Radiometry, Radiosity

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Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 1 / 34 Global illumination, radiosity

 based on physics – energy transport (light transport) in simulated environment – first usage of radiosity in image synthesis: Cindy Goral (SIGGRAPH 1984)

➨ radiosity is able to compute diffuse light, secondary lighting, .. ➨ basic radiosity cannot do sharp reflections, mirrors, ..  time consuming computation – Radiosity: light propagation only, RT: rendering

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 2 / 34 Radiosity - examples

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 3 / 34 Basic radiometry I

Radiant flux, Radiant power d Q Φ = [ W ] d t

Number of photons (converted to energy) per time unit (100W bulb: ~1019 photons/s, eye pupil from a monitor: 1012 p/s)

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 4 / 34 Basic radiometry II

Irradiance, Radiant exitance, Radiosity d Φ( x) E (x) = [ W/m2 ] d A( x) Photon areal density (converted to energy) incident or radiated per time unit

dA dt dA

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 5 / 34 Basic radiometry III

Radiance d 2 Φ(x ,ω) 2 L( x ,ω) = ⊥ [ W/m /sr ] d Aω ( x) d σ(ω)

Number of photons (converted to energy) per time unit passing through a small area perpendicular to the direction w. Radiation is directed to a small cone around the direction w.

Radiance is a quantity defined as a density with respect to dA and with respect to solid angle ds(w).

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 6 / 34 Radiance I

➨ received/emitted radiance in direction : 2 – Lin() (Le(), Lout()) [ W/(m · sr) ] 2 N  d  L  x,   out dA d cos d dB   out d cos x dI dA  dA cos

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 7 / 34 Solid angles

dA  r2 sin d d d dA d   sin d d r  r2  d [] .. steradian (sr) the whole sphere .. 4 sr

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 8 / 34 Radiance II

Φ(x ,ω) ∝ d σ(ω)

sensor lens d σ(ω) x

ω

pixel

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 9 / 34 Radiance III

⊥ Φ(x ,ω) ∝ d Aω (x)

⊥ sensor d Aω x lens ω

 n dA  θ pixel ω ⊥ d Aω = d A⋅cosθ

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 10 / 34 Energy preservation law (ray / fiber)

L1() L1 d1 dA1  L2 d 2 dA2

r emitted received d1 dA1 power power

L2()

d 2 dA2

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 11 / 34 Energy preservation law (ray / fiber)

L1() L1 d1 dA1  L2 d 2 dA2

r T  d dA  d dA  d 1 1 2 2 dA 1 1 dA dA  1 2 r2 L2() ray capacity

L1  L2 d 2 dA2 ray ... radiance L

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 12 / 34 Light measurement

➨ measured quantity is proportional to radiance from visible scene

sensor: aperture: area A2

area A1

R    Lin A,   cos d dA  Lin  T A2 

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 13 / 34 BSDF (Local transfer function) („Bidirectional Scattering Distribution Function“, older term: BRDF)  L ( ) n i i

Lo(o) dswi  o i

d L (ω ) d L (ω ) f (ω →ω ) = o o = o o s i o d E (ω ) ⊥ i Li (ωi) cosθi d σ (ωi)

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 14 / 34 Helmholtz law (reciprocity) ➨ for real surfaces (physically plausible):

f in   out   f  out  in 

➨ general BSDF needs not be isotropic (invariant to rotation around surface normal) – metal surfaces polished in one direction, ..

f in,in,out,out   f in,in  ,out,out  

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 15 / 34 Local rendering equation

 L (x, ) n i i

Lo(x,o) ray-cast dswi

Ls(x,o)  xi vacuum:

Li(x,) = Lo(xM(x,w),-) Le(x,o) = L (y,-) x o

own emission at x

Lo( x ,ωo) = Le ( x ,ωo) + ⊥ +∫ Lo( y ,−ωi)⋅f s( x ,ωi →ωo)⋅d σ x (ωi)

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 16 / 34 Radiance received from a surface

N y dA y yo Nx  xi cosyo dA di  x  y 2 x

cos cos Geometric term: G y,x  yo xi x  y 2

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 17 / 34 Radiance received from a surface

Lo  x, o   integral over all incoming directions

 Le  x, o    f x,i   o   Li  x,i   cosxi di  

 Le  x, o    f x,i   o   Lo  y,i   G y,x dA S integral over an emitting surface

(assumption: the whole surface S is visible from x)

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 18 / 34 Reflected light

N z z N N y dA  x zi y yo  xo xi

x if y sees x

Terminology: L y,x  Lo  y,x  y  Li  x, y  x f y,x, z  f x,  y  x   z  x 

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 19 / 34 Indirect radiance equation

1 if y sees x V y,x  0 else

L x, z  Le  x, z   f y,x, z  L y,x  G y,x  V y,x dA S

BRDF geometric own (emitted) terms radiant exitance

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 20 / 34 Radiosity equation ➨ assumption – ideal diffuse (Lambertian) surface: – BRDF is not dependent on incoming/outgoing angles – outgoing radiance L(y,) independent on direction 

L x, z  Le  x, z  f x   L y,x  G y,x  V y,x dA S

L x, z  B x , Le  x, z  E x , f x   x 

G y,x  V y,x B x  E x   x  B y  dA   S

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 21 / 34 Discrete solution B x  E x   x   B y  g y,x dA S G y,x  V y,x where g y,x  

 solution B is infinit-dimensional ➨ discretization of the task: – Monte-Carlo ray-tracing (dependent on camera) – classical radosity (finite/boundary elements FEM)

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 22 / 34 General radiosity method

 object surfaces divided into set of elements

 definition of knot points on elements – radiosity will be computed there

 choice of an approximation method and error metric – basis functions for convex blend from knot points

 coefficients of linear equation system – “form-factors”

 solution of linear equation system – result: radiosity in knot points

 reconstruction of values on whole surfaces – linear blends using basis functions and knot point radiosities

 rendering of results (arbitrary view) – light is proportional to radiosity

 step  is performed in algorithm design phase – does not appear in an implementation

 some advanced methods do not strictly follow the sequence  to  – sometimes a computation flow goes back to some previous phase, some phases can be iterated,..

constant bilinear quadratic (knots in (knots in (more knots centers) vertices) in centers..)

➨ on every element Ai constant reflectivity is assumed , radiosity B – average of B(x):

– terminology: i, Bi for i = 1 .. N average over B x  E x   x   B y  g y,x dA area Ai S   1 N B  E     B g y,x dA  dA i i i A   j  j i i j1 Ai  Aj 

radiosity received in point x (lying on Ai)

switching sum and integral: N 1 Bi  Ei  i   Bj  g y,x dAj dAi Ai   j1 Ai Aj

geometric term – form factor Fij

(part of energy irradiated from Ai received directly by Aj)

N B  E    B F W i i i  j ij  m2  j1

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 28 / 34 Intuitive derivation N BiAi  EiAi  i   BjAj Fji  W j1 emitted power = own power + reflected power

reciprocal rule: Aj Fji  Ai Fij

N 1 BiAi  EiAi  i   Bj Fij Ai  Ai j1 N B  E    B F W i i i  j ij  m2  j1

Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 29 / 34 System of linear equations N Bi  i   Bj Fij  Ei i  1.. N j1

1 1F1,1 1F1,2 .. 1F1,N  B1  E1         2F2,1 1 2F2,2 .. 2F2,N  B2  E2          ......   ..   ..         NFN,1 NFN,2 .. 1 NFN,N BN EN

vector of unknown vars [Bi]

➨ for planar (convex) surfaces: Fii = 0 – the diagonal contain only unit values

➨ nondiagonal items are usually very small (abs value) – matrix is “diagonally dominant”  system is stable and can be solved by iterative methods (Jacobi, Gauss-Seidel)

➨ for light change (light sources) [Ei] system needs not to be fully re-computed, only reverse phase could be done

Even in constant element approach usage of some color interpolation method is recommended (Gouraud)

(B +B )/2 1 2 B2 B1

B B 2 1 ... B4

B3 B1 ... (B1+B2 +B3)/3

C. M. Goral, K. E. Torrance, D. P. Greenberg, B. Battaile: Modeling the Interaction of Light Between Diffuse Surfaces, CG vol 18(3), SIGGRAPH 1984 A. Glassner: Principles of Digital Image Synthesis, Morgan Kaufmann, 1995, 871-937 M. Cohen, J. Wallace: Radiosity and Realistic Image Synthesis, Academic Press, 1993, 13-64 J. Foley, A. van Dam, S. Feiner, J. Hughes: Computer Graphics, Principles and Practice, 793-804