curves b Rotating the projected gradients by creates a

vector eld whose ow lines are isolux contours Local maxima are then encircled by

clockwise loops

ow lines are isolux contours See Figure b Combining these observations we

dene the isolux dierential equation by

r Pr r r

r where with the initial condition r

Pr Rn r I n r n r

and Rz is a rotation by ab out the vector z The matrix Pr is constant

for planar surfaces The solution of this ODE is an isolux contour with irradiance

c r

Solving the Isolux Dierential Equation

Anytechnique for solving rstorder ordinary dierential equations can b e applied

to solving the isolux dierential equation The overriding consideration in selecting

an appropriate metho d is the number of irradiance values and gradients used in

taking a step along the curve Obtaining this information involves a global clipping

op eration which is generally the most exp ensive part of the algorithm

Multistep metho ds are particularly appropriate for solving the isolux ODE

since they make ecient use of the recent history of the curve For example

Milnes predictorcorrector metho d is a multistep metho d that predicts the point

r rs by extrap olating from the three most recent gradients and function

k k

values using a parab ola When the matrix P is xed Milnes predictor is given by

r r P g g g

k k k k

where g denotes the gradientatthepoint r andh is the step size Given the

k k

predicted value a corrector is then invoked to nd the nearest p oint on the curve

Because the contour is the zero set of the function r c the correction can be

p erformed very eciently using Newtons metho d Beginning with the predicted

point r a Newton corrector generates the sequence r r by

k k k

T i

h i

r r

i i i

r r c r

k k

jj r r jj

which converges quadratically to a point on the curve The iteration is rep eated

until

c r

where is a preset tolerance With this corrector accurate p olygonal approxi

mations can be generated for arbitrarily long isolux contours This would not be

p ossible with the traditional Milne corrector for example whichwould eventually

drift away from the curve With a go o d predictor very few correction steps are

required which saves costly gradientevaluations

Examples of Isolux Contours

The predictorcorrector metho d describ ed ab ove was used to compute isolux con

tours for simple test cases with b oth uno ccluded and partially occluded sources

Figure A family of isolux contours for three unoccluded sources

The step size h and the tolerance for the corrector were usersupplied parameters

Use of the Newton corrector made the curve follower fairly robust even abrupt

turns at or near derivative discontinuities in the irradiance function were automat

ically comp ensated for

To generate a family of curves depicting equal steps in irradiance similar to

a top ographic map we must nd starting points for each curve with the desired

used to nd a irradiance values c c c The Newton corrector can be

pointonthek st curveby nding a ro ot of the equation r c b eginning

at any p oint on the k th curve The curve families in Figures and were

automatically generated in this way Figure shows a family of isolux contours

resulting from three uno ccluded rectangular sources Three distinct families were

generated starting at each of the three lo cal maxima which were found by the

ascent metho d describ ed in section Figure shows a family of isolux

contours resulting from a rectangular source and a simple blo cker These contours

surround both a p eak and avalley

Because distinct isolux contours cannot cross any collection of closed contours

has an obvious partial ordering dened bycontainment To display lled contours

as shown in Figures a and b the regions can be painted in backtofront

order after sorting according to the partial order

Figure Isolux contours on a planar receiver due to a rectangular source and simple

blocker above the plane of the receiver

Because the isolux contours describ ed in the previous section are generated by

direct computation rather than by p ostpro cessing an image they may b e used in

the image generation pro cess For example isolux contours can b e used to drivea

meshing algorithm for global illumination

The idea is similar to that of discontinuity meshing which can iden

tify imp ortant discontinuities in the radiance function over diuse surfaces Isolux

contours provide additional information ab out radiance functions and can b e em

ployed for mesh generation either in a prepro cessing step for mo deling direct illu

mination or as part of a radiosity p ostpro cess to create a highquality mesh for

rendering a nal image

Figure Fil led isolux contours corresponding to the previous gures Each region

is shaded according to the constant irradiance of its contour

Figure Isomeshes generated from families of isolux contours using constrained

Delaunay triangulation

To b est exploit the information in the contours the mesh elements of an iso

mesh must follow the contours Constrained Delaunay triangulation may b e used to

generate a mesh with this prop erty from a sequence of segments that approximate

isolux contours In earlier work Lischinski et al used the same technique to

generate meshes that incorp orated irradiance discontinuities In the present work

the constraints force the edges of the mesh elements to coincide with the isolux

contours rather than crossing them Another advantage of Delaunay triangulation

is that it creates triangles with go o d asp ect ratios Figure shows the result

of applying this algorithm to the families of isolux contours shown in Figure

Meshes of varying coarseness can be generated by selecting subsets of the p oints

along the contours

Chapter

Irradiance Tensors

In this chapter we generalize the concept of irradiance to higherorder forms and de

rive an extended version of Lamb erts formula that applies to nondiuse surfaces

The generalization of irradiance consists of an innite sequence of tensors that in

cludes b oth the scalar and vector forms of irradiance as sp ecial cases Each tensor

in the sequence called an irradiance tensor consists of moments of a radiance

distribution function f r A large class of emission and scattering distributions

can b e characterized bycombinations of these moments which makes them useful

in the simulation of nondiuse phenomena Applications include the computation

of irradiance due to directionallyvarying area light sources reections from glossy

surfaces and transmission through glossy surfaces

The techniques develop ed in this chapter apply to any emission reection

or transmission distribution expressed as a p olynomial over the unit sphere As

e derive expressions for a simple but versatile subset of a concrete example w

these p olynomial functions called axial moments which are closely related to the

Phong reection mo del Using the extended version of Lamb erts formula we

derive closedform expressions for moments of all orders in p olygonal environments

Complete algorithms for ecient evaluation of the new expressions are presented

and demonstrated by simulating Phonglike emission and scattering eects

NonLambertian Phenomena

Rendering algorithms are frequently quite limited in the surface reectance func

tions and luminaires they can accommo date particularly when they are based

on purely deterministic metho ds To a large extent this limitation stems from

the diculty of computing multidimensional integrals asso ciated with nondiuse

phenomena such as reections from surfaces with directional scattering While nu

merous closedform expressions exist for computing the radiative exchange among

uniform Lamb ertian diuse surfaces with simple geometries these

expressions rarely apply in more general settings Currently the only approaches

capable of simulating nondiuse phenomena are those based on Monte Carlo

hierarchical sub division or numerical quadrature

Few metho ds exist at present for computing semicoherent reections of a scene

in a nearlysp ecular or glossy surface The earliest examples of glossy reection in

computer graphics are due to Amanatides and Co ok Amanatides used cone

tracing to simulate glossy reections for simple scene geometries and reectance

functions Co ok intro duced a general Monte Carlo metho d for simulating such ef

fects that was later extended to path tracing byKajiya and applied to realistic

surfaces by Ward Wallace et al approximated Phonglike directional

scattering by rendering through a narrow viewing ap erture using a zbuer Aup

p erle et al devised the rst general deterministic metho d using threep oint

transfers coupled with viewdep endent hierarchical sub division

This chapter presents the rst analytic metho d for computing direct lighting

eects that involve b oth directional emission or scattering and area light sources

these include illumination from nondiuse luminaires and reections in nondiuse

surfaces The metho d handles a wide range of emission and scattering distributions

from ideal diuse to sharply directional which greatly extends the rep ertoire of

eects that can b e computed in closed form As an example the simulation shown

in Figure depicts a stained glass windowdesignbyElsaSchmid reected in

Figure A stained glass window reected in a Phonglike glossy surface The reec

tion was computed analytical ly at each pixel using a boundary integral for each luminaire

The expressions were derived using irradiance tensors

a nearlysp ecular surface with a Phonglike reectance function Reections of this

typ e are challenging to simulate using previous metho ds but are straightforward

using the techniques develop ed in this chapter

The fundamental building blo ck of this chapter is the irradiance tensor a tensor

representation of irradiance whose elements are angular moments that is weighted

integrals of radiance with resp ect to direction Metho ds based on angular mo

ments have a long history in the eld of radiative transfer but are applied

here in a fundamentally dierentway In classical radiative transfer problems only

worder moments are relevant since detailed reections that o ccur at surfaces the lo

can generally be neglected For image synthesis however where the light

reected from surfaces is paramount highorder moments can be used to capture

the app earance of a nondiuse surface or the distribution of light emitted from a

boundary polygonal integral luminaires

reflective surface scattering

distribution

Figure Reectedradiance is determined by the scattering distribution integrated over

each polygonal luminaire Each surface integral can be replaced by a boundary integral

directional luminaire

Angular moments are an extremely general to ol for representing radiance distri

bution functions for instance they apply to all emission and reectance functions

that are dened in terms of p olynomials over the sphere However the sp ecic al

gorithms presented in this chapter address only a small class of these p olynomials

corresp onding to Phong distributions These p olynomials have a fundamen

tal connection with irradiance tensors and admit practical closedform expressions

in p olyhedral environments The resulting closedform expressions were used in

p olygonal computing the reection in Figure where the contribution of each

window elementwas computed analytically at each p oint of the reecting o or by

rst converting it to a b oundary integral See Figure

The new expressions are computed in much the same way as Lamab erts formula

for irradiance which requires that we rst compute the visible contours of

each luminaire when o cclusions are present When applicable the new expressions

oer a number of advantages over previous metho ds including ecientevaluation

exact answers in the absence of roundo error relative ease of implementation

and symb olic evaluation of related expressions such as derivatives Finally the

statistical error or noise that accompanies Monte Carlo metho ds is eliminated

The remainder of the chapter is organized as follows Section intro duces

basic concepts that motivate the denition of an irradiance tensor which is fur

ther develop ed in section These tensors are quite general with applications

in image synthesis well b eyond those explored in this thesis Section describ es

an extension of Lamb erts formula based on irradiance tensors that makes it ap

plicable to nondiuse environments In section we use the tensor version of

Lamberts formula to derive expressions for axial moments a convenient form of

moment with immediate applications In section we fo cus on p olygonal lumi

naires and derive the closedform expressions and algorithms that are subsequently

applied to three dierent nondiuse simulations in section Finally section

describ es two extensions nonplanar luminaires and spatially varying luminaires

Preliminaries

This section intro duces the physical and mathematical concepts that motivate the

denition of irradiance tensors and summarizes some of the to ols from dierential

geometry that are needed to derive expressions for the new tensor quantities We

b egin by describing a collection of tensors with immediate physical interpretations

and then extend this collection to an innite family The memb ers of the family

are of interest b ecause they embody higherorder moments of radiance distribu

in characterizing the directional variation of such tion functions which are useful

functions

Three Related Quantities

Most radiometric quantities can be dened in terms of weighted integrals of ra

diance We shall examine three such quantities that lead naturally to irradiance

tensors First the monochromatic radiation energy density at the p oint r IR

denoted urand dened by

joules

f r u d u ur

c m

is the radiant energy per unit volume at r Here c is the sp eed of light in the

medium A similar quantity known as the scalar irradiance p is given

by cur which has the units wattsm b ecause of the constant The vector

irradiance at a point r which app eared in chapter is dened by the closely

related vector integral

watts

r u f r u d u

The scalar quantity r v is the net ow of radiant energy through a surface at

r with normal v Finallythe radiation pressure tensor at r denoted by

r is a symmetric matrix found by integrating the outer pro duct uu

joules

uu f r u d u r

c m

The physical meaning of the bilinear form w rv is the rate at which photon

momentum in the direction w ows across a surface at r with normal v This tensor

is exactly analogous to the stress tensor encountered in the theory of elasticity

Thus each of the ab ove integrals has a dierent meaning and provides distinct

information ab out the radiance distribution function at the p oint r

Note that in equations and the integral is in eect distributed across

the elements of the vector or matrix In equation each elementofthevector is

aweighted integral of f r withaweighting function that is a rstorder monomial

on the sphere that is one of the co ordinate functions x y orz where

u e u x

y u e u

z u e u

for all u S and fe e e g is the canonical basis for IR Alternatively the

functions x y and z can b e viewed as the direction cosines of the vector u In equa

tion the weighting functions are the secondorder monomials x y z xyxz

and yz Thus the scalarvalued integrals embodied in equations and

resp ectively corresp ond to rst and secondorder angular moments of the radiance

distribution function

Although the three quantities dened ab ove are distinct they are also interre

lated For example since x y z itfollows that

trace rur

for all r IR which demonstrates a connection between the nd and thorder

moments Relationships of this nature also exist among higherorder analogues

and will b e an essential elementintheirconstruction

Polynomials over the Sphere

While the integrals dening vector irradiance and the radiation presure tensor em

b o dy rst and secondorder weighting functions resp ectivelywenow motivate the

extension of this idea to higher order monomials First we observe that p olynomi

als over the sphere based on the co ordinate functions x y and z can approximate

avery large class of functions in exact analogy with p olynomials over the real line

ell suited to approximating a class Secondly we show that the p olynomials are w

of functions that arise in simulating nondiuse phenomena

The general approximation prop erty can be shown very easily using a non

constructive argument Observe that any collection of functions over a common

domain denes an asso ciated vector consisting of the algebraic closure of the set

under p ointwise addition and scalar multiplication that is

f u f u

f g u f ug u

Figure The three direction cosines x y and z have graphs consisting of two

spherical lobes one positive and one negative These functions generate an algebra of

functions over the sphere that is dense in L

where is a scalar The resulting collection may b e further extended byintro ducing

a third op eration pointwise multiplication dened by

f g u f u g u

The collection of all functions obtained from the initial set by means of nite

pro ducts and linear combinations forms an algebra Let A denote the algebra

generated by the realvalued functions x y andz dened on S which are depicted

in Figure The set Awhich is algebraically closed under the usual vector space

op erations augmented by p ointwise multiplication denes the p olynomials over the

sphere

Wenow show that the elements of this algebra can approximate any L function

on the sphere with p that is any function in the space L S consisting

of all measurable functions f S IR with resp ect to such that

d u jf uj

That p olynomials over S can uniformly approximate this large class of functions

follows easily from the StoneWeierstrass theorem p stated b elow

Figure a A cosine lobe dened by a thorder monomial and b the same lobe

about an arbitrary axis which represents a thorder polynomial over the sphere

Theorem StoneWeierstrass If X is a compact set and N is an algebra

of continuous functions over X that separates the points of X and contains the

constant functions then N is uniformly dense in the continuous functions over X

points if for any A collection of functions over a set X is said to separate its

two p oints x and y in X there exists a function f in the collection such that

f x f y To see that the theorem applies to the algebra A of p olynomials over

the sphere we observe that the unit sphere is a compact set the algebra

generated by x y andz contains the constant functions since x y z and

the functions x y and z separate the points of S since distinct p oints must

dier in at least one co ordinate

Finally the approximation prop erty extends to all L functions with p

b ecause the set C S of continuous functions on S is dense in L S with

resp ect to the L norm p That is any L function on the sphere may

be approximated to within at almost every p oint by a continuous function

It follows that A is dense in L S

Of greater practical signicance is the ease with which these p olynomials can

approximate certain functions that arise in the simulation of nondiuse emission

and scattering As a simple example consider the monomial z whose graph con

sists of two elongated lob es as shown in Figure a The same shap e can be

dened along any axis by a thorder p olynomial for this reason the represen

tation is sometimes referred to as steerable See Figure b Scattering

distributions can be approximated by sup erp osing lob es along dierent axes and

of dierent orders Figure shows ahyp othetical BRDF constructed from three

lob es Directed lob es have other prop erties that make them a convenient to ol for

representing radiance distributions in general as we show in section

Figure A hypothetical bidirectional reectance distribution function dened as a

polynomial over the sphere the polynomial is formed from the superposition of three

directed lobes

In the remainder of this section we establish several basic facts ab out these

integrals that will b e useful in integrating p olynomials over subsets of the sphere In

particular we consider the normalization function for monomials over the sphere

which we dene by

i j k

u i j k x y z d

This function plays an imp ortant role in the development of irradiance tensors For

example it is the key to the o ddeven behavior of p olynomials over the sphere

that is a fundamental dierence b etween monomials whose exp onents are all even

and those with at least one o dd exp onent Wenow state the twocentral theorems

concerning the function i j k

Theorem Let i j and k be nonnegative integers Then i j k if and

only if at least one of i j and k is odd

Pro of See App endix A

Where it is nonzero the normalization function can b e succinctly expressed in

two very dierent forms The rst is in terms of the double factorial dened by

n n if n is even

n n if n is o dd

The second representation of i j k involves the multinomial coecient

i j k

i k

where n i j k This co ecient gives the number of distinct permutations

among n ob jects of three distinguishable typ es i of the rst typ e j of the second

typ e and k of the third typ e Using these denitions wenow state the second

theorem

Theorem Let i j and k be nonnegative even integers Then

i j k

i k

where n i j k and the primes denote division by

See App endix A Pro of

Elements of Dierential Geometry

To study higherorder generalizations of irradiance we shall employ some basic

machinery from dierential geometry In particular we shall use the language of

tensors and dierential forms This section summarizes the relevant facts that will

be needed in the remainder of the chapter

Dierential forms are a formalism for representing integrands of multiple in

tegrals The algebraic prop erties of dierential forms subsume the classical dif

ferential op erators of divergence gradient and curl and extend immediately to

arbitrary dimensions To emphasize the sp ecial behavior of multidimensional in

tegrands the classical notation dy dx is replaced with the wedge pro duct dy dx

which is referred to as a form higherorder forms are constructed from multi

ple wedge pro ducts The collection of algebraic rules asso ciated with dierential

forms is known as the exterior calculus it describ es the interaction of the wedge

pro duct and the exterior derivative op erator d in terms of classical derivatives

The identities from the exterior calculus that we require are these

dx dy dy dx

dfddf d

f f f

dx dy dz df

x y z

where f is a realvalued function dened on IR Several intrinsic prop erties of

dierential forms are also imp ortant In particular a pform is said to b e closed

if d and exact if there exists a p form such that d The principle

motivation b ehind the development of the exterior calculus was to determine when

dierential forms are exact The concept of an exact dierential form is crucial

to the following development and will further explain the o ddeven behavior of

p olynomials over the sphere

Tensors are another imp ortant to ol in the study of multidimensional integra

ya large role in what follows The concept of a tensor may tion and they will pla

be dened in a number of ways in the present work an nthorder tensor or n

tensor shall mean a multilinear functional dened on the nfold cartesian pro duct

V V of a vector space V In the following discussion we shall only

consider the case where V is the Euclidean space IR

Tensors of any order can b e formed using the tensor pro duct op erator denoted

by which constructs bilinear mappings on the cartesian pro duct of two vector

spaces If A and B are both linear functionals on IR then A B is a bilinear

functional on IR IR or IR dened by

A Bx y Ax By

for all x y IR The tensor pro duct op eration is asso ciative so pro ducts of the

form A B C are unambiguous

Every linear functional A IR IR is uniquely represented as an inner pro duct

with some vector which in turn may b e sp ecied by its co ordinates with resp ect to

some basis Thus the tensor A may b e identied with its cartesian components

The comp onents of an ntensor will be denoted denoted by A for i

using either n subscripts or a single multiindex consisting of n subscripts The

tensor pro duct can therefore be written in comp onent form as A B A B

i j

for i j where A and B are b oth tensors

For notational convenience we shall adopt the summation convention so that

summation is implicit over every index o ccurring twice in a term For example

A B A B

ij jk ij jk

is a tensor This convention will not apply to subindices that is to indices

of other indices The tensors that we shall derive will dep end largely on two

fundamental tensors the Kronecker delta and the LeviCivita symbol also called

the permutation symb ol resp ectively dened by

if i j

otherwise

if i j k is an even permutation

if i j k is an odd permutation

ij k

if i j k is not a permutation

In the derivations that followwe employanumb er of useful relationships that exist

among these tensors The basic identities that we shall require are summarized

b elow without pro of

ij ij

ij k lj k il

pj l kml pk jm pm jk

A B A B

ij i j

A B A B

j k i

ij k

Here and denote the standard dot and cross pro ducts resp ectively Finallywe

require the generalized Stokes theorem which maybe written

Z Z

A A

p pk

where A is a subset of IR or more generally a pmanifold in IR and is a

p form Here A denotes the b oundary of A Equation is an extension

of the fundamental theorem of calculus which subsumes b oth the classical Stokes

theorem and Gausss theorem see for example Bishop and Goldb erg or Spi

to ol by which we shall reduce vak The generalized Stokes theorem is the

the area integrals corresp onding to irradiance tensors into b oundary integrals and

in some instances obtain closedform expressions

Generalizing Irradiance

The physical quantities dened in the previous section can b e extended to higher

order forms using the formalism of tensors Doing so will provide a means of

characterizing radiance distributions in terms of highorder moments

Denition of Irradiance Tensors

Aside from the constant c which has the units secm all of the integrals

describ ed in section are multilinear functionals of the form

watts

u u f r u d u

where denotes a tensor pro duct that is eachoftheintegrals denes a mapping

from IR IR to IR that is linear in each of the vector arguments inde

p endently Viewed in this way radiation energy density vector irradiance and

the radiation pressure tensor are tensors of order and resp ectively with

equation providing the natural extension to tensors of all higher orders

When the function f represents radiance the family of integrals in equation

subsumes the notion of vector irradiance and each memb er p ossesses the units of

irradiance wattsm We therefore refer to this family of multilinear functionals as

irradiance tensors Although highorder irradiance tensors do not have immediate

physical interpretations p they are nevertheless useful vehicles for inte

grating p olynomial functions over the sphere such as those representing emission

or scattering distributions

In this work we shall restrict f r to be piecewise constant or more gener

ally piecewise p olynomial over the sphere Angular moments of f then reduce

represent these to p olynomials integrated over regions of the sphere To concisely

integrals we intro duce a simplied form of irradiance tensor dened by

T A d u u u

n factors

where A S and n isaninteger The integrand of such atensorcontains all

i j k

monomials of the form x y z where x y z S and i j k nthus T A

consists of nthorder monomials integrated over A

In what follows the region A S will represent the spherical pro jection of

P a luminaire P IR Using the notation of chapter we shall write A

where again we assume without loss of generality that the origin is the center of

pro jection The tensors T A allow us to p erform several useful computations

for example wemay compute angular moments of the illumination due to uniform

Lamb ertian luminaires or the irradiance due to directionally varying luminaires

Irradiance tensors of all orders are dened by surface integrals and in some

instances may b e reduced to onedimensional integrals by means of the generalized

Stokes theorem The resulting b oundary integrals can be evaluated analytically

for certain patch geometries such as p olygons The foregoing approach extends

the classical derivation of Lamb erts formula for irradiance yielding more

complex b oundary integrals in the case of higherorder tensors

The Form Dierential Equation

In this section weinvestigate some of the prop erties of the tensor T A and show

that it may b e expressed in terms of a b oundary integral and a term involving the

solid angle of A which is A

One of the fundamental to ols of this chapter is a representation of solid angle in

terms of a surface integral If eachray through the origin meets a surface P IR

at most once then the solid angle subtended by P is

P d

where the form d is given by

dy xdy dz y dz dxz dx

x y z

over the domain IR fg Here x y and z denote co ordinate charts over the

manifold P For the following development it is convenient to omit the absolute

value signs in equation thus allowing the solid angle to b e signed according

to the orientation of the surface The negative sign in equation is included

so that solid angle will be positive for surfaces whose orientation is p ositive or

n(r ) θ r dΑ r

dω

Figure The geometrical relation between the forms d and dA interpreting them

as innitesimal quantities

counterclockwise with resp ect to the origin Note that the denition of the form

d corresp onds to the familiar change of variable

cos

dA d

where d and dA are interpreted as innitesimal solid angle and innitesimal area

resp ectively and r and are as shown in Figure More formally equation

is the pullbackofthe solid angle form to the volume element dA of the surface

The connection between the measure and the form d is simply

Z Z

A d u d

A A

where A S and A is p ositively oriented We shall always assume that A is

suciently wellb ehaved that integrals of b oth forms are dened that is A is b oth

measurable and rectiable Which representation we use will dep end on the

context In general we adopt the language of dierential forms d for computations

that involve the integrand and the measuretheoretic notation when the emphasis

is on the integral as a whole either as a function or as a transformation

Our strategy for obtaining closedform expressions for each of the irradiance

tensors is to rst convert the surface integrals into integrals over the b oundaries

The resulting b oundary integrals can then be expressed in closedform for simple

geometries such as p olygons This approach is analogous to classical derivations

of formulas for irradiance To extend the approach to more general

settings we seek a form which is itself an nthorder tensor such that

Z Z

u d

A A

where u r jj r jj and u denotes the nfold tensor pro duct of u that is

u u u

n terms

Thus each elementofu is a pro duct of n direction cosines

u u u u

i i i

The relationship between the dierential forms asso ciated with the two integrals

in equation is provided by the generalized Stokes theorem from which it

follows that

d u d

In terms of individual comp onents equation is equivalent to the equations

d u u d

i i i i

n n

where i fork n As we shall see in some instances no form

satises the ab ove equation consequently aslight mo dication of equation

will be required in order to integrate u d over A Because the pro duct of the

n scalars on the right of equation commute the tensor p ossesses a large

number of symmetries Of its elements only

n n

are unique Equation gives the number of ways to distribute n indistinguish

able ob jects exp onents among three bins x y and z

Now we shall consider the question of whether there exists a form that

satises equation which is equivalent to determining when the form

u u d is exact We rst observe that if is an exact form dened on

i i

S then

This is a direct consequence of Stokes theorem for if is exact then d for

some form dened on S It follows that

Z Z Z

S S

since S has no b oundary An imp ortant implication of this fact is that the form

d is not exact since

which is the surface area of the unit sphere Hence no form can satisfy

equation when n which implies that the solid angle subtended by A

cannot be expressed as an integral over its outer contour as seen from the origin

Nevertheless it is common practice to denote the form for solid angle by d

with the apparent implication that it is the dierential of some form

Because d is not exact we require a direct means of computing T A which

corresp onds to solid angle since it cannot b e reduced to a b oundary integral When

A is a p olygon this is easily accomplished using an elementary result of spherical

trigonometry as shown below

i j k

Theorem implies that the form x y z d cannot b e exact if i j and k are

all even which indicates that the complication noted ab ove for computing solid

angle must also arise with all the other evenorder tensors Fortunately the general

case is no more dicult to handle than the thorder case To see this we add a

constant to each monomial that forces equation to hold which is a necessary

condition for exactness That is we intro duce constants c such that

ij k

h i

i j k

x y z c d

ij k

for all i j and k Clearly c i j k so equation maybe replaced by

ij k

h i

i j k

d x y z i j k d

i i

where i j and k are the number of o ccurrences of the indices and resp ec

tively in i i i To show that the new form on the right of equation

is now exact by De Rhams theorem p it suces in this instance to show

that it is closed To show that the right hand side of equation is closed

i j k

we show that all expressions of the form x y z d are closed See App endix A

Therefore we are guaranteed that a form satisfying equation exists in

all cases To express equation in tensor form we dene

I I I I

where I is the multiindex i i and denotes the numb er of o ccurrences of

the index i in I Equation can now be written as

n n

d d u

which is the fundamental equation for This equation is an example of a form

dierential equation since the unknown is the dierential form p

Given a form satisfying equation the ntensor T A is given by

T A A

Thus the tensor is a comp onent of the nal expression for T A which makes

is the explicit the role of solid angle in all evenorder tensors Equation

generalization of Lamb erts formula that we develop further and apply in the re

mainder of this chapter

To make equation more concrete the following theorem provides an

expression for in terms of elementary tensors

Theorem When n is odd When n is even

j j j j

JS I

where S I is the set of al l permutations of the multiindex I i i

Pro of See App endix A

Listed b elow are the rst few cases of equation which have been simplied

by combining equivalent terms

ij kl ik jl il jk

ij kl pq ij kp lq ij kq pl ik jl pq ik jp lq

ik jq pl il kj pq il kp jq il kq pj ip kl jq

lj iq kj pl ip kj lq ip kq jl iq kl pj iq kp

While many of the terms of the expansion in equation can be combined

the number of distinct terms nevertheless grows rapidly as is already evident for

n The number of terms remaining after simplication is the number of

distinct groupings of the n indices into pairs To count them consider the number

of ways to complete a sequence of n pairs removing the indices one at a time

without replacement this number is n Consequently equation is

useful for symb olic purp oses but quickly b ecomes impractical for computation as

n increases On the other hand as n increases b ecomes increasingly sparse

vanishing completely whenever n is o dd When n is even only

n n

of the elements are nonzero This is the number of ways to distribute the n

powers of among x y andz or equivalentlythenumb er of distinct arrangements

of n ob jects and two partitions

Extending Lamberts Formula

In this section we address the problem of nding the form that satises equa

tion The derivation will b e completely general applying to irradiance ten

sors of all orders We shall show that the higherorder tensors can b e conveniently

expressed by means of a recurrence relation of the form

n n

T G T H A

n n

with T and T A A Thus each tensor of order or higher can be

constructed from two comp onents a function of the b oundary and a function of a

lowerorder tensor We shall start by examining the secondorder tensor as it will

illustrate some of the steps that also apply in the general case

The Radiation Pressure Tensor

We rst demonstrate the approach by deriving an expression for the secondorder

tensor T A which is prop ortional to the radiation pressure tensor dened on

page We then derive a closedform expression for this tensor when A is a

spherical p olygon

First let A S be an arbitrary region It follows from equation and

theorem that the form corresp onding to T A is a matrix such that

I d d uu

This is the form dierential equation asso ciated with the radiation pressure tensor

Letting r denote an arbitrary p oint on the surface P IR wemay rewrite form

d dened in equation as

r dr dr

qlm

q l m

jj r jj

where jj jj is the Euclidean norm Henceforth we denote the scalar jj r jj by r

From equations and it follows that d may b e written

d Q dr dr

ij l m

where we have intro duced the tensor Q dened by

ij l m

r r r r

ij i j qlm q

ij l m

For each i and j the form can be written as a linear combination of basic

forms which implies that there exists a tensor A such that

A dr

ij ij l l

Dierentiating b oth sides of equation and simplifying we have

d A dr dr

ij ij l m m l

where the index following the comma indicates a partial derivative Therefore the

problem of nding reduces to nding a tensor A that satises

ij l

dr A dr dr Q dr

m ij l m m l l

ij l m

for all i and j in f g Since r r r r equation is equivalentto

i j j i

A Q

kml ij l m klm

ij l m

for all i j and k in f g Note that the transp osition of dr and dr is accounted

l m

for by transp osing the indices of one of the factors Substituting equation

into the ab ove equation and using identity equation becomes

r r r r

j k ij i

kml ij l m

∂A

Figure For any region A S the unit vector n is normal to the boundary A

directed outward and tangent to the sphere

for all i j andk in f g This is the fundamental equation asso ciated with the

tensor T given any tensor A satisfying equation we can then express the

tensor T A in terms of a boundary integral and a term involving surface area

In particular it follows from equation that

A A dr T A

ij ij l l

The following theorem provides a formula for T A in which the tensor A

app ears as an expression involving the outward normal to the b oundary curve A

as shown in Figure We then nd a closedform expression for the b oundary

integral app earing on the right of equation when A is a spherical p olygon

Theorem For any A S the tensor T A is given by

un ds T A I

where ds denotes integration with respect to arclength and n is the outward normal

to the curve A

Pro of We need only show that if un ds A dr then A satises equa

ij l l ij l

tion We pro ceed by expressing the integral in equation in terms of

unit sphere

. u boundary curve n

Figure The outward normal to the boundary curve is obtained from u and its

derivative with respect to arclength denoted by u

the p osition vector r and its derivative From Figure we see that n u u

where u denotes the derivative with resp ect to arclength duds Therefore

n ds u du

I uu r

r r

r dr

It follows that we may replace un ds with the expression A dr where

ij l l

r r

jpl p i

ij l

Toshow that the ab ove expression satises equation we compute its partial

derivativeand multiply by First note that

kml

k k

jj r jj jj r jj

for any n Thus the partial of A with resp ect to r is

ij l m

r r r r r

i p m im p pm i

ij l m pj l

r r

Multiplying b oth sides of equation by r identity yields

kml

r r r r r r r r A

i p m im p pm i kml ij l m pm jk pk jm

Expanding and simplifying using identities and we have

r A r r r r r r r r r r r

kml ij l m jk i jk i jk i i j k ij k jk i

r r r r

ij i j k

Dividing by r yields equation which proves the theorem

Corollary Let A S be a spherical polygon with vertices fu u u g

Then the secondorder irradiance tensor T A is given by

T A I sin b n

where is the angle between u and u and

i i i

u u

i i

jj u u jj

i i

u u

i i

jj u u jj

i i

are the unit normals and unit bisectors respectively

Pro of Let be the ith edge of A That is is the great arc connecting vertices

i i

u and u We compute the b oundary integral along using two facts the

i i i

normal n is constant along each edge and the integral of u along the arc results

in a vector that bisects the angle See Figure Thus

Z Z

T T

u ds n un ds

i i

b n cos d

b n sin

Substituting the expression ab ove into equation gives the result

c bi

ui ui+1 ζ i

Θi

Figure The integral of u over a great arc is a vector proportional to the bisector of

the arc This fol lows from symmetry

Corollary extends Lamberts formula to the secondorder form In itself

this generalization is of little utility other than illustrating some of the machinery

needed for the general form which is the topic of the next section

A General Recurrence Relation

In this section we show that equation can be extended to all higher orders

by means of a recurrence relation expressing each irradiance tensor in terms of

lowerorder tensors

We intro duce some sp ecial notation b efore giving the general theorem for ir

radiance tensors If I is the multiindex i i i we dene I to be the k th

n k

index of I and for any integer k n we dene Ik to be the n element

multiindex obtained by deleting the k th index of I That is

I k i i i i i

k k n

Using the new notation w enow state and prove the central theorem of the chapter

Theorem Let n be an integer and let A S be a measurable set Then

the tensor T A satises the recurrence relation

n n

T A T A u n ds

j I j

I j I

with T A A and T A where I is an n index ds denotes

integration with respect to arclength and n is the outward normal to the curve A

Pro of The strategy will b e to transform the b oundary integral in equation

into a sum of irradiance tensors obtaining the other two terms in the recurrence

relation The pro of will pro ceed in four steps Step one is to express the b oundary

integral in terms of r and dr which will make it easy to manipulate Step two

is to convert this b oundary integral into a surface integral by means of Stokes

theorem Step three which is the most lab orious is to express the surface integral

in terms of solid angle The fourth and nal step is to show that the resulting

For the surface integral is equivalent to the remaining terms in equation

purp ose of the derivation we shall let P denote any p olygon whose pro jection is

the spherical p olygon A That is A P

Step Rewrite the b oundary integral

We b egin by expressing n ds in terms of the p osition vector r and its derivative

Note that n is a function of u since it is an element of the tangent space of S at

u From equation we have n ds r dr r Therefore we may write

Z Z

r dr

jpl

p l

u n ds

n n r r

A P

A dr

I jl

where we have intro duced the n order tensor A which is dened by

r r

jpl

Ijl

n r

Step Convert the b oundary integral to a surface integral

This step follows immediately from Stokes theorem and the fact that

d f r dr f r dr dr

i i im m i

which is a consequence of equations and Applying Stokes theorem

to equation followed by equation we have

Z Z Z

n n n

A dr d A dr A dr dr

l l m l

I jl Ijl I jlm

P P P

Computing the partial derivativeofA with resp ect to r yields

Ijl

n n

r r r

I m I m p

pj l

Ijlm

n n

r n r

By substituting equation into equation we obtain a surface integral

although it is not yet in the desired form This is remedied in the next step

Step Express the surface integral in terms of solid angle

Rewriting the surface integral in terms solid angle will be done in several steps

by exploiting the anticommutativityof the wedge We rst obtain a factor of

kst

pro duct and applying identity in doing so we intro duce additional dummy

indices k s and t Thus

Z Z

dr dr dr dr

m l l m

n n

A dr dr A

m l

I jlm Ijlm

P P

tl ms tm sl

dr dr A

s t

Ijlm

h i

dr dr

s t

kst

kml

I jlm

Note that the presence of makes the nal factor on the right very similar to

kst

d it diers only by a factor of r r To complete it we obtain the missing factor

after expanding the other bracketed expression Multiplying equation by

and simplifying we have kml

n n

r r r

I I m m p

pm jk pk jm kml

Ijlm

n n

r n r

n n n

r r r r n r

r r r

k m

I j j k Im I

n n n

n r r n r

The ab ove expression simplies further since the nal term vanishes this follows

from the observation that

n n

X X

n n n n

r r r r r r n r

m I

m m I

Im Ik Ik I

k k

Incorp orating this simplication equation maynow b e written as

Z Z

r r

r r r

dr dr

s t

j k Ij

kst

dr A dr

l m

Ijlm

n n

n r r

P P

r r

I j j

n n

n r r

where the factor of r r has b een used to complete the solid angle form d The

result is an integral over solid angle as desired

Step Convert to asumof irradiance tensors

It is now straightforward to express equation in terms of irradiance ten

sors Breaking equation into two terms and expanding r as we did in

equation we have

Z Z Z

X r r

j I

I k

d d A dr

I jl

n n

n r r

P P P

Finally using equation to replace the expression of the left and identifying

the two integrals on the right of equation as irradiance tensors of orders n

and n resp ectively we have

n n

u n ds A T A T

j I

k I k

n n

which proves the theorem

From theorem it can be seen that each tensor of the form shown in equa

tion can b e reduced to a b oundary integral and a term constructed from the

tensor of two orders lower the latter being added along generalized rows and

columns of T Note that the recursive formulation subsumes the expression for

given in theorem

Another consequence of the theorem is that T A can be computed analyt

ically whenever the corresp onding b oundary integrals and base case can be In

particular when A is the spherical pro jection of a k sided p olygon and n

equation yields

T n n ds A

j j

where is the angle subtended by the ith edge of the p olygon and n is its

outward normal This is the vector form of Lamb erts wellknown formula

Theorem denes a family of closedform expressions that provides a natural

extension of Lamb erts formula Although equation is impractical com

putationally for moments of order three and higher it succinctly expresses the

relationship among all the tensors For instance it is apparent that all evenorder

tensors incorp orate solid angle T A while the o ddorder tensors do not Wenow

and apply them to problems of derive several useful formulas from this equation

image synthesis

Angular Moments

With equation as a starting p oint wemay obtain expressions for individual

moments or sums of moments without explicitly constructing the tensors This

is of great practical imp ortance since the size of T A grows exp onentially with

n yet only On of its elements are distinct In particular we shall nd that the

p olynomials corresp onding to the steerable lob es describ ed in section can be

integrated over p olygonal domains with the aid of irradiance tensors These lob es

dene a sp ecial class of weighting functions over the sphere that are fundamentally

related to irradiance tensors and lead to a numb er of useful formulas by means of

equation

Axial Moments

We b egin by considering the sp ecial case of moments ab out an axis which denes

a simple class of p olynomials ov er the sphere Given an arbitrary subset A S

and a unit vector wwe dene the nth axial moment of A ab out w by

A w w u d u

More precisely equation is a momentofthe characteristic function of Athat

is the function dened on S that is one on A and zero elsewhere As a cosine

to a power the p olynomial weighting function within is essentially a Phong

around the direction w as shown in Figure a These distribution centered

functions are steerable in the sense that they to b e reoriented without raising their

order The ease of controlling the shap e and direction of the lob e makes this

p olynomial function useful in approximating reectance functions as Ward did

with Gaussians or an exact representation of simple Phonglike functions

To obtain a closedform expression for we b egin by expressing the integrand of

equation as a comp osition of tensors

w u u u w w

I I

0.6 0.4

0.4 0.3

0.2 0.2 0.1

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6

-0.1

-0.2

n n

Figure Crosssections of the weighting functions w u and w u v u where v

is the vertical axis moment orders are and starting from the outer curves

The summation convention applies to all rep eated pairs of indices which includes

all subindices of I in equation It follows that

A wT Aw w

which asso ciates the nth axial moment with the nthorder tensor Using equa

tion to expand equation and simplifying by means of equation

we obtain a recurrence relation for the axial moment A w

n n

w u w n ds n w w n

where the function arguments have been omitted for brevity Equation is a

recurrence relation for with base cases A and A A When

n the recurrence relation reduces to a single b oundary integral involving a

p olynomial in w u Since w isaunitvector we have

n q n n

n w u w u

w u w n ds

where q if n is even and q if n is odd This expression is useful as a

comp onent of more general expressions such as doubleaxis moments

DoubleAxis Moments

An imp ortant generalization of equation is to allow for moments with re

sp ect to multiple axes simultaneously this will prove useful for handling radiant

exchanges involving pairs of surfaces We dene the doubleaxis moment of A with

resp ect to w and v by

n m

A w v w u v u d u

A recurrence relation for can also be obtained from equation by ex

pressing the integrand as a tensor comp osition with n copies of w and m copies of

the vector v We shall only consider the case where m which corresp onds to

T Aw w v and yields the formula

n A w v n w v A w w u v n ds

shap e of the weighting Figure a shows how an additional axis can change the

function Note that when v w equation reduces to the n order axial

momentgiven by equation Recurrence relations for with mcanbe

obtained in a similar manner although the resulting b oundary integrals are more

dicult to evaluate

Evaluating equations and in closed form is the topic of the next

section In section we showhow these moments can b e applied to the simulation

of nondiuse phenomena

Exact Evaluation of Moments

Equations and reduce tensors and moments to onedimensional in

tegrals and in the case of even orders solid angle This section describ es how

b oth of these comp onents can b e evaluated in closed form Thus far no restrictions

have been placed on the region A S however we shall now assume that A is

the spherical pro jection of a p olygon P IR which may be nonconvex The

B Q β E A α B A γ C C D

(a) (b)

Figure a The solid angle of a spherical triangle is easily obtainedfrom the internal

angles b Nonconvex spherical polygons can behandled by spoking into triangles from

an arbitrary point Q and summing the signed areas

resulting pro jection is a geodesic or spherical p olygon whose edges are great arcs

that is segments of great circles

When P is a p olygon the computation of solid angles and b oundary integrals

are b oth greatly simplied First and most imp ortantlythe outward normal n is

constant along each edge of a spherical p olygon whichallows the factors of w n and

v n to b e moved outside the integrals in equations and resp ectively

simplication emerges in the parametrization of the b oundary integrals A second

as we shall see below

Solid Angle

The solid angle subtended by A is given by a simple surface integral Because

the corresp onding dierential form is not exact however it cannot b e simplied

further to a b oundary integral p Fortunately the solid angle subtended

by a p olygon can be computed directly in another way If P is a triangle in IR

its pro jection P is a spherical triangle AB C Girards formula for spherical

triangles p then states that

AB C

where and are the three internal angles as shown in Figure a The

internal angles are the dihedral angles between the planes containing the edges

For instance the angle in Figure a is given by

B A A C

cos

jj B A jjjj A C jj

Equation generalizes immediately to arbitrary convex p olygons p

however nonconvex p olygons require a slightly dierent approach The solid an

gle subtended by a nonconvex p olygon can b e computed bycovering its spherical

pro jection with n triangles one per edge all sharing an arbitrary vertex Q S

See Figure b The solid angle is then the sum of the triangle areas signed ac

cording to orientation In the gure QAB QC D andQD E are all p ositive

while QB C is negative according to the clo cksense of the vertices This metho d

P into triangles avoids the complication of decomp osing

Boundary Integrals

The b oundary integrals in equations and can b e approximated bynu

merical quadrature or evaluated analytically in terms of On elementary functions

p er edge We shall only describ e analytic evaluation and a related approximation

b oth based up on a sumofintegrals of the form

F x n cos d

Integrals of this form may be evaluated exactly using a recurrence relation given

b elow To express the integral in equation in terms of F x n when A is

a spherical p olygon we pro ceed by parametrizing the great arc dened by each

edge as

s cos t sin u

ui+1

t ζ i Θi u

s i

Figure The vectors used to parametrize the arc dened by a polygon edge

where s and t are orthonormal vectors in the plane containing the edge and the

origin with s directed toward the rst vertex See Figure

To simplify the line integral over the great arc S let be the angle

subtended by the arc which is also the length of and let

a w s

b w t

a b c

From the parametrization given ab ove it follows that

Z Z

w u ds a cos b sin d

n n

c cos d

c F n F n

where is the angle satisfying cos ac and sin bc The function F x n

c steps by means of the recurrence relation may then be evaluated in bn

h i

F x n cos x sin x n F x n

where F x sin x and F x x Recurrence relation follows easily

from equation after integrating by parts Finally the complete integral in

equation is a weighted sum of the integrals shown in equation with a

sequence of dierent exp onents which are all even or all o dd By computing this

sequence of integrals incrementally the complete sum can be expressed in terms

of On elementary functions as demonstrated in the following section

Algorithms for Ecient Evaluation

Wenowshow that nthorder axial moments of a k sided p olygon may b e computed

exactly in the absence of roundo error in Onk time and provide the complete

algorithm in pseudoco de The algorithm evaluates equation in On time

for each of the k edges using recurrence relation The steps are conve

niently broken into three pro cedures that build up on one another CosSumIntegral

LineIntegral and BoundaryIntegral The key to ecient evaluation is the pro ce

dure CosSumIntegral which computes the sum

k k n

c F x k c F x k c F x n

where k m if m n is even and k m otherwise for a given integer

m The reason for the parameter m will b ecome apparent later it is included

so that the pro cedure CosSumIntegral can accommo date b oth single and double

axis moments

Because recurrence relation generates integrals of cosine with increasing

powers all integrals in expression may b e generated as easily as the last term

F x n This strategy is embodied in the pro cedure CosSumIntegral which is c

shown below

real CosSumIntegral real x c integer m n

integer i if evenn then else Loop counter

real F if evenn then x else sin x Cosine integrals

real S Accumulates the nal sum

while i n do

if i m then S S c F

F cos x sin x i F i

i i

endwhile

return S

end

tegral corresp onding to a The next pro cedure LineIntegral reduces the line in

p olygon edge into a sum of cosine integrals the steps corresp ond to equation

summed over a sequence of exp onents from m to n

real LineIntegral vector A B w integer m n

if n or w A and w B then return

vector s Normalize A

vector t Normalize I ss B Component orthogonal to A

real a w s

real b w t

real c a b

real angle b etween A and B

real sign b cos ac

return CosSumIntegral c m n CosSumIntegral c m n

end

The next pro cedure BoundaryIntegral computes the complete b oundary integral

for a given k sided p olygon P by forming a weighted sum of k line integrals The

weight asso ciated with each edge is the cosine of the angle between its outward

normal and the second vector v which may coincide with w

real BoundaryIntegral p olygon P vector w v integer m n

real b Accumulates the contribution from each edge

for each edge ABinP do

vector n Normalize A B

b b n v LineIntegral A B wmn

endfor

return b

end

With these three basic pro cedures wemaynow dene the pro cedure AxialMoment

which computes the nth axial moment of a p olygon P with resp ect to the axis w

Evenorder moments also require the computation of a signed solid angle which

is handled by this pro cedure Equation then corresp onds to the simple

pro cedure AxialMoment which follows

AxialMoment p olygon P vector w integer n real

real a BoundaryIntegral P w w n

if evenn then a a SolidAngle P

return an

end

The function SolidAngle returns the solid angle subtended by the p olygon P using

the metho d describ ed in section Because the sign of the b oundary integral

dep ends on the orientation of the p olygon the solid angle must b e similarly signed

Thus SolidAngle is p ositive if the vertices of P are counterclo ckwise as seen from

the origin and negative otherwise

Finally the pro cedure DoubleAxisMoment computes the nthorder momentof

apolygonP with resp ect to the w axis and the storder moment with resp ect to

the v axis Equation then corresp onds to the pro cedure DoubleAxisMoment

en next which is giv

real DoubleAxisMoment p olygon P vector w v integer n

if n then return AxialMoment P vn

real a AxialMoment P wn

real b BoundaryIntegral P w vnn

return n a w v b n

end

It is easy to see that b oth AxialMoment and DoubleAxisMoment require Onk

time assuming that trigonometric and other elementary functions are evaluated

in constant time Note that the redundancy in calling b oth AxialMoment and

ed however this do es not aect the order BoundaryIntegral can easily be remov

of the algorithm

Normalization

In applying the ab ove metho ds it is frequently useful to normalize the resulting

distributions while ignoring negative lob es Wenow showhow this can b e done for

distributions dened in terms of doubleaxis moments

The two planes orthogonal to the axes of a doubleaxis moment partition the

sphere into four spherical lunes also known as spherical digons let L and L

be the two lunes in the p ositive halfspace dened by v as shown in Figure

o normalize the weighting function asso ciated with a doubleaxis moment for T

example we must compute the integral

N w vn w u v u d u

L w v

where v is the surface normal thus the integrand of equation is p ositive

for all exp onents n on the lune L All luminaires must be clipp ed by the planes

dening this lune when computing the moments

Integral can b e evaluated analytically using equation which results

in two b oundary integrals corresp onding to the arcs C and C in Figure The

w v

sp ecial nature of the b oundaries greatly simplies the computation In particular

we have w u and n w on the curve C while v n on the curve C

w v

Substituting equation into equation with A C C and applying

v w

the aforementioned simplications we have

n N w vn w u ds

w v L w jq j ds

i h

n q

ds w u w u w v

where q if n iseven and q if n is odd

To apply the pro cedure CosSumIntegral to the computation of the integrals

ab ove we split the lune L into twoidentical spherical triangles Letting denote

one of the halves by symmetry it follows that

N w vn w v

The computation within the lo op of pro cedure CosSumIntegral is greatly simplied

by the fact that x which eliminates the trigonometric terms Consequently

the terms of the sum can be generated using the recurrence formula

t c t

i i

starting with t ifn is even and with t c if n is o dd where

w v c

To compute w we note that the area of the lune L is twice its

internal angle and cos w v Thus

cos w v

w L2 Cw

Figure Normalizing the weighting function of a doubleaxis moment reduces to

computing a boundary integral along the two half circles C and C This boundary

v w

denes the spherical lune L on which the moment is positive for al l n

Combining these facts and the minor dierences due to paritywe obtain a pro ce

dure for evaluating integral given arbitrary unit vectors w and v and any

integer n The optimized algorithm which requires On time is shown b elow

ector w v integer n real N v

real S Accumulates boundary integral along C

real d w v

d real c

real t if evenn then else c

real A if evenn then else cos d

integer i if evenn then else

while i n do

S S t

t t c i i

i i

endwhile

return t d A d S n

end

The three terms t d A and d S in the nal expression of the ab ove pro cedure

corresp ond to the rst second and third terms of equation resp ectively

Optimizations

We conclude this section on exact evaluation by describing some simple optimiza

tions to the pro cedures for computing moments and by showing how the exact

formulas can lead to ecient approximations with error b ounds

For clarity the pseudoco de in the previous section do es not depict a number

of simple optimizations For instance the powers in pro cedure CosSumIntegral

maybe computed incrementally by rep eated multiplication and no trigonometric

functions need be evaluated in the inner lo op Also in computing doubleaxis

moments a great deal of redundant computation ma y be avoided by allowing

pro cedure CosSumIntegral to return one additional term in the series as well as by

computing b oth endp oints simultaneously These optimizations do not change

the time complexityofthe algorithms but can signicantly reduce the constant

While there are generally b enets to obtaining closedform expressions they

are not always practical computationally An imp ortant means of sp eeding the

computation is to settle for an approximation To arrive at one such approximation

note that the terms in equation decrease in magnitude monotonically since

rapidly we jF j jF j for all k and c When the terms approach zero

k k

may therefore obtain an accurate approximation with little work Moreover by

b ounding the tail of the series it is p ossible to guarantee a given tolerance For

example to compute a doubleaxis moment to a relative accuracy of the lo op in

CosSumIntegral may b e terminated immediately up on up dating S if the condition

n k

v n c F c F

jS j

u v u n c

is met In this case the tail of the series and the nal integral in equation

may be dropp ed Early termination of the lo op is most useful with high orders

Because the test is costly it should not b e p erformed at every iteration of the lo op

v P -w v P

w r r

(a) (b)

Figure a Computing the irradiance at the point r due to a directional lyvarying

area light source P is equivalent to b computing a doubleaxis moment of a uniform

Lambertian source of the same shape

Applications to Image Synthesis

For ideal diuse surfaces irradiance is sucient to compute reected radiance The

situation is dramatically dierent for nondiuse surfaces however In the extreme

case of ideal sp ecular surfaces all features of the incident illumination app ear

again in the reection For surfaces that are neither ideal diuse nor ideal sp ecu

lar highorder moments can b e used to quantify additional features of the incident

a power series expansion For p olygonal environments with illumination much as

emission and reection distributions dened in terms of simple p olynomials the

pro cedures given in the previous section may b e applied to the simulation of direc

tional luminaires glossy reections and glossy transmissions These applications

are describ ed in the following sections

Directional Luminaires

Metho ds for simulating the illumination due to diuse area sources and di

rectional p oint sources are well known however directional area sources are

problematic for deterministic metho ds In this section we shall see how a class of

Figure Examples of directional luminaires At each point the irradiance is com

puted analytical ly from the area light source The moment orders are and

respectively

directional luminaires can be handled using doubleaxis moments

Let P b e a p olygonal luminaire whose emission distribution is spatially uniform

but varies directionally according to a Phong distribution that is as a cosine to a

power For instance the direction of maximum radiance may be normal to

the plane of the luminaire and fall o rapidly in other directions as shown by the

distribution in Figure a The irradiance at the p oint r is then given by

ju wj cos du

where P is the luminaire translated by r and is the angle of incidence of u

that is cos v u where v is the surface normal Observe that this computation

is equivalent to a doubleaxis moment of a Lamb ertian source P where the nth

moment is taken with resp ect to w and the factor of cos is accounted for bythe

second axis v See Figure b Therefore pro cedure DoubleAxisMoment can b e

used to compute the irradiance due to directional luminaires of this form exactly

Figure shows a simple scene illuminated by an area source with three dif

ferent directional distributions Note that the areas directly b eneath the luminaire

get brighter with higher orders increasing the contrast with the surrounding ar

eas which get darker Polygonal o cclusions are handled by clipping the luminaire

P v w u' u

Figure A simple BRDF dened by an axial moment around the mirror reection w

of u By reciprocity the radiance in the direction u due to P reduces to a doubleaxis

moment of P with respect to w and v

against all blo ckers and computing the contribution from each remaining p ortion

precisely as Nishita and Nakamae handled Lamb ertian sources

Glossy Reection

A similar strategy can be used for computing glossy reections of p olygonal Lam

b ertian luminaires Let r be a point on a reective surface Then the reected

radiance at r in the direction u due to luminaire P is given by

r u u u f r u cosdu f

where is the BRDF and is the angle of incidence of u Now consider a simple

BRDF dened in terms a Phong exp onent Let

T T

u u c u I vv u

where c is a constant and v is the surface normal Note that the Householder

matrix I vv p erforms a reection through the tangentplaneatr This BRDF

denes a cosine lob e ab out an axis in the direction of mirror reection as shown

in Figure Because ob eys the recipro city relation u u u u

the radiance reected in the direction u can be found by integrating over the

distribution shown in the gure To ob ey energy conservation the constant c must

be b ounded by n

Figure Analytical ly computed glossy reection of a convex polygon From left to

right the moment orders are and

Figure Analytical ly computed glossy reection of a stained glass window From

left to right moment orders are and

When the luminaire P is Lamb ertian the function f r is constant Therefore

the integral in equation reduces to a doubleaxis momentofP with resp ect

to the surface normal v and the vector

w I vv u

Pro cedure DoubleAxisMoment may therefore be used in this context as well to

compute the glossy reection of a diuse area light source The technique is demon

strated in Figure where the order of the moment is and in Figures

and which depict a sequence of moment orders to demonstrate surfaces with

varying nishes

More complex BRDFs may be formed by sup erp osing lob es of dierent orders

andor dierent axes Other eects such as anisotropic reection and sp ecular

reection near grazing can be simulated by allowing c and n to vary with the

incident direction u doing so do es not alter the moment computations however

recipro cityisgenerally violated

Glossy Transmission

As a nal example we note that glossy transmission can be handled in much the

same way as glossy reection the dierence is in the choice of the axes w and v

which must now exit from the far side of the transparent material Figure

shows a sequence of images depicting frosted glass with dierent nishes corre

sp onding to axial moments of dierent orders This eect was rst demonstrated

byWallace et al using a form of sto chastic sampling in Figure a similar

eect has b een computed analytically using pro cedure DoubleAxisMoment

Figure Afrosted glass simulation demonstrating glossy transmission From left to

right the moment orders are and

Further Generalizations

We conclude the chapter by describing two further generalizations of the metho ds

considered thus far We shall see that the two most fundamental constraints can

in fact be relaxed and in some instances still yield analytic solutions These

generalizations which are taken up in the following two sections are luminaires

with nonp olygonal geometry and luminaires with spatially varying brightness

Spherical Luminaires

Formulas and apply to arbitrary regions A S and therefore to all

luminaire geometries However when the luminaire is nonp olygonal the spherical

pro jection will generally not be b ounded by great arcs so we can no longer take

advantage of the simplications exploited earlier In particular the normal to the

boundary A will generally vary continuously and therefore cannot be removed

from the b oundary integral Nevertheless there are instances in which closedform

solutions can still b e obtained the simplest example b eing spherical luminaires

We shall consider the problem of computing the axial moment A w when

A S is the uno ccluded pro jection of a sphere As with p olygonal luminaires

the computation of A w entails a sequence of integrals of the form

w u w n ds

which follows from equation Since the b oundary A is a circle it is easily

radius of A is sin where is the halfangle of the circular parametrized The

cone dened by the luminaire To conveniently represent the path corresp onding

to A we intro duce a set of orthogonal vectors fa b cg where a points toward

the center of the spherical luminaire Then the dot pro ducts w u and w n can

be expressed parametrically as

u w a cos b cos c sin sin

w a sin b cos c sin cos v

where The function v can also b e expressed in terms of u as

w a

u cos v

sin

It follows that

Z Z

w u w n d sin u v d

Z Z

n n

w a u d cos u d

where the additional factor of sin on the right results from reparametrizing in

terms of arclength Thus the normal vector n is incorp orated into the integral

rather than factored out as in the case of p olygonal luminaires We next con

centrate on evaluating the integrals on the right of equation To simplify

notation we dene the following scalar quantities

a w a cos

b w b sin

c w c sin

With these denitions we may write

Z Z

n n

u d a b cos c sin d

d a h cos

b c and is the angle such that cos bh and sin ch where h

When the integral is taken over the range which is true for an uno ccluded

sphere we may set Integrals of this form may be evaluated using the

following lemma which generalizes recurrence relation

Lemma Let a and h be real numbers let n be an integer and dene

F a h cos d

Then F satises the recurrence relation

nF ha h cos sin an F n h a F

n n n

where n F and F

The result follows from integration by parts See App endix A Pro of

The complete algorithm for evaluating equation for a spherical luminaire

is quite similar to the corresp onding algorithm for p olygons The integral

h i

n n q

w u w u w u w n ds

maybe reduced by means of equations and to the expression

w a F F F cos F F F

n n q n n q

is dened as in lemma and q if n is even and q if n is o dd where F

The ab ove expression is analogous to expression Note that this expression

incorp orates all of the integrals F F F which are generated naturally bythe

recurrence relation in lemma and that F also app ears in the lefthand expression

when n is odd The following pseudoco de summarizes the algorithm

real AxialMoment sphere S vector w integer n

real halfangle of cone dened by S

if n then return cos Solid angle

real a normalized vector toward center of S

real a w a cos

real h w a sin

real F

real F a

real S if evenn then else Sum terms to n

real T Sum terms to n

for i n do

if eveni n

then T T F

else S S F

a F i F a i F i h

i i i

endfor

real A w a S cos T

if evenn then A A cos Add solid angle

return An

end

This pro cedure easily generalizes to spheres that are partially o ccluded by p oly

hedra or other spheres In such a case the outer contour of the visible p ortion of

a given sphere is a collection of lines and circular arcs

Spatially Varying Luminaires

This section describ es several fundamental observations ab out the problem of com

directional distributions are a function of puting moments of luminaires whose

positionwe refer to such luminaires as spatial ly varying Handling this typ e of lu

minaire is an imp ortant generalization with immediate applications to highorder

nite element metho ds for simulating global illumination We shall demonstrate

Y' X' Q h z w r u 0 y

Figure A polynomial over the plane may be represented as a rational polynomial

over the sphere

that the problem is equivalent to integrating a class of rational functions over

the sphere moments of spatially varying luminaires are interrelated via a gen

eralization of recurrence relation and even for p olygonal luminaires with

p olynomially varying brightness cases arise in which no solution in terms of ele

mentary functions is p ossible

We shall only consider luminaires with p olynomially varying radiant exitance

this typ e of variation can b e accommo dated by a simple generalization of irradiance

tensors In particular w e intro duce tensors whose elements are a restricted form

of rational p olynomials integrated over regions of the sphere

To see how rational p olynomials arise in this context consider a p olygonal lu

minaire P IR in a plane not containing the origin Supp ose that the radiant

exitance of P varies according to a p olynomial u v with resp ect to some orthog

onal co ordinate system X Y and origin Q in the plane To compute angular

moments of f the radiance distribution function at the origin we rst ex

direction cosines Let w S denote the vector press this function in terms of

orthogonal to the plane and let h denote the distance between the plane and the

origin as shown in Figure

Let r IR be an arbitrary point on the plane and let x y denote its lo cal

co ordinates with resp ect to the imp osed co ordinate frame That is

x r Q X

y r Q Y

To express these co ordinates in terms of direction cosines we employ the usual

notation u r jj r jj and note that h jj r jj u w Then

r u

u w

It follows that the p olynomial function dened on the surface of the luminaire may

be expressed as a rational p olynomial over the sphere Sp ecically

u X u Y

x y h t h t

x y

u w u w

where t Q X and t Q Y By expanding the ab ove expression and

x y

regrouping terms it follows that the radiance distribution function at the origin

maybe expresses as

f u x y z

where x y and z are the direction cosines of r and is the rational p olynomial

The form of is constrained however as only powers of u w app ear in the

denominator Tointegrate p olynomials of this form over regions A S suchasa

spherical p olygon weintro duce a tensor whose elements are integrals of appropriate

rational functions We dene

u u

d u T A w

u w

Then the integral of x y z over A and all of its moments can b e formed from

nq n

elements of the new tensors Note that T reduces to T when q We shall

show that the new tensors with rational elements satisfy a recurrence relation of

the form

nq nq nq

T G T T H A

nq nq

where all comp onents implicitly dep end on the axis w and the region A S

Equation is thus an extension of equation

Theorem Let n and q be integers with n q and let A S be a

measurable set Then the tensor T A w satises the recurrence relation

n u

nq nq

T q w T T ds

j I

I j

n q u w

where I is an n index ds denotes integration with respect to arclength and

n is the outward normal to the curve A

The pro of parallels that of equation See App endix A Pro of

By theorem we see that all highorder tensors T with nqcan b e reduced

to a sequence of rational boundary integrals and lowerorder tensors with n q

While this provides a means of reducing the order of these tensors two problems

remain computing the b oundary integrals and computing the base cases

We shall pursue the second point in the remainder of this section and exp ose one

of the diculties asso ciated with spatially varying luminaires

A complication that arises in dealing with spatially varying luminaires is that

cannot always be reduced to el integrals of rational p olynomials over the sphere

ementary functions even when the domain of integration is a spherical p olygon

in this resp ect the problem of spatially varying luminaires is intrinsically more

dicult than directional luminaires One form of integral that arises is

log sec d

where and which has no elementary solution in general

Figure shows the graph of for several values of For some parameters

or instance equation reduces immediately to known denite integrals F

0.2 0.4 0.6 0.8 1 1.2 1.4

Figure The function plotted as a function of over the interval

Here and starting from the bottom curve

when we may apply the formula

a b

log a cos b sin d log

due to Grobner and Hofreiter vol II pp and derived in a dierent

manner by Carlson to show that

log

For other parameters however evaluation of involves at least one sp ecial

function In particular in App endix A we show that integral can be

expressed either in terms of the dilogarithm with a complex argument or in terms

of the Clausen integral It is interesting to note that the dilogarithm also

app ears in computing the form factor between two arbitrary p olygons as shown

by Schroder and Hanrahan

We now show that evaluating integral is a prerequisite for solving the

the consequence of general problem of p olynomially varying luminaires and not

1 β α

o y

Figure The irradiance at the origin due to a triangular luminairewithquadratical ly

varying radiant exitance is given by a

a particular solution strategy To show this we reduce the problem of computing

x y to that of computing the irradiance due to a twoparameter family of

luminaires with spatially varying radiant exitance

Let denote the triangle in Figure which has two parameters the angle

and the edge length Let A denote the spherical pro jection of the triangle

and let denote the irradiance at the origin due to this luminaire

en by with radiance distribution giv

f r u r r

x y

Thus the radiance varies quadratically with p osition but is indep endent of direc

tion It follows that the irradiance at the origin due to this luminaire is

u cos u f

x y

z u

z z

The problem therefore reduces to integrating the rational function z over the

region A S where A dep ends on the parameters and To evaluate this

integral werstconvert it from an integral over solid angle to an integral over the

area of the luminaire By equation we have

cos r

Using this relationship and the fact that z cos in this conguration the integral

in equation can be written in Cartesian form as

Z Z

dx dy

x y

Finallychanging variables once again to p olar co ordinates this time the integral

reduces to after simplication We have

Z Z Z

sec

u rdrd

z r

sec

log r d

Therefore any metho d capable of computing irradiance due to p olygonal lumi

naires with p olynomially varying radiant exitance must also be capable of evalu

ating the function over its entire range of parameters

These results constitute a start toward obtaining closedform expressions for

moments of luminaires whose radiantexitancevaries p olynomially over the surface

However the meaning of closedform in this context must b e expanded to include

at least one sp ecial function such as Clausens integral or the function

Chapter

Comparison with Monte Carlo

Monte Carlo is a versatile simulation metho d that can b e applied to virtually any

problem of numerical approximation In this chapter we describ e several Monte

Carlo integration metho ds for computing irradiance tensors and axial moments

of all orders The results are compared with those generated by the algorithms

of chapter to provide indep endent evidence of their correctness This metho d

of validation is app ealing b ecause the Monte Carlo algorithms are comparatively

simple and rely on entirely dierent principles making the chance of a systematic

error remote

consists in converting a deterministic prob A typical Monte Carlo strategy

lem to a problem of statistical parameter estimation such as estimating the mean

value of a random variable The eectiveness of the metho d hinges on the costs

and statistical prop erties of the random variable so considerable eort is usually

warranted in devising random variables with the correct mean and low variance

Monte Carlo metho ds are particularly well suited to problems of multidimensional

integration largely b ecause of the close connection between integration and sta

tistical exp ectation Consequently Monte Carlo integration app ears throughout

computer graphics examples include estimating form factors visibility

and illumination from complex or partially o ccluded luminaires

We shall describ e two Monte Carlo strategies for estimating irradiance tensors

and doubleaxis moments one strategy is based on rejection sampling and the other

on stratied sampling The stratied sampling is p erformed using a new algorithm

for generating uniformly distributed random samples over spherical triangles which

has applications b eyond those explored in this chapter

Sampling by Rejection

The rejection method is the simplest and most general metho d for generating ran

dom samples according to a given distribution and is particularly useful in

dealing with irregular domains The idea is to embed the desired distribution in

a larger domain that can be sampled conveniently and ignore samples that land

outside the target domain The remaining samples are then distributed appropri

ately To obtain a Monte Carlo estimator of T A based on the rejection metho d

we observe that

T A u d u

d u

X u u u

where d denotes integration with resp ect to the measure weighted by

and X is the characteristic function of the set A dened by

if x A

X x

otherwise

The measure is p ositive and normalized to one since S which

makes it a probability measure Consequently equation may b e viewed as the

exp ected value of a random tensor variable The essence of Monte Carlo lies in

this shift of fo cus by which equation is given a probabilistic interpretation

To make this interpretation precise let denote a random vector in S dis

tributed in such away that

Prob

for any subset S Since the measure is invariant under rotations the

probability densityof is constantover the domain S wesaythat is uniformly

distributed over the sphere The relationship b etween and will b e denoted by

When the random variable is uniformly distributed equation implies that

D E

T A X

where h x i denotes the mean or expected value of the random variable x The

random variable X is called a primary estimator of T A and may

be used in constructing other estimators Obtaining a primary estimator by

expanding the domain of integration as we have done ab ov e is the basis of the

Monte Carlo rejection metho d Since the estimator is zero whenever A it is

only a crude approximation in itself esp ecially when A is small

To obtain a more desirable secondary estimator one with lower variance we

dene a new statistic by

A m X

i i i

where are identically distributed random vectors chosen uniformly

over the sphere S The new random variable is the sample mean for a sample of

size m which has the same exp ected value as the primary estimator Therefore

T Ah A m i

for any m By the law of large numb ers we have

A m T A lim

with probability whenever the variance is nite Thus the random vector

A misanunbiased estimator for T A the variance of whichiscontrolled by

the parameter m In practice we use A m with a xed m to estimate T A

n n

A m T A

Similarlywe may estimate the doubleaxis moment A w v directly by

X w v A w v m

i i i

are and rather than estimating the corresp onding tensor The estimators

easily computed provided that we have some means of generating the random

vectors

A wellknow metho d for simulating the random vector is based on the

following fact a horizontal section through the unit sphere corresp onding to

z a b has surface area b a which implies that sections of equal

height have equal area Consequently given two random variables and that

are uniformly distributed in the unit interval the threetuple

Normalize cos sin

is uniformly distributed over the sphere Here selects the z co ordinate uniformly

from and selects the rotation ab out the z axis uniformly from

Equation denes a transformation of the unit square onto the sphere

that preserves uniform distributions We construct a similar mapping for spherical

triangles in the following section

Stratied Sampling

One of the most common metho ds of improving the statistical eciency of a Monte

Carlo algorithm is stratied sampling within the computer graphics literature

this technique is also known as jitter sampling The idea behind stratied

sampling is to partition the region b eing sampled into disjoint subregions or strata

which are then sampled indep endently Used in conjunction with Monte Carlo

integration stratied sampling reduces the variance of the estimator when the

integrand varies slowly over p ortions of the domain p Hence Monte

Carlo integration of continuous functions will generally b enet from stratication

Let us rst see how to reformulate the previous estimators when there is only

one stratum which corresp onds to the entire region A S In this case we have

d u

T A A u u

A h i

with

where j is the measure restricted to A Thus is uniformly distributed over

the region A S This leads to the new secondary estimators

A m

i i

A w vm w v

i i

where are indep endent random vectors distributed uniformly over

A w vm which are based on m the set A The estimators A m and

strata are dened similarly

A m A

i i

A w vm A w v

i i

where A A A is a partition of the domain A and the random vector is

uniformly distributed over the ith subregion of A That is

A i

If the subregions are reasonably shap ed meaning convex with asp ect ratios near

one then the samples will b e more evenly spread over the domain This prop erty

signicantly reduces the clumping that inevitably o ccurs with uniform sampling

and generally reduces the variance of estimators based on the samples Of course

this metho d can only be applied if we have a means of generating samples from

and this requires sampling regions of the the given strata In the case of

sphere We now address this problem for the sp ecial case of spherical triangles

which can be applied to arbitrary spherical p olygons

An Algorithm for Sampling Spherical Triangles

The general problem of generating uniformly distributed samples over a given n

dimensional domain may be solved by constructing a onetoone mapping from a

rectangular domain onto the given domain that preserves uniform distributions

Such a mapping f X must have the following prop erty If S and S

olumes then f S and f S are subsets of are two subsets of with equal v

X with equal volumes Obtaining a mapping with this prop erty is generally di

cult b ecause it involves the inversion of one or more cumulative marginal density

functions a step that frequently entails numerical ro otnding

Given such a mapping however we may then generate the samples over the

simple domain and then transform them to the complex domain X In contrast

to rejection this metho d guarantees that all samples will fall within the desired

region Therefore one advantage of this approach is that no samples are wasted

A far greater advantage is that it leads to a simple means of stratied sampling

The mapping of the unit square onto an arbitrary spherical triangle may be

constructed using elementary spherical trigonometry Let T b e the spherical trian

gle with area A and vertices A B and C Let a bandc denote the edge lengths

of T and let and denote the three internal angles which are the dihe

dral angles between the planes containing the edges See Figure To generate

uniformly distributed samples over T we seek a bijection f T with the

prop erty describ ed ab ove The function f can be derived using standard Monte

Carlo metho ds for sampling bivariate functions as describ ed by Spanier and Gel

bard and Rubinstein To apply these metho ds to sampling spherical

triangles we require the following three identities

cos cos cos sin sin cos c

cos cos cos sin sin cos b

The rst is Girards formula whichwehave already encountered in chapter and

the other two are spherical cosine laws for angles In all there are six versions

of the spherical law of cosines three for angles and three for edges

To generate a sample from T we pro ceed in two stages In the rst stage we

b b

randomly select a subtriangle T T with an area A that is uniformly distributed

between and the original area A In the second stage we randomly select a p oint

along an edge of the new triangle Both stages require the inversion of a probability

distribution function

b b

The subtriangle T is formed by cho osing a new vertex C on the edge between

A and C and the sample p oint P is then chosen from the arc between B and C

as shown in Figure The point P is determined by nding its distance from

b b

B as well as the length of the new edge b the values b and are computed by

inverting the distribution functions

F b

cos

F j b

cos a

where equation is the conditional probability distribution of given b Note

that b oth A and a are taken to be functions of b in the equations ab ove Given

two random variables and uniformly distributed in we rst nd b by

c β β α a a A P γ γ b

C C

b b

Figure The vertex C is determined by specifying the area of the subtriangle AB C

and the point P is chosen from the arc between C and B

b b b

solving F b which is equivalent to Ab A Then b will corresp ond

to random subtriangles of T whose areas are uniformly distributed b etween and

b b

F j b A Having found b in this way the next step is to nd such that

Then will b e distributed along the edge BC with a density that increases toward

C more precisely the density will be prop ortional to cos d which is the

area of the isosceles triangle with dierential angle d and height

We now carry out these two steps explicitly To nd the edge length b that

pro duces a subtriangle of area A we use equations and to obtain

b b

cos cos cos

cos b

sin sin

b b

where A Equation completely determines b since b From

equations and it follows that

b b

u cos v sin

where

u cos cos

v sin sin cos c

P C

θ 0 C A

[C | B ]

Figure The point P is chosen using a coordinate system with vertex B dening the

vertical axis The height z of P is uniformly distributed between C B and

Solving equation for the sine and cosine of we have

v u

b b

p p

sin cos

u v u v

where the sign is determined by the constraint however the sign will b e

immaterial in what follows b ecause only ratios of these quantities will b e needed

Simplifying equation using the ab ove expressions we obtain

v cos u sin cos v

cos b

v sin u cos sin

b b

Note that cos b determines b which in turn determines the vertex C

As the nal step we select a point along the arc connecting C and B After

cho osing a convenient co ordinate system as shown in Figure a random point

is selected on the vertical Baxis with a height z uniformly distributed b etween

C B and The p oint P is then constructed by pro jecting horizontally to the

arc between C and B as shown in Figure These steps corresp ond to solving

for z cos using equation and the equation cos a C B The complete

algorithm is given in the next section

Implementation and Results

In this section we present the complete sampling algorithm as well as the com

putation required for setup First the pro cedure DeneTriangle computes all the

required quantities asso ciated with a spherical triangle from the three vertices

which are assumed to b e unit vectors

DeneTriangle vector A B C

Compute the internal angles

cos Normalize B A Normalize A C

cos Normalize C B Normalize B A

cos Normalize A C Normalize C B

Compute the edge lengths

a cos B C

b cos A C

c cos A B

the triangle Compute the area of

end

We shall assume that the values computed by this pro cedure are available to

the sampling algorithm so DeneTriangle is a required prepro cessing step To

succinctly express the sampling algorithm we shall let x j y denote the normalized

comp onent of the vector x that is orthogonal to the vector y That is

x j y Normalize x x yy

The algorithm for mapping the unit square onto the triangle T may then be

wovariables summarized as shown b elow The input to the pro cedure consists of t

and each in the unit interval and the output isapoint P T S

p oint SampleTriangle real real

Use one random variable to select the new area

A A

Save the sine and cosine of the angle

s sinA

t cosA

Compute the pair u v that determines

u t cos

v s sin cos c

Let q be the cosine of the new edge length b

v v t u s cos

v s u t sin

Compute the third vertex of the subtriangle

C q A q C j A

Use the other random variable to select cos

z C B

Construct the corresponding point on the sphere

z C j B P z B

return P

end

To generate uniformly distributed points over the triangle we supply and

that are indep endent random variables uniformly distributed over For

example these values might be supplied by a pseudorandom number generator

such as the algorithm prop osed by Haas

Note that cos sin cos c and C j A in pro cedure SampleTriangle need

only be computed once per triangle not once per sample Consequently the cost

of computing these values as well as invoking pro cedure DeneTriangle can be

amortized when many samples are to b e generated from the same triangle When

few samples are required from a given triangle the cost p er sample is much higher

Figure a A spherical triangle sampled randomly with the new algorithm b By

applying the transformation to stratied samples in the unit square we obtain stratied

samples over the triangle with much better statistical properties

Using pro cedure SampleTriangle which provides a mapping of the unit square

onto any given spherical triangle it is straightforward to p erform stratied sam

pling This can be done simply by stratifying the input arguments in

riangle is oneto the unit square Because the mapping implemented by SampleT

one the stratication is carried over to the triangle T The sampling algorithm

may also b e applied to spherical p olygons by decomp osing them into triangles and

p erforming stratied sampling on each comp onent indep endently This strategy

provides an eective means of sampling the solid angle subtended by a p olygon

and is analogous to the metho d prop osed by Turk for planar p olygons

Figure shows the results of the new algorithm applied in two dierentways

On the left the samples are uniformly distributed resulting in a pattern that is

equivalent to that obtained by a rejection metho d Each of the samples is guaran

teed to land within the triangle however which is not the case with rejection The

by rst stratifying the input arguments using pattern on the right was generated

a regular grid which dramatically improves the characteristics of the pattern

Verication of Assorted Formulas

Results generated by the closedform expressions describ ed in chapter were com

pared with the Monte Carlo estimators describ ed in this chapter In the rst test

analytically computed irradiance tensors of orders and were compared with the

corresp onding Monte Carlo estimates The tensors were computed at a sequence

of p oints along a line that was illuminated by two diuse quadrilateral lumi

naires The analytic results were compared with the Monte Carlo estimates in two

ways

Figure shows the maximum deviation among all the tensor elements for

the thorder tensor The results of b oth uniform and stratied sampling are

shown The vertical axis is the error in the matrix element with the largest absolute

deviation from the exact value and the horizontal axis corresp onds to p osition

The plot clearly shows the large reduction in variance from uniform sampling to

stratied sampling and also provides indep endent verication of the closedform

expression

In Figures and a single element was chosen from each tensor and com

pared to the corresp onding element of the Monte Carlo estimator Figure is

the graph of xy z integrated over the solid angle subtended by the luminaires and

stratied Figure is a plot of x y z over the same range of p ositions Only

sampling is shown

0.04 Stratified Uniform 0.03

0.02

0.01

-0.01

-0.02

-0.03

-0.04

0 20 40 60 80 100 120

Figure The signed maximum deviation of the Monte Carlo estimate of T A

from the analytic solution Both stratied and uniform sampling with samples are

compared

Analytic Stratified 0.1

0.05

-0.05

-0.1

-0.15

0 20 40 60 80 100 120

Figure The exact element T and a Monte Carlo estimate using samples

0.025 Analytic Stratified

0.02

0.015

0.01

0.005

0 20 40 60 80 100 120

Figure The exact element T and a Monte Carlo estimate using samples

As a second test the formulas for doubleaxis moments were veried using the

corresp onding Monte Carlo estimators The solid line in Figures and is the

graph of the thorder doubleaxis moment with resp ect to two p olygons as the

ma jor axis is rotated across them

Figure also shows the result of an estimator based on uniform sampling at

each of p ositions of the axis while Figure shows the result of stratied

sampling The same scenario is shown in Figures and where the moment

een the analytic order has been increased to The plots show agreement betw

solutions and the Monte Carlo estimators and also demonstrate the eectiveness

of stratied sampling in reducing variance

An advantage of unbiased Monte Carlo estimates is that the indep endent tests

taken together provide additional evidence of correctness This follows by observing

that the estimates are approximately evenly split among p ositive and negative

deviations from the analytic solution

Analytic Uniform 0.2

0.15

0.1

0.05

0 20 40 60 80 100 120 140 160 180 200

Figure Uniform sampling of a doubleaxis moment of order with samples

Analytic Stratified 0.2

0.15

0.1

0.05

0 20 40 60 80 100 120 140 160 180 200

Figure Stratied sampling of a doubleaxis moment of order with samples

0.02 Analytic Uniform

0.015

0.01

0.005

0 20 40 60 80 100 120 140 160 180 200

Figure Uniform sampling of a doubleaxis moment of order with samples

0.02 Analytic Stratified

0.015

0.01

0.005

0 20 40 60 80 100 120 140 160 180 200

Figure Stratied sampling of a doubleaxis moment of order with samples

Chapter

Op erators for Global Illumination

Thus far we have considered only direct illumination that is pro cesses relating

to light emission shadowing and rstorder reection In this chapter we con

sider the global il lumination problem which includes the simulation of b oth direct

and indirect comp onents of illumination and therefore accounts for interreections

among surfaces This problem has received much attention in the elds of radia

tive transfer illumination engineering and computer graphics Within the eld

of computer graphics global illumination encompasses all illumination problems

that include indirect lighting while the term radiosity refers to the sp ecial case of

environments with only Lamb ertian surfaces The latter terminology originated in

transfer where problems of precisely the same nature arise engineering heat

The principles of geometrical optics are appropriate for simulating global illu

mination b ecause the relevant interactions o ccur only at large scales and involve

incoherent sources of light thus eects due to interference and diraction are neg

ligible Physical or wave optics eects may enter as well but only at the level of

surface emission and scattering which is incorp orated into the b oundary conditions

of the global illumination problem

Because light transp ort is a ow of energy it is sub ject to thermo dynamic con

straints In particular the rst and second laws of thermo dynamics have a direct

b earing on the pro cess of light scattering These laws require energy to be con

served and imp ose a recipro city condition on surface reectance The recipro city

law for surface reection can b e demonstrated directly by means of a hyp othetical

conguration inside an isothermal enclosure p or deduced as a sp ecial

case of the general principle attributed to Helmholtz which applies to all forms of

electromagnetic energy propagation

The physical principles governing global illumination can be emb o died in a

single equation most commonly formulated as a linear integral equation In this

regard surfacebased radiation problems dier fundamentally from those of conduc

tion or convection which are most naturally p osed as dierential equations

The linearity of the equation is a consequence of the simplifying assumptions im

plicit in geometrical optics and discussed in chapter For instance we neglect

energy transfer between wavelength bands due to absorption and reemission at

surfaces This assumption is valid to extremely high accuracy for illumination

problems only at very high temp eratures do es reemission become signicant in

the visible part of the sp ectrum

Equivalent formulations of the governing integral equation have app eared in nu

merous contexts In illumination engineering Mo on p gave a restricted

of version of the equation in terms of luminosity the photometric counterpart

radiant exitance which holds for diuse environments In thermal engineering

Polyak posed the integral equation in terms of radiance and in a form that

holds for arbitrary surface reectances Within computer graphics Immel et al

presented an equation equivalenttoPolyaks and Ka jiya oered a new formu

lation known as the rendering equation expressed in terms of multipoint transport

quantities but otherwise equivalent to Polyaks version Ka jiyas formulation has

provided a theoretical foundation for all global illumination algorithms since its

intro duction in

In this chapter we reformulate the governing equation for global illumination in

r' n(r) u u' dω' θ θ'

φ' φ r Y

Figure The local coordinates of a bidirectional reectance function The incident

vector u and the reected vector u are in world coordinates The angles dene a local

coordinate system

terms of linear op erators whichwe then study in depth using standard techniques

of functional analysis The analysis naturally subsumes direct illumination as a

sp ecial case The new formulation claries the roles of various physical constraints

in particular those of the rst and second laws of thermo dynamics We identify

appropriate function spaces for surface radiance functions and show that the global

lie within the same space as the solution satisfying the op erator equation must

surface emission function Finallywe demonstrate that the new op erator equation

is equivalent to Ka jiyas rendering equation and give an alternate interpretation

of transp ort intensity which is the quantity he used to express the equation For

simplicitywe assume mono chromatic radiation throughout

Classical Formulation

Let M denote a piecewisesmo oth manifold in IR corresp onding to the surfaces

of an environment and assume for now that M forms an enclosure Let X denote

avector space of realvalued surface radiance functions dened on MS thatis

functions dened over all surface points and outward directions in the unit sphere

S Given a surface emission function f X which sp ecies the origin and

directional distribution of emitted light the global illumination problem consists

in determining the surface radiance function f X that satises the linear integral

equation

f r uf r u k r u u f r u cos d u

where is the hemisphere of incoming directions with resp ect to r M k is

a directional reectivity function is the polar angle of the incoming direction

vector u and r is a p oint on a distant surface determined by r and u See

Figure This equation is essentially the formulation p osed by Polyak

which embodies the same physical principles as Ka jiyas rendering equation

Equation is essentially a Fredholm integral equation of the second kind

precisely conform to the standard denition because of the however it do es not

implicit function r which dep ends on the argument r and the dummyvariable u

This seemingly minor dierence is resp onsible for the most imp ortant distinguish

ing feature of radiative transfer problems the nonlo calness of the interactions

This feature is crucial to retain although it is dicult to work with as it app ears

in equation One solution is to rephrase the integral in equation in

terms of direct transfers among surfaces which eliminates the implicit function

This was the approach taken by Ka jiya An alternative is to represent the inu

t surfaces using an explicit linear op erator The latter approachhas ence of distan

several advantages as demonstrated below

To precisely dene the function r app earing in equation we intro duce

two intermediate functions similar to those employed by Glassner First the

boundary distance function r u is dened by

r u inf fxr xu Mg

which is the distance from r to the nearest p oint on the surface M in the direction

of u When M do es not form an enclosure it is p ossible that no such p oint exists

in such a case r u by denition Next we dene the closely related ray

casting function p r u by

pr u r r uu

whichisthepointofintersection with the surface M when one exists the function

is undened if no such p oint exists Thus we have

r r u pr u

The kernel function k app earing in equation is the bidirectional reectance

distribution function BRDF at each surface p oint phrased in terms of worldspace

direction vectors rather than angles as it more commonly app ears That is

k r u u

where u and u denote incident and reected directions resp ectively and is the

conventional radiometric denition of a BRDF which is expressed in terms

of four angles relative to a lo cal co ordinate system at the point r as shown in

Figure Here and denote the polar and azimuth angles of the

and reected directions resp ectively By denition k is insensitive to the incident

sign of its vector arguments the angles formed with resp ect to the lo cal co ordinate

system are the same whether the vectors are directed toward or away from the

surface This convention is convenient for opaque surfaces where thereisnoneed

to disambiguate two distinct incident hemispheres

Note that the BRDF may be an arbitrary function of p osition on M cor

resp onding to changing materials or surface prop erties and that all scattering is

assumed to take place at the surface The latter assumption is an idealization since

the physical pro cess of reection mayinvolve some amount of subsurface scatter

ing Consequently the p oints of incidence and reection need not coincide

for real materials We shall ignore this eect here

The function k has two crucial prop erties that follow from the thermo dynamic

principles of energy conservation and recipro city By conservation we have

k r u u cos du

where u and is the outgoing hemisphere Both and implicitly

i o i o

dep end on the surface p oint r Equation states that the energy reected from

a surface cannot exceed that of the incidentbeam Recipro city states that

k r u uk r u u

for all u and u These facts play a ma jor role in the following analysis

i o

Note that the implicit function r in equation is the means of constructing

the distribution of energy impinging on a surface from the distribution of energy

leaving distant surfaces that is it constructs lo cal eld radiance from distant

surface radiance The connection aorded by r r u corresp onds to the intuitive

notion of tracing a ray from r in the direction u This simple coupling is a

consequence of steadystate radiance being invariant along rays in free space in

the presence of participating media the coupling is replaced by the equation of

transfer along the ray which turns equation into an integrodierential

equation

Equation which rst app eared in this general form in thermal engineering

is a continuous balance equation governing the direct exchange of mono chromatic

radiant energy among surfaces with arbitrary reectance characteristics In current

radiative heat transfer literature this governing equation is used only when par

ticipating media are ignored In current computer graphics literature this

equation and its equivalent formulations are the foundation for most physically

based rendering algorithms

G K

rrr

(a) (b) (c)

Figure The actions of G and K at a single point r The operator G converts a

distant surface radiance directed toward r into b local eld radiance where c it is

again mapped into surface radiance by K

Linear Op erator Formulation

Integral equations such as equation can be expressed more abstractly as

operator equations Op erator equations tend to b e more concise than their integral

linearity and counterparts while capturing essential algebraic prop erties such as

asso ciativity of comp osition as well as top ological prop erties such as b oundedness

or compactness The abstraction aorded by op erators is appropriate

when the emphasis is on integrals as transformations rather than on the numerical

asp ects of integration Op erator equations were rst applied to global illumination

by Ka jiya although their connection with integral equations in general has

been studied for nearly acentury

Equation can be expressed as an op erator equation in numerous ways

Here we shall construct the central op erator from two simpler ones eachmotivated

by fundamental radiometric concepts The representation given here has twonovel

features rst it cleanly separates notions of geometry and reection into distinct

linear op erators and second it employs an integral op erator with a cosineweighted

measure Both of these features simplify the subsequent analysis

We rst dene the local reection operator denoted by K whichisanintegral

op erator whose kernel k accounts for the scattering of incident radiant energy

at surfaces This op erator is most easily dened by sp ecifying its action on an

arbitrary eld radiance function h

Khr u k r u u hr u du

Equation is essentially the denition of a kernel operator with the minor

dierence that the p osition r acts as a parameter of the kernel consequently the

integration is over a prop er subset of the domain of h The K op erator maps

the eld radiance function h to the surface radiance function after one reection

Figures bc The op eration is local in that the transformation o ccurs at each

surface point in isolation of the others In equation the measuretheoretic

notation allows us to intro duce a new measure that incorp orates the cosine

weighting The measures and are related by

E ju nrj d u

which holds for any measurable E S henceforth the dep endence on the p osition

r will be implicit The new notation eliminates the proliferation of cosine factors

that app ear in radiative transfer computations and more imp ortantly emphasizes

that the cosine is an artifact of surface integration By asso ciating the cosine with

the integral op erator and not the kernel the function k retains the recipro city

prop erty of reectance functions Moreover this observation is vital in dening

appropriate norms and inner pro ducts for radiance functions

a linear Next we dene the eld radiance operator denoted by G which is

op erator expressing the incident eld radiance at eachpoint in terms of the surface

radiance of the surrounding environment Figures ab Showing the action of

G on a surface radiance function h we have

hpr u u when r u

Ghr u

otherwise

The G op erator expresses the transp ort of radiant energy from surface to surface

as a linear transformation on the space of surface radiance functions The symbol

chosen for this op erator is intended as a mnemonic for global or geometry

Dening G in this way allows us to factor out the implicit function r r u from

the integral in equation It is easily veried that KG is equivalent to the

original integral and that both K and G are linear Therefore we may write

equation as

f f KG f

which is a linear op erator equation of the second kind This formulation retains the

full generality of equation yet is more amenable to some typ es of analysis To

complete the denition of equation it is necessary to sp ecify the function space

X over which the op erators are dened We return to this p oint in section

To emphasize the linear relationship b etween surface radiance and surface emis

sion equation can b e written more concisely as

M f f

where the linear op erator M is dened by

M I KG

and I is the identity op erator The op erator M embodies all information ab out

the surface geometry and surface reectance However since G is determined by

manifold M a global illumination problem is completely sp ecied by the tuple

M Kf of surfaces lo cal reection op erator and surface emission function

In the following sections we explore fundamental prop erties of the op erators de

ned ab ove In chapter these prop erties will b e used to study numerical metho ds

for solving equation and identify sources of error

Normed Linear Spaces

In this section weintro duce the basic to ols of analysis that will b e used in deriving

error b ounds for approximate solutions for global illumination Quantifying error

requires a notion of distance b etween the memb ers of the space X which implies

a metric of some form We shall imp ose metric prop erties on X by making it a

normed linear space denoted by the ordered pair X jjjj where jj jj is a real

valued norm dened on X We summarize a few of the essential prop erties of

normed linear spaces that p ertain to radiative transfer More complete treatments

are given by Rudin and Kantorovic h for example

Function Norms

Innitely many norms can b e dened on a space of radiance functions each imp os

ing a dierent top ology on the space as well as adierent notion of size distance

and convergence Since equation requires only that radiance functions be

integrable with resp ect to the kernel k in order to be well dened it is natural to

consider the L norms and their corresp onding function spaces The appropriate

denition for the L norm of a radiance function is

Z Z

jj f jj j f r uj du dmr

M S

where m denotes area measure and p is a real number in Here f will

typically denote either a surface or eld radiance function which is zero over one

hemisphere Denition is also meaningful for p but the corresp onding

norms are not strictly convex and are normally excluded The function space

L MS m is then dened to b e the set of measurable functions over MS

with nite L norms Three of the L norms are of particular interest The L and

p p

L norm p ossesses L norms have immediate physical interpretations while the

algebraic prop erties that make it appropriate in some instances for example when

the structure of a Hilb ert space is required to dene pro jectionbased metho ds

In the limiting case of p the L norm reduces to

k f k ess sup ess sup j f r u j

rM uS

where ess sup is the essential supremum that is the least upp er b ound obtain

able by ignoring a subset of the domain with measure zero More precisely

if h is a real function dened on a set A then

ess sup hx inf f m j hx m for almost every x A g

Thus the L norm ignores isolated maximal p oin ts for example As the maximum

radiance attained or approached over all surface points and in all directions

k f k has the dimensions of radiance wattsm sr In contrast k f k is the total

power of the radiance function f and consequently has the dimensions of power

watts The L norm is related to vector irradiance by

k f k j r nr j dmr

where risthevector irradiance at r due to the the radiance distribution function

and nris the surface normal at r f r

To assign a meaning to the distance between two radiance functions f and f

in a normed linear space X we may use the induced metric jj f f jj Similarly

we may dene the distance between a function and a subspace Y X by

dist f Y inf jj f f jj

f Y

where inf denotes the greatest lower b ound When the subscript on the norm is

omitted it implies that the denition or relation holds for any choice of norm

Op erator Norms

Toinvestigate the eects of p erturbing the op erators as we shall do in the following

chapter we must also endow the linear space of op erators with a norm If X and

Y are normed linear spaces and A X Y is a linear op erator then the operator

norm of A is dened by

jj A jj sup fjjAh jj jj h jj g

where the norms app earing on the right are those asso ciated with Y and X resp ec

tively The op erator norm is said to be induced by the function norms Although

the theory of linear op erators closely parallels that of matrices there are imp ortant

dierences for instance matrix norms are necessarily nite while op erator norms

need not b e We therefore distinguish the class of bounded op erators as those with

nite norm

Equation implies that jj Ah jj jj A jj jj h jj for all h X Op erator

norms also satisfy jj AB jj jj A jj jj B jj whenever the comp osition AB is mean

ingful this prop erty makes op erator norms compatible with the multiplicative

structure of op erators Additional b ounds p ertaining to inverse op erators can

b e deduced from these basic prop erties For instance given a b ounded op erator A

with an inverse any op erator B suciently close to A is also invertible with

k A k

k k

jj A B jj k A k

This holds whenever jj A B jj k A k Inequality is known as Ba

nachs lemma p A useful corollary of this result is the inequality

jj A B jjkA k

k k

A B

k jjA B jjkA

which holds under the same conditions as inequality p

Norms of Sp ecial Op erators K G and M

Wenow compute b ounds for the op erators K G and M which will b e essential

in deriving all subsequent bounds From denitions and we

may deduce an explicit formula for k K k A straightforward computation yields

k K k ess sup ess sup k r u u d u

rM u i

This norm is the least upp er b ound on the directionalhemispherical reectivityof

any surface in the environment disregarding isolated points and directions and

other surfaces of measure zero Equation guarantees that k K k is b ounded

ab ove by one in any physically realizable environment If we disallow p erfect

reectors as well as sequences of materials approaching them and ignore the wave

optics eect of sp ecular reection near grazing the norm has a b ound strictly less

than one that is

k K k

When the b ound is less than one it is p ossible to use a Neumann series to derivea

number of related b ounds that would otherwise be dicult to obtain Therefore

the motivation for imp osing the ab ove restrictions is convenience of analysis rather

than physical constraints Note that when transparent surfaces are involved the

phenomenon of total internal reection must also b e ignored to maintain a b ound

that is less than one

The L norm of K can be computed in a similar fashion In precise analogy

with matrix norms the L norm merely exchanges the roles of the two directions

which are the arguments of the kernel

k r u u d u k K k ess sup ess sup

rM u

of both of these norms By the reci See App endix A for complete derivations

pro city relation shown in equation we have k K k k K k which implies

that reected radiance is everywhere diminished by at least as much as the total

power It is interesting to note that the L b ound on K implies that it is thermo

dynamically imp ossible for a passive optical system to increase the radiance of its

input this fact was previously proven by Milne using a very dierent argument

The two b ounds on K given ab ove can be extended to a bound on k K k for

all p The connection is provided by the following theorem

Theorem If Ais a kernel operator then k A k max fkA k k A k g

Pro of See App endix A

Although theorem applies to standard kernel op erators it can be extended

slightly to accommo date the lo cal reection op erator K which is kernel op erator

with an extra parameter We state the resulting b ound as a theorem

Theorem If K is a physical ly realizable local reection operator for an envi

ronment with directionalhemispherical reectance bounded away from then

k K k

for al l p where

Pro of See App endix A

The eld radiance op erator G is also b ounded but we shall arrive at its norm

in an entirely dierent manner To study various prop erties of this op erator and

to compute its norm we rst prove three elementary identities which are collected

in the following lemma

Lemma Let f and g be surface radiance functions and let f g denote their

pointwise product Then the eld radiance operator G satises

G f g Gf Gg

G f G f

G jf j j G f j

where p is a positive integer and jf j denotes the pointwise absolute value

Pro of To prove equation let r M and u S and supp ose that

r p r u is dened Then

G f g r u f g r u

f r u g r u

Gf r u Gg r u

Gf Gg r u

When pr u is undened all of the ab ove functions are zero so equality holds

trivially Equation follows by forming powers of f through rep eated appli

cation of equation Finally equation follows by observing that Gf

preserves every value assumed by the function f Changing the sign of f at any

point of its domain causes the corresp onding sign change at a single p oint in the

range of Gf

G can now b e obtained using the ab ove lemma and the principle The b ound on

that radiance remains constant along straight lines in free space which is applicable

here since we are assuming that there is no participating medium present The

bound follows immediately from the following theorem

Theorem If G is the eld radiance operator associated with the manifold M

then k Gf k kf k for al l p and for al l surface radiance functions f

p p

Furthermore when M forms an enclosure equality holds for al l f and when M

does not form an enclosure then either k Gf k k f k for some f or G

p p

Pro of Let us b egin by assuming that M forms an enclosure which is synonymous

with G I Then at any r M the complete radiance distribution function

over S is given by f r uf pr u u The net ux through the surface at r

enclosure M G

(a) (b)

Figure The net ux at the point r is a function of a the surface radiance and b

the eld radiance constructed using G

can then be obtained by integrating over S giving

f r uf pr u udu r nr

in Fig But f pr u u can be expressed in terms of the G op erator as shown

ure Integrating b oth sides over the entire surface M we have

Z Z Z

f r u Gf r udu dmr r nr dmr

M S M

But by Gausss theorem it follows that

Z Z

r nr dmr rr dv r

M V

where the nal equality is a result of equation which states that the lighteld

is solenoidal in empty space This holds b ecause we have excluded participating

media from the volume V enclosed by M Thus from equation we have

Z Z

f G f

for all surface radiance functions f where the integration is taken to b e over MS

and with resp ect to the measure m From equation and lemma we

have the sequence of equalities

Z Z Z Z

p p p p

jf j G jf j G jf j jGf j

which implies that k f k k Gf k for all f Thus G preserves all L norms

p p p

when M forms an enclosure Toshow that the result also holds for nonenclosures

we note that removing anypartofM simply reduces the supp ort of Gf which can

only decrease k Gf k Consequently k Gf k remains b ounded ab ove by k f k

p p p

Furthermore this b ound can always be attained by a function f that denes a

thin b eam from one surface to another When M is convex no such b eams exists

and G otherwise k Gf k k f k

p p

Theorem implies that k G k for all p except when M is

convex This is a weaker statement however which do es not subsume the theo

rem Given the L b ounds on K and G we may now b ound the op erator M

which maps surface emission functions to surface radiance functions at equilibrium

It follows from equation and theorem that M exists and can be ex

pressed as a Neumann series p Taking norms and summing the resulting

geometric series we have

k M k

n n n

for all p since k KG k k KG k As a direct consequence of

this b ound we have the following theorem

Theorem The space L M S m is closed under global il lumination

Pro of Supp ose that f M f where f L Because M is b ounded in all

L norms it follows that k f k kM k k f k Therefore the equilibrium

p p p p

solution f is also in L p

Related Op erators

In this section we show that K and G can also be used to dene and analyze

other op erators that arise in global illumination In particular we show that the

adjointofM with resp ect to the natural inner pro duct follows directly from basic

prop erties of K and G We also establish the exact relationship between these

op erators and a similar op erator decomp osition prop osed by Gershb ein et al

Adjoint Op erators

Adjoint op erators have b een studied in the context of global illumination for the

purp ose of computing solutions over p ortions of the environment such as the sur

faces that are visible from a giv en vantage point This application of the adjoint

transp ort op erator has been used in simulating diuse environments by Smits et

al and in nondiuse environments by Aupp erle et al The approach is

known as importancedriven global illumination

The problem of computing viewdep endent solutions for global illumination is

similar to the ux at a p oint problems that arise in reactor shielding calculations

where it is necessary to compute the ux of neutrons arriving at a small detec

tor Such problems are solved much more eciently using the adjoint transp ort

equation For global illumination the direct and adjoint solutions can be

used in conjunction by computing them simultaneously This approach allows

viewdep endent solutions to be computed far more eciently than either comp o

nent in isolation In this section we consider only the algebraic prop erties of

adjoint op erators and show how they relate to the op erators K and G

Because the adjoint of an op erator is most naturally dened with resp ect to

an inner pro duct we may pro ceed most easily in the setting of a Hilb ert space

Consequently we shall restrict our attention to the space L since neither the

norm nor the L norm can b e dened in terms of an inner pro duct The latter L

fact can be shown very simply by observing that the paral lelogram law

k f g k k f g k k f k k g k

holds for all f and g in a normed linear space if and only if the norm k k is

compatible with an inner pro duct However equation is violated by

both the L norm and the L norm for example this is so when f and g are

disjoint unit stepfunctions

Both K and G have a natural symmetry and they b ecome selfadjoint in L

if we imp ose a simple identication In particular if K and G are to be identical

with their Hilb ert adjoints K and G it is necessary that they be mappings of a

space into itself Therefore we must identify surface and eld radiance functions

so that they maybe viewed as the same space

To establish the required identication we note that the space of eld radiance

functions is naturally isomorphic to the space of surface radiance functions We

where denote the obvious isomorphism by H

Hf r u f r u

is the linear op erator that simply reverses the direction asso ciated with each radi

ance value From this denition it is clear that H maps surface radiance functions

to eld radiance functions and vise versa and that H H Using this isom

etry we may formally dene new versions of b oth K and G that have additional

symmetry Sp ecicallywe dene

K KH

G HG

isometric isomorphism with resp ect to Then T KG KG Note that H is an

all L norms that is k f k k Pf k for all pwhich implies that k K k k K k

p p p p p

for all p Using the new op erators the space of surface radiance functions can be

mapp ed onto itself without the intermediate space of eld radiance functions The

actions of the various op erators are depicted in the simple commutative diagram

shown in Figure By virtue of the isometry H it is clear that the distinction

between the two typ es of radiance functions is only sup ercial although it is an

aid to visualizing the physical pro cess In the context of Hilb ert adjoints however

wemust forgo this distinction so the op erators may b e dened on the same space

G Xs Xf

G T K

X X

s K s

Figure Operators connecting the spaces of surface and eld radiance functions

which are denoted by X and X respectively

We shall now show that both K and G are symmetric with resp ect to the

natural inner pro duct on the space L M S m which is given by

Z Z

r h f j g i f r u g r u du dm

M S

D E D E D E D E

More preciselywe shall show that Gf g f j Gg and Kf g f j Kg

or equivalently that K K and G G Op erators with this prop erty are said

to b e selfadjoint We state and prove the result as a theorem

Theorem The operators K and G are selfadjoint in the Hilbert space L

that is they are selfadjoint with respect to the inner product h j i dened in

equation

Pro of That K is selfadjointin L follows from the symmetry of the kernel and

Fubinis theorem

Z Z Z

D E

f j Kg f r u k r u u g r u du du dmr

o o

Z Z Z

g r u k r u u f r u du du dmr

o o

D E

Kf g

Therefore K K for any collection of surfaces M whose reectance functions

ob ey recipro city To obtain the corresp onding result for G it is convenient to b egin

by assuming that M forms an enclosure Using the fact that G I followed by

equations and we obtain the sequence of equalities

Z Z

D E

Gf g Gf g Gf G g

Z Z

D E

Gg f Gg f j Gg G f

Here the integration is once again taken to b e over M S as in equation

Hence G is selfadjointwhen M forms an enclosure Toshow that the result holds

for arbitrary M we rst express the space L as a direct sum of two orthogonal

subspaces generated by G In particular we dene the sets X and X by

n o

X f L G f f

o n

Gf X f L

It is easy to verify that X and X are subspaces of L such that L X X

p p

G G on X and GX X Moreover the functions of X and X have

disjoint supp orts so X X Therefore for all f g L we have

D E D E D E D E

Gf g Gf g f j Gg f j Gg

where f and g denote the X comp onents of f and g resp ectively From equa

tion we see that G is selfadjoint on all of L p

It follows immediately from the previous theorem and basic prop erties of adjoint

op erators that the adjointof M is

I GK M I KG

Therefore wehaveshown that the op erators K and G suce to form b oth M and

its adjoint This is another indication that partitioning of the integral op erator in

equation into the op erators K and G is an extremely natural one

Op erators for Diuse Environments

Gershb ein et al prop osed an op erator decomp osition similar to equation

but sp ecically tailored to diuse environments Their op erators were chosen to

exploit the very dierent behaviors exhibited b y irradiance functions and surface

reectance functions the former are nearly always smo oth with resp ect to p osition

whereas the latter mayvary rapidly due to highfrequency textures Weshowthat

these op erators are simply related to K and G

We b egin by employing the notation of this chapter to rephrase the two essential

c c

op erators used by Gershbein et al which we shall denote here by G and K The

op erator G is the irradiance operator which is dened by

Gbr bp r u du

where b M IR is a surface radiosity function wattsm and p is the ray

casting function dened on page The factor of results from the conversion

of radiosity to equivalent surface radiance on surrounding surfaces The op erator

K is the reectance operatorwhich is dened by

Kbr r br

y function that is r is the constantof where M IR is the surface reectivit

prop ortionalitybetween irradiance and reected radiosity at a p oint r on a diuse

c c

surface In analogy with equation the op erators K and G lead to a governing

radiosityG irradiance K radiosity

U U -1 A U U -1

surface G field K surface

radiance radiance radiance

b b

Figure The operators K and G play a role similar to K and G

equation for global illumination in diuse environments where the unknown is now

a radiosity function

c c

To illustrate the connections among K G K and G we dene op erators

that convert between the corresp onding radiometric quantities used in diuse and

nondiuse settings First we dene the local irradiance operator A by

Ahr hr u du

where h is a eld radiance function The op erator A converts eld radiance to

irradiance which is the rst moment ab out the surface normal at each p oint r M

Next to convert between radiosity functions and surface radiance functions we

dene a prolongation op erator U by

br Ubr u

The op erator U elevates a radiosity function to the equivalent radiance function

which is indep endent of direction Using the new op erators we may express G

as AGU in a diuse environment Similarly at diuse surfaces the kernel k sat

ises k r u u r In this case K simply averages over eld radiance

and pro duces a directionindep endent surface radiance function It follows that

U K KA where U denotes the restriction op erator that takes a direction

indep endent radiance function to the equivalent radiosity function We are lead to

the following simple relationship among the op erators

c c c

KG K AGU

h i

KA GU

U KGU

which holds only for entirely diuse environments The actions of all the op erators

are shown in Figure where the dashed arrows indicate transitions in which

information is lost except in the case of diuse environments

The Rendering Equation

The rendering equation formulated by Ka jiya is the form in which global

illumination problems are most frequently p osed we nowshow that the rendering

equation is equivalent to equation Toconveniently express the new equation

Ka jiya intro duced a number of multipoint transport quantities that dier from

standard radiometric quantities The rendering equation is

b b b b

k r r r f r r dmr f r r f r r g r r

where r r M and the nonstandard transp ort quantities are denoted by hats

the corresp onding terminology and physical units are summarized b elow

Quantity Name Physical Units

f r r transp ort intensity watts m

f r r transp ort emittance watts m

k r r r scattering function dimensionless

g r r geometry term m

The geometry term g in equation is dened by

jj r r jj if r and r are uno ccluded

g r r

otherwise

r''r r

θ''' θ u' θ'' θ' u

Figure The three points and four angles used in dening the multipoint transport

quantities appearing in the rendering equation

This function enco des p ointtop oint visibility and is closely related to the ray

casting function p dened earlier The most signicant dierence between equa

tion and equation is that the former is expressed exclusively in terms

of surface points whereas the latter uses points and solid angle The physical

interpretation of equation is that radiant energy ows from the p oints r on

ard the p oint r with some fraction reaching r after scat surrounding surfaces tow

tering See Figure The rendering equation can b e derived from equation

or equivalently from equation by a sequence of steps that include a change

of variable The derivation also leads naturally to the transp ort quantities

Derivation from the Classical Formulation

Because the rendering equation is expressed in terms of surface p oints instead of

directions it is convenient to intro duce the converse of the ray casting function

We dene the twopoint direction function by

r r

qr r

k r r k

r' M

u S2

Figure The form dened on the surface M is the pul lback of the solid angle

form The pul lback is dened in terms of the twopoint direction function the inverse

of the ray casting function

At a xed point r the functions q and p are inverses of one another Thus if we

dene M pr which is the subset of M visible to the point r then

qr M f qr r r Mg

Note that M is actually a function of r The twop oint direction function is the

key to deriving the rendering equation from the corresp onding classical version

f r uf r u k r u u f r u cos d u

where the variables app earing in this equation and in the following discussion are

angles will enter into the equation depicted in Figure Note that only p olar

as these relate to surface geometry and therefore to integration over surfaces All

dep endence on the azimuth angles on the other hand is hidden within the kernel

function k which may corresp ond to an anisotropic BRDF

Now let R r u denote the radiance at r in the direction u due only to reected

light That is

k r u u f r u cos d u R r u

qrM

where we have used equation to express the domain of integration in terms

of q this was done in anticipation of a change of variables which will express

the integral in terms of the surface M instead of solid angle This transition is

equivalenttochanging a form dened on the manifold S into a form dened on

the manifold M The formal mechanism for accomplishing this is the pul lback

dened by q which is denoted by q Since q M S the op eration q d

pulls d back to M where d is the solid angle form In particular we have

cos

q d dr

jj r r jj

where dr denotes the volume elementon M which denes the measure m by

dr mE

for all E M Equation is a wellknown formula for converting from solid

angle to surface integration Performing the change of variables we have

cos cos

R r u k r qr r u f r qr r dmr

jj r r jj

But the integral on the rightmaybe expressed in terms of the complete manifold

M byintro ducing the geometry term This step can b e viewed as constructing the

characteristic function for the set M in terms of the function g r which has

the added b enet of absorbing the factor of k r r k Thus we may write

R r u k r qr r u f r qr r g r r cos cos dmr

k r qr r u f r r dmr

where the last equality follows from the denition of the twop oint transp ort in

w Thus equation may b e written tensity function f which is given belo

f r u f r u k r qr r u f r r dmr

Since u is a free parameter it may be expressed in dierent terms without alter

ing the rest of the equation In particular we may obtain u from r and a new

parameter r M by means of the twop oint direction function Thus we have

f r qr r f r qr r k r qr r qr r f r r dmr

which now puts the equation in terms of r and r Finally to obtain the form of

equation we multiply both sides of the equation by g r rcos cos and

intro duce the multip oint transp ort quantities which are summarized b elow

Quantity Dening expression

f r r cos cos fr qr r g r r

f r r cos cos f r qr r

cos cos kr qr r qr r k r r r

These quantities corresp ond to the denitions given byKajiya but are phrased

here in terms of the twop oint direction function q Note that the multip oint

quantities could be dened more symmetrically if the emission term f were to

include a factor of g r r this would give both transp ort emission and transp ort

intensity the same physical units as well as the same behavior with resp ect to

o cclusion that is b oth would b e zero for o ccluded paths

MeasureTheoretic Denition of Transp ort Intensity

t transp ort intensity f r r is analogous to radiance f r u and can The twop oin

be dened by means of a measuretheoretic argument similar to that given in

chapter We givean outline of the argument here

Let A and B b e subsets of Mandletm denote Leb esgue measure over M We

then dene two natural measures over the pro duct manifold M M one that is

purely geometrical and one that characterizes the transfer of radiant energy from

M to itself in steady state To obtain twop oint transp ort intensitywe dene the

completed pro duct measure m m Let geometrical measure to be simply the

F denote the measure dened bytheow of radiant energy from M to itself that

is F is the p ositive set function that assigns to each set A B M M the

amountofpower leaving A and reaching B directly The p ositivity and additivity

of the set function F satisfy the axioms of a measure on MM The construction

of F is similar to that used for the measure E dened in chapter but here the

emphasis is on surfaces and the semantics of energy is present from the start

Since either mAormB implies that F A B the measures are

related by absolute continuity denoted by F m m This is the only connection

we require between the measures to infer the existence of a density function By

the RadonNiko dym theorem there exists a function f MM IR such that

Z Z

f r r dmr dmr F A B

A B

The function f is the twop oint transp ort intensity In view of equation

this quantity also corresp onds to the RadonNiko dym derivative of the measure F

with resp ect to the pro duct measure m m This relationship is denoted by

dm m

As in chapter the dimensional relationship follows from this yielding the dimen

sions of wattsm for twop oint transp ort intensity

Equation suggests that wemay also dene radiance directly from surface

measures To do so we pro ceed as ab ove but with a dierent geometric measure

over M M By replacing the pro duct measure m m with

Z Z

cos cos

F A B dmr dmr

A B

jj r r jj

where the angles corresp ond to those shown in Figure we obtain

where the density function f now corresp onds to radiance The quantity F A B

can b e interpreted as the numb er of lines passing through b oth A and B whichis

an imp ortant concept in the study of radiative transfer In particular the ratio

F A B

is the fraction of all lines passing through A that also meet B which is precisely

the form factor or conguration factor from surface A to surface B Thus

radiance is the RadonNiko dym derivative of radiantpower with resp ect line mea

sure which is closely related to the areatoarea form factors that app ear in both

radiative heat transfer and computer graphics

Chapter

Error Analysis for Global

Illumination

In this chapter we identify sources of error in global illumination algorithms and

derive b ounds for each distinct category Errors arise from three sources inac

curacies in the b oundary data discretization and computation Boundary data

consist of surface geometry reectance functions and emission functions all of

which may be p erturb ed by errors in measurement or simulation or by simpli

cations made for computational eciency Discretization error is intro duced by

a nitedimensional lin replacing the continuous radiative transfer equation with

ear system usually by means of b oundary elements and a corresp onding pro jection

metho d Finally computational errors p erturb the nitedimensional linear system

through roundo error and imprecise form factors inner pro ducts visibility etc

as well as by halting iterative solvers after a nite number of steps Using the

error taxonomyintro duced in this chapter we examine existing global illumination

algorithms

There are many practical questions concerning error in the context of global

illumination For instance in simulating a given physical environment p erhaps

under varied lighting conditions how accurately must the reectance functions

be measured Or when simulating radiant transfer among diuse surfaces how

imp ortant is it to use higherorder elements Can we exp ect higher accuracy by

using analytic areatoarea instead of p ointtoarea form factors Finallyhow ac

curate must visibility computations b e for global illumination While the analysis

presented in this chapter cannot provide denitive answers to these questions it

provides a formalism and a starting p oint for determining quantitative answers

We shall only consider a welldened class of global illumination problems

Sp ecically we address the problem of approximating solutions to a form of the

rendering equation given imprecise data for geometry reectance functions

and emission functions We further assume that the approximation is to b e assessed

quantitatively by its distance from the theoretical solution

Given these restrictions we derive error bounds in terms of p otentially known

quantities such as b ounds on emission and reectivity and bounds on measure

menterror Toboundtheerrorofanumerical solution using this typ e of informa

tion wedraw up on the general theory of integral equations aswell as the more

abstract theory of op erator equations

Pro jection Metho ds

By far the most common metho ds for solving global illumination problems are

those employing surface discretizations which are essentially boundary element

more abstract terms b oundary element metho ds are themselves metho ds In

projection methods whose role is to recast innitedimensional problems in nite

dimensions In this section we p ose the problem of numerical approximation for

global illumination in terms of pro jections This level of abstraction will allowus

to clearly identify and categorize all sources of error while avoiding the details of

sp ecic implementations

The idea b ehind b oundary element metho ds is to construct an approximate so

lution from a known nitedimensional subspace X X where the discretization n

parameter n typically denotes the dimension of the subspace For global illumi

nation the space X may consist of n b oundary elements over which the radiance

function is constant Alternativelyitmay consist of fewer b oundary elements but

with internal degrees of freedom such as tensor pro duct p olynomials spher

ical harmonics or wavelets In any case each element of the function

space X is a linear combination of a nite number of basis functions u u

n n

That is

X spanfu u g

n n

Given a set of basis functions we seek an approximation f from X that is close

n n

to the exact solution f in some sense By virtue of the nitedimensional space

nding f is equivalent to determining n unknown co ecients suchthat

n n

f u

n j j

There are many p ossible metho ds for selecting such an approximation from X

each motivated by a sp ecic notion of closeness and the computational require

ments of nding the approximation

approaches is that they op A universal feature of discrete b oundary element

erate using a nite amount of information gathered from the problem instance

For pro jection metho ds this is done in the following way We select f X by

n n

imp osing a nite number of conditions on the residual error which is dened by

r M f f

n n

Sp ecically we attempt to nd f such that r simultaneously satises n linear

n n

constraints Since we wish to make the residual small we set

i n

for i n where the X IR are linear functionals The functionals and

basis functions together dene a pro jection op erator as we show in section

Any collection of n linearly indep endent functionals denes an approximation f

by pinning down the residual error with suciently many constraints to uniquely

determine the co ecients However the choice of functionals has implications for

the quality of the approximation as well as the computation required to obtain it

Combining equations and we have

M u f

j j i

for i n which is a system of n equations for the unknown co ecients

By exploiting the linearity of and M we may express the ab ove

n i

equations in matrix form

M u Mu f

Mu M u f

n n n n n

As w e shall demonstrate below most global illumination algorithms describ ed in

recent literature are sp ecial cases of the formulation in equation Sp ecic

pro jectionbased algorithms are characterized by the following comp onents

A nite set of basis functions u u

A nite set of linear functionals

Algorithms for evaluating Mu for i j n

i j

Algorithms for solving the discrete linear system

These choices do not necessarily coincide with the sequential steps of an algorithm

Frequently information obtained during evaluation of the linear functionals or dur

ing the solution of the discrete linear system is used to alter the choice of basis

functions The essence of adaptive meshing lies in this form of feedback Regardless

which the steps are carried out the approximation generated ulti of the order in

mately rests up on sp ecic choices in each of the ab ove categories Consequently

we can determine conservative bounds on the accuracy of the nal solution by

studying the impact of these choices indep endently

A Taxonomy of Errors

In the previous section wecharacterized the fundamental features that distinguish

pro jectionbased global illumination algorithms In this section we intro duce a

higherlevel organization motivated by distinct categories of error this subsumes

the previous ideas and adds the notion of imprecise problem instances Assuming

that accuracy is measured by comparing with the exact solution to equation

all sources of error incurred by pro jection metho ds fall into one of three categories

Perturb ed Boundary Data

Both the operator M and the source term f may be inexact due to measur e

ment errors andor simplications made for eciency

Discretization Error

The nitedimensional space X may not include the exact solution In ad

dition satisfying the constraints may not select the best possible

approximation from X

Computational Errors

The matrix elements Mu may not be computed exactly thus perturbing

i j

the discrete linear system Final ly the perturbed linear system may not be

solved exactly

It is imp ortan t to note that the ab ove categories are mutually exclusive and ac

count for all typ es of errors incurred in solving equation with a pro jection

metho d The conceptual error taxonomy is shown schematically in Figure In

the remainder of this section we illustrate each of these categories of error with

examples from existing algorithms

Perturbed Boundary Data Space of Radiance Exact balance Functions X equation Emission/Reflection * f Geometry

Perturbed balance Discretization Error equation f Basis Functions

f n Discrete Projection linear system ~ Computational Error f Perturbations Approximate T Xn coefficients Inexact Solution

(a) (b) (c)

Figure a The conceptual stages for computing an approximate solution b The

exact solution at each stage is an approximation for the previous stage c Approxima

tions specic to each stage introduce new errors

Perturb ed Boundary Data

The idealized problems that we solve in practice are rarely as realistic as wewould

like As a rule we settle for solving near by problems for several reasons First

the data used as input may only be approximate For instance reectance and

emission functions obtained through simulation or empirically through mea

surement of actual materials or light sources are inherently contaminated

y error b

A second reason that b oundary data may b e p erturb ed is that use of the exact

data may be prohibitively exp ensive or even imp ossible Thus a surface may be

treated as a smo oth Lambertian reector for the purp ose of simulation although

the actual geometry and material exhibits directional scattering Regardless of the

source of the discrepancy the nearby problem can be viewed as a perturbation of

the original problem

Discretization Error

To make the problem of global illumination amenable to solution by a digital

computer we must recast the problem in terms of nitedimensional quantities

and nite pro cesses This transition is referred to as discretization In general

the discrete nitedimensional problem cannot entirely capture the b ehavior of

the innitedimensional problem and the discrepancy is called discretization er

ror This typ e of error is particularly dicult to analyze as it inherently involves

innitedimensional spaces and their relationship to nitedimensional subspaces

Consequently most global illumination algorithms rely up on heuristics rather than

error b ounds to p erform discretization tasks such as adaptive meshing

Wenowlookathowtwo asp ects of discretization have b een treated in various al

gorithms the choice of basis functions which determines the nitedimensional

space containing the approximation and pro jection the nite pro cess by which

the approximation is selected from this space

Basis Functions

For many global illumination algorithms the basis functions are completely deter

mined by the geometry of the b oundary elements This is true of any piecewise

constant approximation such as those employed by Goral et al Cohen et

al and Hanrahan et al Often a smo othing step is applied to piecewise

constant approximations for display purp oses although this do es not necessarily

improve the accuracy of the approximation When nonconstant elements are em

ployed such as tensor pro duct p olynomials or wavelets the degrees

of freedom of the elements add to the dimension of the approximating subspace

For nondiuse environments the nitedimensional space must also account

for the directional dep endence of surface radiance functions several avenues have

been explored for doing this Immel et al used sub divided cub es centered

at a nite number of surface p oints to simultaneously discretize directions and

p ositions Sillion et al used a truncated series of spherical harmonics to

capture directional dep endence and a quadrilateral mesh of surface elements for

the spatial dep endence As a third contrasting approach Aupp erle et al

used piecewiseconstant functions dened over pairs of patches to account for b oth

directional and spatial variations

The accuracy of the approximation is limited by the space of basis functions

in general the exact solution cannot be formed by a nite linear combination of

p olynomials or other basis functions The error can b e reduced either by expanding

the subspace X or by selecting basis functions that t the solution more closely

Discontinuity meshing is an example of the latter strategy

Pro jections

Given a set of basis functions the next conceptual step is to construct an ap

proximation of the exact solution from it We now demonstrate how the ma jor

pro jection metho ds follow from sp ecic choices of the linear functionals

describ ed in section The rst technique is col location which follows by

dening to b e an evaluation functional at the ith collo cation p oint that is

h hx

i i

where x x are distinct p oints in the domain of the radiance functions chosen

x Given these functionals the ij th element of the matrix on so that detu

j i

the left of equation has the form

u x KGu x

j i j i

Collo cation has b een widely used in global illumination because of the relative

simplicity of evaluating expressions of this form In the case of constant basis

functions over planar b oundary elements the resulting matrix has a unit diagonal

with p ointtoarea form factors o the diagonal which is a widely used radiosity

formulation In general metho ds based on a nite number of p ointtoarea

interactions are collo cation metho ds

A second technique is the Galerkin method which follows by dening to be

an inner pro duct functional with the ith basis function that is

h hu j hi

i i

where the inner pro duct h j i denotes the integral of the pro duct of two functions

The ij th element of the resulting linear system has the form

h u j u i hu j KGu i

i j i j

The Galerkin metho d was rst employed in global illumination by Goral et al

using a uniform mesh and constan t basis functions The use of higherorder basis

functions was investigated byHeckb ert and later by Zatz and Troutman

and Max For diuse environments with constant elements the second inner

pro duct in equation reduces to an areatoarea form factor

There are other p ossibilities for the linear functionals For instance

the inner pro ducts in equation may be taken with resp ect to a dierent set

of basis functions If we set

h hM u j hi

i i

we obtain the least squares metho d With the ab ove functional the solution to

residual error that is orthogonal to the the linear system in equation has a

space X which minimizes the residual error with resp ect to the L norm How

ever the matrix elements for the least squares metho d include terms of the form

h KG u j KG u i which are formidable to evaluate even with trivial basis func

i j

tions As a result there are currently no practical global illumination algo

rithms based on the least squares metho d

Computational Errors

Given a particular metho d of discretization error may b e incurred in constructing

the discrete nitedimensional linear system and once formulated wemayfailto

solveeven the discrete problem exactly These facts illustrate a third distinct class

of errors Because these errors arise from the practical limits of computational

pro cedures this class is called computational errors Computational errors

p erturb the discrete problem and then preclude exact solution of the p erturb ed

problem We now lo ok at examples of each of these

Perturbation of the Linear System

The most computationally exp ensive op eration of global illumination is the eval

uation of the matrix elements in equation this is true even of algorithms

that do not store an explicit matrix Furthermore only in very sp ecial cases

can the matrix elements be formed exactly as they entail visibility calculations

coupled with multiple integration Consequently the computed matrix is nearly

always p erturb ed by computational errors

A common example of an error in this category is imprecise form factors The

algorithm intro duced by Cohen and Greenberg for diuse environments com

puted form factors at discrete surface p oints by means of a hemicub e whichintro

duced a numb er of errors sp ecic to this approach Errors are also intro duced

when form factors are approximated through ray tracing or by using sim

These errors can be mitigated to some extent by the use of pler geometries

analytic form factors yet there remain many cases for which no analytic

expression is known particularly in the presence of o ccluders

For nonconstant basis functions the form factor computations are replaced by

more general inner pro ducts which require dierent approximations For instance

the matrix elements in the Galerkin approach of Zatz required fourfold inte

grals which were approximated using Gaussian quadrature Nondiuse environ

ments pose a similar diculty in that the matrix elements entail integration with

reectance functions

Another form of matrix p erturbation arises from simplications made for the

sake of eciency For example small entries may be set to zero or the entire

matrix may b e approximated by one with a more ecient structure such asablock

matrix or awavelet decomp osition

Inexact Solution of the Linear System

Once the discrete linear system is formed we must solve for the co ecients

This has been done in a number of ways including Gaussian elimi

nation GaussSeidel Southwell relaxation which is also known

as shooting and Jacobi iteration In any such metho d there will be some

error intro duced in the solution pro cess direct solvers like Gaussian elimination

are prone to roundo error whereas iterative solvers like GaussSeidel must halt

after a nite number of iterations

In approaches where the matrix is constructed in advance such as full matrix

radiosity or hierarchical radiosity the iterative solution can b e carried out

to essentially full convergence Other approaches such as progressive radiosity

construct matrix elements on the y and then discard them The cost of computing

convergence making this source of these elements generally precludes complete

error signicant

Error Bounds

With the results of the previous section we can obtain bounds for each category

of error However b ecause the metho ds employed here take account of very little

information ab out an environment the bounds tend to b e quite conservative We

now examine each source of error shown in Figure

Bounding Errors Due to Perturb ed Boundary Data

We solve global illumination problems using inexact or noisy data in hop es of

obtaining a solution that is close to that of the original problem But under

what circumstances is this a reasonable exp ectation To answer this question we

analyze the mapping from problem instances M Kf to solutions f We shall

assume that any input to a global illumination problem may be contaminated by

error and determine the impact of these errors on the solution W e shall show

that the problem of global illumination is wellp osed in all physically realizable

environments that is small p erturbations of the input data pro duce small

errors in the solution in physically meaningful problems

To b ound the eects of input data p erturbations we examine the quantity

jj f f jj where f is the solution to the exact or unp erturb ed system and f is

the solution to the p erturb ed system

M f f

where p erturb ed entities are denoted with tildes

Perturb ed Reectance and Emission Functions

e rst investigate the eect of p erturbing the reectance and emission functions W

The former is equivalent to p erturbing the lo cal reection op erator K Consider a

f f

p erturb ed op erator K and a p erturb ed emission function f such that

and

f f

Since jj G jj and G is assumed to be exact it follows that inequality

also applies to the corresp onding p erturb ed M op erator

f f f

jj G jj

K M K

G M KG K k

Intuitively the ab ove inequality holds b ecause the worstcase behavior over all

p ossible eld radiance functions is unaected by G which merely redistributes

radiance

To bound the error jj f f jj due to p erturbations in the reection and emis

sion functions according to the inequalities and we write

jj f f jj M f M f

f f

M M f M f f

From inequalities and we have

M M

Combining the ab ove and noting that f jj f jj by inequality we

arriveat the bound

jj f jj

g k g

jj f f jj

whichcontains terms accounting for p erturbations in the reectance and emission

functions individually as well as a secondorder term involving Note that the

k g

reectance term requires indicating that the problem may b ecome less

stable as the maximum reectivity approaches For highly reective environ

ments the results may b e arbitrarily bad if the input data are not corresp ondingly

accurate In general the worstcase absolute error in f dep ends up on the maxi

mum reectivityoftheenvironment the p erturbation of the reection functions

and the error in the emission function

k g

Perturb ed Surface Geometry

The eects of imprecise surface geometry are more dicult to analyze than those

due to imprecise reection or emission While the space of radiance functions has a

linear algebraic structure that underlies all of the analysis no analogous structure

exists on the set of p ossible surface geometries The analysis must therefore pro ceed

along dierent lines

One p ossible alternative is to study imprecise surface geometry indirectly by

means of the G op erator That is we can express the eect of surface p erturbations

on the eld radiance at each point as a p erturbation of G Given a p erturb ed

op erator G and a b ound on its distance from the exact G the same analysis used

for p erturb ed reectance functions can b e applied Such an approachmay b e useful

for analyzing schemes in which geometry is simplied to improve the eciency of

global illumination such as the algorithm prop osed by Rushmeier et al

However relating changes in geometry to bounds on the p erturbation of G is an

op en problem and solutions to this problem are likely to b e normdep endent

Bounding Discretization Error

In this section we study discretization errors intro duced by pro jection metho ds

Clearly the discretization error jj f f jj is bounded from below by dist f X

n n

the distance to the b est approximation attainable within the space X To obtain

an upp er bound we express equation using an explicit pro jection op erator

P r

n n

where represents the zero function and P is the pro jection op erator corre

with P P and sp onding to the subspace X That is P is a linear op erator

n n n

P h h for all h X Such a pro jection can be dened in terms of the basis

n n

functions u u and the linear functionals describ ed in section

n n

The form of P is particularly simple when

i j ij

which is commonly the case For instance this condition is met whenever there is

exactly one collo cation p oint within the supp ort of each basis function or when or

thogonal p olynomials are used in a Galerkinbased approach When equation

holds the pro jection op erator P is given by

P h hu

n i i

for any function h X It is easy to see that equation denes a

pro jection onto X and that P h if and only if h for i n

n n i

that is equation is a valid replacement for equation To pro duce the

desired b ound we write equation as

P M f f

n n

then isolate the quantity jj f f jj Adding I P f f to b oth sides of

n n n

equation and simplifying using the fact that I P f we arrive at

n n

I P KGf f I P f

n n n

When the op erator on the left of equation is invertible we obtain the b ound

jj f f jj I P KG jj f P f jj

n n n

A more meaningful b ound can b e obtained by simplifying b oth factors on the right

hand side of the ab ove equation Since I P KG is an approximation of the

op erator M let be such that

jj M I P KG jj

n P

which simplies to

jj K P K jj

n P

Because M is invertible so is I P KG when is suciently small Banachs

n P

lemma then provides the b ound

I P KG

The second norm on the right of inequality can b e simplied as follows Let

h X Noting that P h h we have

n n

jj f P f jj jj f hh P f jj

n n

jj f h jj jj P h f jj

jj P jj jj f h jj

Since h X was chosen arbitrarily the inequality holds for the greatest lower

bound over X which results in the b ound

jj f P f jj jj P jj distf X

n n n

From the ab ove inequalities we obtain the upp er and lower bounds

dist f X

dist f X jj f f jj jj P jj

n n n

where the constant is such that

jj K P K jj

n P

The b ounds in equation dep end on b oth the subspace X and the pro jection

metho d Note that if f is in the space X then dist f X and the upp er

n n

bound implies that f f Thus all pro jection metho ds nd the exact solution

when it is achievable with a linear combination of the given basis functions On

when dist f X is large then the lower bound implies that the the other hand

approximation will be poor even when all other steps are exact Unfortunately

this distance is dicult to estimate a priorias it dep ends on the actual solution

The dep endence on the typ e of pro jection app ears in the factor of jj P jj

and in the constant For all pro jections based on inner pro ducts jj P jj

P n

when the basis functions are orthogonal p For other metho ds such

as collo cation the norm of the pro jection may be greater than one p

The meaning of the constant is more subtle The norm in equation is a

measure of how well the pro jection P captures features of the reected radiance n

Bounding Computational Errors

The eects of computational errors can be estimated by treating them as p er

turbations of the discrete linear system The analysis therefore parallels that of

p erturb ed b oundary data although it is carried out in a nite dimensional space

We shall denote the linear system in equation by A b In general the ma

trix elements as well as the vector b will b e inexact due to errors or simplications

We denote the p erturb ed system and its solution by

A b

Although the exact matrix is unknown it is frequently p ossible to b ound the error

present in each element For instance this can be done for approximate form

factors and blo ck matrix appro ximations

Given elementbyelement error bounds the impact on the nal solution can

be b ounded From the element p erturbations we can nd and such that

A b

A A

and

b b

for some vector norm and the matrix norm it induces Also since the p erturb ed

matrix is known the norm of its inverse can be estimated

With the three b ounds describ ed ab ove essentially the same steps used in b ounding

the eects of p erturb ed b oundary data can be applied here and yield the bound

jj jj

b b

The form of this b ound is somewhat dierent for relative errors which are more

conveniently expressed in terms of condition numb ers p

Computing values for and that are reasonably tight is almost always

A b

dicult requiring error b ounds for each step in forming the matrix elements There

is no universal metho d by which this can be done each approach to estimating

visibility or computing inner pro ducts for example requires a sp ecialized analysis

In contrast the b ound in equation is more accessible as it is purely a

problem of linear algebra

Given that equation generally cannot be solved exactly the solution

pro cess is yet another source of error The result is an approximation of which

we denote by This last source of error is b ounded by

e f

e e

e e e

b A

When A and b are stored explicitly the ab ove expression can be used as the

stopping criterion for an iterative solver

To relate errors in the co ecients to errors in the resulting radiance

function consider the mapping T IR X where

Tx x u

j j

As a nitedimensional linear op erator T is necessarily b ounded its norm supplies

R and functions in X Observe that the connection between co ecients in I

f f T T

jj T jj

e e e

jj T jj

Equation relates the computational error present in the nal solution to

inequalities and The value of jj T jj will dep end on the basis functions

u u and the choice of norms for b oth IR and X which need not b e related

n n

The Combined Eect of Errors

In the previous sections we derived inequalities to b ound the errors intro duced into

the solution of a global illumination problem Using these inequalities we can now

b ound the distance b etween the exact solution and the computed solution By the

triangle inequalitywe have

e e

f f f f f f f f

n n

which is the numerical analogue of the chain of approximations shown in Figure

The terms on the right corresp ond to errors arising from p erturb ed b oundary data

discretization and computation sections and provide b ounds for

each of these errors The rst and third terms can be reduced in magnitude by

decreasing errors in the emission and reectance functions and in the computational

metho ds for forming and solving the linear system

Chapter

Conclusions

Summary

Chapter presented a measuretheoretic development of phase space density the

abstract counterpart of radiance The approach identied the physical and math

ematical principles up on which radiance is based and provided a simple pro of of

the constancy of radiance along rays in free space

Chapter presented a closedform expression for the irradiance Jacobian due

to p olygonal sources of uniform brightness in the presence of arbitrary p olygonal

blo ckers The expression can be evaluated in much the same way as Lamb erts

formula for irradiance When blo ckers are present a minor extension of standard

p olygon clipping is required Several applications that make use of the irradiance

Jacobian were demonstrated including the direct computation of isolux contours

and lo cal irradiance extrema b oth in the presence of p olygonal o ccluders

Chapter presented a number of new closedform expressions for computing

illumination from luminaires area light sources with directional distributions as

well as reections from and transmissions through surfaces with a wide range of

nondiuse nishes The expressions can b e evaluated eciently for arbitrary non

is related to the directionality of the convex polygons in Onk time where n

luminaire or glossiness of the surface and k is the numb er of edges in the p olygon

These are the rst such expressions available

The new expressions were derived using a prop osed generalization of radiance

called irradiance tensors These tensors were shown to satisfy a simple recurrence

relation that generalizes Lamberts wellknown formula expressions for axial mo

ments and doubleaxis moments of p olygonal luminaires were derived directly from

this recurrence relation The latter quantities have direct applications in simulat

ing nondiuse phenomena whose distributions are dened in terms of moments

The formulas give rise to ecient and easily implemented algorithms The new al

gorithms were veried by means of Monte Carlo in chapter where a new metho d

was derived for generating stratied samples over spherical triangles

Chapter intro duced a new op erator equation describing the transfer of

mono chromatic radiant energy among opaque surfaces The new formulation is

appropriate for global illumination and has several theoretical advantages over

previous formulations First it is based on standard radiometric concepts which

allows for the direct application of thermo dynamic constraints Secondly it is

wellsuited to the a priori error analysis presented in this thesis

Chapter identied three sources of error in global illumination algorithms

inaccuracies in the input data discretization and computational errors Input

noise in measurement or simulation or from simplications errors result from

Discretization errors result from restricting the space of p ossible approximations

and from the metho d of selecting the approximation Computational errors form

a large class that includes imprecise form factors and visibility as well as errors

intro duced by blo ck matrix approximations and iterative matrix metho ds To

pro duce a reliable solution each of these sources of error must be accounted for

Using standard metho ds of analysis we have derived worstcase bounds for each

based on prop erties of the new op erators category

Future Work

The techniques of chapter can be applied to other problems and extended For

instance the expression for the irradiance Jacobian can be applied to geometric

optimization problems involving illumination that is in selecting optimal shap e

parameters for luminaires and blo ckers with resp ect to some lighting ob jective

function The b enet of analytic Jacobians over nite dierence approximations

increases with the number of parameters Also the algorithms that incorp orate

irradiance gradients can be extended to apply to nondiuse luminaires by means

of the expression for double axis moments given in chapter

There are a number of natural extensions to the work presented in chapter

Given axial moments with resp ect to two or more axes all of arbitrary orders it

is p ossible to combine the eects demonstrated in the chapter For example it

would b e p ossible to compute glossy reections of directional luminaires Another

application would b e the simulation of nondiuse surfaces illuminated by skylight

using the p olynomial approximation of a skylight distribution prop osed by Nimro

et al According to equation all moments of this form admit closed

do not yet form expressions however practical algorithms for their evaluation

exist Another imp ortant extension is to accommo date more general p olynomial

functions over the sphere in particular p olynomial approximations of realistic

BRDFs obtained from theory or measurement

The analysis of chapter would be more practical if it were based on more

detailed information ab out environments constants such as maximum reectivity

are much to o coarse to obtain tight b ounds A deciency of the L function norms

employed in the chapter is that they do not adequately handle the wave optics eect

of sp ecular reection at grazing angles Other norms should b e explored including

those that are in some sense p erceptuallybased Finally reliable b ounds are needed

for a wide assortment of standard computations such as form factors betw een

partially o ccluded surfaces and inner pro ducts involving higherorder elements

App endix A

Additional Pro ofs and Derivations

A Pro of of theorem

To prove theorem on page we rst imp ose a parametrization on S bymeans

of the standard co ordinate charts

x sin cos

y sin sin

z cos

which map onto S Then d sin dd and

Z Z

k ij i j

cos sin d A cos sin d i j k

To simplify this expression we intro duce the function pi j dened by

i j

pi j cos sin d A

which is p ositive for all i and j and the function e i dened by

if i is even

A ei

otherwise

We now employ symmetries of sine and cosine to simplify the integrals on the

right hand side of equation A First b ecause sine is an odd function we have

sin sin It follows that

Z Z

i j i j

cos sin d cos sin d ej

Similarlycosineisodd so cos cos It follows that

Z Z

i j i j

cos sin d ei cos sin d

By means of these reductions b oth factors on the right hand side of equation A

maybe expressed in terms of the function pi j yielding

ei ej ek pi j pk i j A i j k

The right hand side of equation A is clearly zero if and only if at least one of

i j or k is odd

A Pro of of theorem

In this app endix we supply the pro of of theorem on page which gives a

closedform expression for i j k We rst state and prove two useful lemmas

Lemma If i and j are nonnegative integers then

pi j pi j A

i j

pi j pi j A

i j

where the function pi j is dened by equation A on page

Integrating pi jby parts we have Pro of

h i

i j

j cos i sin cos sin d pi j

h i

i j

i i j cos cos sin d

i pi j i j pi j

Thus i j pi ji pi j which veries equation A Equa

tion A follows in precisely the same manner

Lemma If i j and k are nonnegative integers and n i j k then

i j k A i jk

and the analogous result also holds for the second and third arguments

Pro of Equation A clearly holds when anyofi j ork are o dd as b oth sides

are then zero When i j and k are all even it follows from equation A that

i j k pi j pk i j A

From equation A and recurrence relations A and A we have

i jk pi j pk i j

i i j

pi j pk i j

i j i j k

i j k A

which proves the lemma

The pro of of theorem follows easily from lemma We start by deriving an

expression for i j k in terms of double factorials Assuming that i j and k are

all even integers rep eated application of recurrence relation A gives

i jk i j k

i i

n n j k

i i j j k k

n n n

i j k

which proves the rst part of the theorem The alternate expression for i j k

based on multinomial co ecients can now b e derived using the two identities

n n n

n n

which follow immediately from the denition of the double factorial Thus

n i j k i j k

n n n i j k

n i j k

n i j k n

i k j k i

where the primes indicate division by two This proves the second part of the

theorem

A Pro of of theorem

In this app endix we provide the pro of of theorem on page

Consider a single element of the ntensor corresp onding to a given multi

index Ii i and let i j k When n is odd it follows from

I I I

theorem that since at least one of i j or k is o dd this veries the rst

part of the theorem Now supp ose that n is even and consider the summation on

the right of equation If an yofi j ork are o dd then at least one function

in each term will have nonmatching indices making as required When

i j and k are all even the only terms in the sum that are nonzero are those

in which the p ermutation j j forms a sequence of matching pairs that is

j j for m n To count the number of times this o ccurs for

m m

the giv en index I observe that there are

j k i

distinct arrangements of the n n matching pairs and each of these arrange

ments o ccurs with amultiplicityof i j k within the set of all n p ermutations of

i i Therefore

i j k

j k i

i j k i k

i j k A

which proves the theorem

i j k

A Pro of that x y z d is closed

i j k

We show that the form x y z d is closed where x y and z are the direction

cosines and d is the solid angle form dened on page We shall show that

i j k

x y z

d d A

where r x y z and n i j k The computation is purely mechanical

By the denition of d we have

i j k i j k i j k i j k

x y z x y z dy dz x y z dz dx x y z dx dy

n n

r r

In computing the dierential of the expression on the right two thirds of the terms

vanish since the wedge pro duct of a form with itself is zero For example the

i j k

expression dx y z dy dz results in only one nonzero term the partial with

resp ect to x Performing all such simplications we have

i j k

i j k x y z

i j k

d x y z dx dy dz d

n n

r r

x y z

i j k

n x y z dx dy dz

Thus the initial form is closed for all exp onents i j and k

A Pro of of lemma

In this app endix we derive the recurrence relation presented as lemma on page

To b egin we write

Z Z

n n

F a b cos d aF b a b cos cos d

n n

Integrating the nal term by parts we have

n n

F aF b a b cos sin n b a b cos sin d

n n

Next we express b sin in terms of a b cos resulting in

b sin b b cos b b cos

b aa b cos b a a b cos

Substituting the ab ove into the expression for F we have

b a b cos sin F aF

n n

i h

n n n

d a b cos aa b cos b a a b cos n

aF b a b cos sin

h i

n F aF b a F

n n n

Simplifying we arriveat

nF an F n b a F a cos sin

n n n

with the base cases n and n following immediately from the denition of

F which proves the lemma n

A Pro of of theorem

In this app endix weprovide the pro of of theorem on page The steps of the

pro of are identical to those used in the pro of of theorem with a crucial dierence

app earing in the formulation of equation To account for the intro duction

of the w u expression in the denominator we dene

nq n

A A A

r w

where A is the nthorder tensor dened in equation We then pro ceed in

the same fashion as b efore but with the new ntensor A First we compute the

partials of A Dierentiating the pro duct we have

q q

r r

n n

A A A A

J m

J Jm

r r w r w

with where J is an nindex As in equation we compute the pro duct of

kml

A for clarity we p erform this step for each of the two terms on the right of

equation A individually Let I denote an n index Using the expression

for A given in equation we have

h i

r qr

n n

A r w r r w A

m m

kml kml

Ijl I jl

r r w r w

q r r

r r w r w

m m

pk jm pm jk

q nq

r w n r

q r r

r r w r w

k j j

q nq

r w n r

n n

r r r w

r q

j j

I I

q nq q nq

n r w r r w r r

Next we consider the second term on the right of equation A Multiplying by

and simplifying we have

kml

n n

q q

r r

i j i j

I I

A r

k kml

Ijlm

n n

r w r w r r

n n n

r r r n r r

i j i j j

I I I

q nq

r w n r r

Note that b oth of the nal expressions ab ove include a factor of r r this factor

will be used to complete the form corresp onding to d thereby converting the

surface integral to an integral over solid angle Thus we have

Z Z

nq nq

A dr A dr dr

l m l

Ijl Ijlm

A A

h i

nq nq nq nq

T w T T T

j I j

I j I Ij

n n

Simplifying and rearranging terms we obtain

u n

nq nq nq

q w T T ds T

j I j

I Ij

n q u w

which proves the theorem

A Reducing to Known Sp ecial Functions

We show that the sp ecial function a x dened on page can be expressed

in terms of the wellknown dilogarithm function denoted by Li z and dened

over the complex plane We shall make use of two intermediate functions

Lobachevskys function p dened by

logcos d A Lx

and atwoparameter generalization of this function denoted by M where

M x logcos cos d A

Both of these integrals can be expressed in terms of the dilogarithm function

Grobner and Hofreiter vol II pp provide the following essential

formulas

i i

Lx x log x A Li e

ix ix

xlog i A i Li e M x i Li e

where the parameter may be an arbitrary complex numb er Since a variety of

distinct denitions of the dilogarithm have app eared in the literature we

note that here Li z is given by the series

Li z A

for all complex z such that jz j An alternate denition of the dilogarithm is

given by the integral

log t

dt A Li z

which is equivalent to the series on the unit disk but extends the denition to the

entire complex plane Only elementary identities are needed to express a x

0.5

1 2 3 4 5 6

-0.5

-1

Figure A Clausens integral over the domain

in terms of the functions L and M ab ove The reduction pro ceeds as follows

log a x d

cos

Z Z

x x

d log cos d log cos a

Z Z

x x

cos a

d logcos d log

M i cosh a x x log Lx A

where the last step follows from the fact that cos ix cosh x for anyrealnumber

x A disadvantage of this formulation is that it involves dilogarithms with complex

arguments which requires atwoparameter sp ecial function p Further

more the arguments have mo dulus greater than one which precludes the use of

the power series in equation A An alternative is to express a x in terms

by Cl and dened by of the Clausen integral denoted

sin nx

Cl x log sin A d

Here it is the series representation that is dened over the entire complex plane

From the series it is clear that the function is a cyclic with Cl x n Cl x

and Cl xCl x A plot of the function over the domain is shown

in Figure A Numerous connections exist among Clausens integral the diloga

rithm and Lobachevskys function For instance from the integral representation

of Clausens function it is easy to see that

Cl x x log A Lx

The counterpart to equation A that we shall use in reducing a x is

F x log sin cos d A

In the mid th century F W Newman showed that this and many closely related

integrals can be reduced to Clausens integral One such identity derived by

Newman p Also see Lewin p is

F x x log sin log tan

Cl xCl x Cl A

where tan sin xtan cos x By means of this identity the reduction of

a x is straightforward

cos a

d x log Lx log a x

Lx F sin x x log

From the ab ove and equations A and A it follows that a x can be

expressed in terms of elementary functions and Clausens integral This represen

tation rests up on a single sp ecial function of a real variable over a nite range

A The L and L Norms of K

To compute the op erator norm jj K jj we begin by considering the function norm

jj Kf jj for an arbitrary function f L MS m The derivation pro ceeds

by manipulating the expression for jj Kf jj to pro duce a pro duct of jj f jj and

a new expression which will be the op erator norm jj K jj Since all quantities

are p ositive we shall drop the absolute value signs Let f denote a eld radiance

function and observe that

Z Z Z

jj Kf jj k r u u f r u du du dmr

Z Z Z

k r u u f r u du du dmr

Z Z Z

f r u k r u u du du dmr

where the rst equality follows from the denitions The second equality holds

by Fubinis theorem p which states that the order of integration can

be changed when the integral exists and is nite In the nal equality we have

isolated the kernel of the integral op erator K to the extent p ossible

intro duce the constant dened by Next we

ess sup ess sup k r u u du A

rM u

which is indep endent of the function f Since this quantity b ounds the eect of

the bracketed expression in the context of the outer double integral we have

Z Z

r jj Kf jj f r u du dm

jj f jj A

whichshows that is an upper bound on jj K jj Toshowthatitisalower b ound

as well we must show that this bound is either attained by some function f or

approached from below by a sequence of functions f f We can accomplish

the latter by a sequence of b eams that approach p erfect collimation ab out the

incident direction u in which

k r u u du A

is maximal If this maximum is not attainable eg the reectivitymay increase as

the incident b eam approaches grazing then we let the sequenct f f approach

p erfect collimation while simultaneously approaching grazing From this it follows

that is also a lower bound on jj K jj Therefore

jj K jj ess sup ess sup k r u u du A

rM u

The computation for the L norm pro ceeds similarly For any f L we have

jj Kf jj ess sup ess sup k r u u f r u du

rM u

jj f jj ess sup ess sup k r u u du

rM u

so the expression in brackets is an upp er b ound on jj K jj Again this b ound can

be approached from b elow by considering a sequence of increasingly collimated

b eams ab out a direction of maximal reectance Therefore we have

jj K jj ess sup ess sup k r u u du A

rM u

which diers from the L norm by exchanging the arguments of the kernel

A Pro of of theorem

In this app endix we provide the pro of of theorem on page Although the

pro of of this theorem is given by Kato for nitedimensional linear op erators p

and is left as an exercise by Dunford and Schwartz p the general

theorem for innitedimensional op erators do es not app ear to be widely known

For completeness a pro of which parallels that given by Kato is included b elow To

simplify notation the theorem is proved using functions of a single real variable

Let K be the integral op erator dened by

k x y uy K ux dy

where u and v are realvalued functions on some domain D is a p ositive measure

on a algebra over D and the kernel k DD IR is such that b oth jj K jj

and jj K jj are nite Letting v Ku we have

jv xj jk x y j

juy j dy

jj K jj jj K jj

juy j d y A

where is the p ositive measure dened by

jk x y j

E dy

jj K jj

for all measurable E D By the denition of jj K jj it follows that D

for all x D Now let p be xed and let denote the function x jxj

Since is convex and D wemay apply Jensens inequality p to

obtain

Z Z

b b

juy j d y uy d y A

x x

D D

for all x D By equation A we may rewrite equation A as

jv xj

juy j d y

jj K jj

for all x D Multiplying b oth sides by jj K jj and integrating we have

Z Z Z

p p p

jv xj dx jj K jj jk x y jjuy j dy dx

D D D

Z Z

p p

jj K jj jk x y j dx juy j dy

D D

p p

jj K jj jj K jj juy j dy

where the interchange of the integrals in the middle equality is justied by the

bounds on K and Fubinis theorem p Identifying the remaining inte

grals as L norms we have

p p p

jj v jj jj K jj jj K jj jj u jj A

p p

Raising b oth sides of equation A to the power p yields

p pp

jj u jj A jj K jj jj v jj jj K jj

p p

for all u Since v Ku equation A implies that

pp p

jj K jj jj K jj jj K jj A

by the denition of an op erator norm Finally for any A B and x

we have

x x

B max fA B g A max fA B g max fA B g

Thus from equation A it follows that

jj K jj max fjjK jj jj K jj g

for any p which completes the pro of

A Pro of of theorem

We shall prove theorem on page by extending theorem slightly to ac

commo date the form of the lo cal reection op erator K To b egin let K denote

the lo cal reection op erator restricted to the point r M That is

K hu k r u u hu du A

where h is a radiance distribution function at r Then K and K are related by

Kf r uK f r u A

From the known b ounds on K it follows that k K k k K k for almost

r r

every r M From theorem it then follows that k K k for almost every

r p

r M and for all p since K is a kernel op erator in standard form

function h is Because the L norm corresp onding to the restricted

jj h jj j hu j du A

we may express the L norm for radiance functions as

jj f jj dmr jj f r jj A

p p

The corresp onding b ound on K now follows easily by observing that

p p

dmr jj Kf jj jj K f r jj

p p

jj f r jj dmr

jj f jj

for all f Therefore jj K jj p

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