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Analytic Methods for Simulated Light Transport

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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 of Yale University in Candidacy for the Degree of Do ctor of Philosophy by James Richard Arvo December c Copyright James Richard Arvo All Rights Reserved 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 ix 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 x 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
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