Random-walk solution of the global illumination problem László Szirmay-Kalos

Expansion solution of the global illuminaton problem ’’ L(x,  ) x   LeLe R(    2 ) =1/S p f r (  ,x,  )cos  ’  f r (  ,y,   )cos  ’ 2 ’2’2  y M (  n L e )   R(    n )L e (    n ) d    d  n

Expansion  =  n   R(    n, x)L e (    n, x) d    d  n dx  1/M  i M S(   (i)  n (i), x (i) ) R(    n,x) = w 1 w 2 …w n w i = f r cos  ’     L e pixel

Why Monte-Carlo expansion? l Avoid the exponential core: –Monte-Carlo or quasi-Monte Carlo integration –Intuitive: R(    n,x)· L e (    n,x) integrand= probability · emission l Integrand is –L 2 : MC finite variance –Not of finite variation, but finite length of discontinuities: QMC

Importance Sampling:  =  M  n L e = =  n  R(    n )L e (    n ) d    d  n dp R(    n, p) = 1/S p w 1 w 2 …w n, w i =f r cos  ’  = M (L e +   L e +   L e + … ) )) = = M (L e +  w 1  L e +  w 2  L e +  L e … ) d   ) d   )

Iterative importance sampling Pr(  i )  w i  L e +  … ) – reflection prob x (direct lighting + indirect lighting) BRDF sampling: Pr(  i )  w i Direct lighting: Pr(  i )  L e Global imp. sampling: Pr(  i )  w i  L in estimate) Adaptive imp. sampling: find a process that converges to Pr(  i )  w i  L e +  … )

BRDF sampling: Pr(  in |  out )  w =f r (  in,  out )cos  ’ l Transformation of random variables: –1. use spherical coordinates: w(  in |  out )d  =w( ,  ’)sin  ’d  d  ’ –2. normalization: p( ,  ’) d  d  ’ = w d  /  f r (  in,  out )cos  ’d  scale: a (  out ) =  f r (  in,  out )cos  ’d  –3. decomposition: p( ,  ’) = p  (  ) p  ’ (  ’) –4. probability distributions: P  ’ (  ’)=   ’ p  ’ (  ) d , P  (  )=   p  (  ) d  –5. Generation of uniform random variables r 1, r 2. –6. Find ,  ’ sample by transforming r 1, r 2 by the inverse probability distributions l  ’= P  ’ -1 ( r 1 ),  = P  -1 ( r 2 )

Diffuse BRDF sampling Diffuse BRDF sampling 1. use spherical coordinates: w(  in |  out )d  = k d cos  ’sin  ’d  d  ’ 2. normalization: albedo= k d   p( ,  ’) = cos  ’sin  ’/  3. decomposition: p( ,  ’) = 1/2  · 2 cos  ’sin  ’= 1/2  · sin 2  ’ 4. probability distributions: P  (  )=  /2 , P  ’ (  ’)= sin 2  ’ 5. Generation of uniform random variables r 1, r Find ,  ’ samples  = 2  r 2,  ’= arcsin  r 1

Diffuse BRDF class class Diffuse { Color Kd; public: SColor BRDF(Vec& L, Vec& N, Vec& V) { return Kd; } double NextDir(Vec& L, Vec& N, Vec& V) { double theta = asin(sqrt( rand() )); double phi = M_PI * 2.0 * rand(); Vec O = N % Vec(0.0, 0.0, 1.0), P = N % O; // cheat !!! L = N*cos(theta)+O*sin(theta)*cos(phi)+P*sin(theta)*sin(phi); return (cos(theta)/M_PI); } };

Diffuse BRDF sampling: geometric approach Diffuse BRDF sampling: geometric approach 1. Generate a uniformly distributed point in a square 2. Reject and generate new if not in the base circle do { x = r 1, y = r 2 } while (x 2 + y 2 > 1) 3. Map to the sphere to find the direction z =  1 - x 2 - y 2 dAdA  dA/cos  rejected samples dd d  cos 

Specular BRDF sampling 1.use spherical coordinates: w(  in |  out ) d  = k s cos n  cos  ’ sin  d  d  2. compromise: p( ,  ’)  k s cos n  sin  3. normalization:  k s cos n  sin  d  d  2  k s /(n+1) p( ,  ) = (n+1)/2  cos n  sin  3. decomposition: p( ,  ) = 1/2  · (n+1) cos n  sin  4. probability distributions: P  (  )=  /2 , P  (  )= 1- cos n+1  5. Generation of uniform random variables r 1, r Find ,  ’ samples:  = 2  r 2,  = arccos r 1 1/(n+1) 7. Radiance is 0 if  ’( ,  degrees

Specular BRDF class class Phong { Color Ks; double shine; public: double NextDir(Vec& L, Vec& N, Vec& V) { u = rnd(), v = rnd(); double cos_V_R = pow(u, 1.0/(shine+1) ); double sin_V_R = sqrt(1.0-cos_V_R*cos_V_R ); Vec O = V % Vec(0.0, 0.0, 1.0), P = O % V; Vec R = O*sin_V_R*cos(2.0*M_PI*v) + P*sin_V_R * sin(2.0*M_PI*v) +V*cos_V_R; L = N * (N * R) * R; if (N * L < 0) return 0; return (shine+1)/2/M_PI * pow(cos_V_R, shine); } };

Ideal reflection sampling w(  in |  out ) is non-zero just for the reflection direction l Select the reflection direction with probability 1.

Infinite dimensional integrals: Russian roulette 1. MC integral of a bounce:  w i  L e +...) d  i = E[w i  L e +...) /p(  i )] = E[ L refl ] 2. Compute the reflected radiance with probability s, otherwise assume that it is 0 3. Compensate the error by dividing the radiance by s Expected value is still correct: E[ L refl* ] = s E[ L refl /s ] + (1-s) 0 = E[ L refl ] Variance is increased D 2 [ L refl* ] = s E[ (L refl /s) 2 ] + (1-s) 0 - E 2 [ L refl ]= (1/s - 1) E[(L refl ) 2 ] + D 2 [ L refl ]

Selection of the termination probability: albedo  BRDF  importance sampling:  w i  L e +...) d  i = E[w i (  i,  i-1 )/p(  i )  L e +...) ] = E[a(  i-1 )  L e +...) ] 2. Continuation with the probability of the albedo: if rnd() < a(  i-1 ) return  L e +... else return 0

Russian roulette with QMC Using less number of samples if the integrand is smaller:  w L in d  ’ Adding an indicator (0/Const) function C(  ’, r): w =  C(  ’, r) dr l Substitution : –  w L in d  ’ =  r   ’  C(  ’, r) L in d  ’ dr

Combined BRDF sampling 1.BRDF is a sum of elementary BRDFs: f r = f 1 + f 2 + … + f n, w = w 1 + w 2 + … + w n 2.  w  L in d  =  w 1  L in d   w n  L in d  3. Evaluate this sum randomly: probability of terms: s1, s2, … sn, probability of no term: (1- s1- s2 - …- sn) if term i is selected it is divided by si Expected value: s1  w 1  L in d  s sn  w n  L in d  sn +(1-..)0=  w  L in d 

Low variance estimator l si is the albedo of the BRDF –1. Select reflection model: l probabilities: a1, a2, … an, l stop with (1-a1-a2-…an) –2. Generate the direction using the reflection model –3. L refl  L in from the selected direction or 0 if stopped  w  L in d  w1  L in d   wn  L in d   w1/p1  L in ]+...+  wn/pn  L in ]   a1  L in ]+...+  an  L in ]

Combined BRDF class class CombMat : Diffuse, Phong, Reflector, Refractor { enum {NO, DIFF, PHONG, REFLECT, REFRACT} selected; public: double SelectModel(Vec& L, Vec& N, Vec& V); double NextDir(Vec& L, Vec& N, Vec& V, BOOL out); Color BRDF(Vec& L, Vec& N, Vec& V); };

SelectModel double CombMat:: SelectModel(Vec& L, Vec& N, Vec& V) { double akd =Diffuse::Albedo(N,V), aks =Phong::Albedo(N,V), akr = Reflector::Albedo(N,V), akt = Refractor::Albedo(N,V); double r = rnd(); if ((r -= akd) < 0) { selected = DIFF; return akd; } if ((r -= aks) < 0) { selected = PHONG; return aks; } if ((r -= akr) < 0) { selected = REFLECT; return akr; } if ((r -= akt) < 0) { selected = REFRACT; return akt; } selected = NO; return 0.0; // Russian roulette }

CombMat::Reflect double CombMat::NextDir(Vec& L, Vec& N, Vec& V, BOOL out) { switch (selected) { case DIFFUSE: return Diffuse :: NextDir(L, N, V); case PHONG: return Phong :: NextDir(L, N, V); case REFLECT: return Reflector :: NextDir(L, N, V); case REFRACT: return Refractor :: NextDir(L, N, V, out); }

CombMat::BRDF Color CombMat :: BRDF(Vec& L, Vec& N, Vec& V) { switch (selected) { case DIFF: return DiffuseMaterial :: BRDF(L, N, V); case PHONG: return SpecularMaterial :: BRDF(L, N, V); case REFLECT: return (N*L>0): kr() / (N*L): Color(0); case REFRACT: return (-N*L>0): kt() / (-N*L): Color(0); }

Requirements of BRDF models l Local illumination and ray-tracing: –Calculation of the L out radiance from L in l Random walk global illumination: –Generation of a random  ’ : –Computation of the albedo l diffuse, ideal reflection: yes l Phong, Ward, Schlick, ideal refraction: almost l Cook-Torrance, He-Torrance: no

Colored scenes Rendering or potential equation should be solved on wavelengths: 1, 2,…, m –Solving on each wavelength separately: l waste of visibility computation –Solving on all wavelength simultaneosly l scalar importance should be defined on the radiance vector Luminance: e.g.: kd eff =  kd  i  weigth( i )

Global importance sampling l Two-pass methods (estimation+random walk) l Preprocessing: photon shooting l Data structure about the radiance (4 5D-table) –5D adaptive tree –photon map –links –wavelets l On the fly importance generation from the data structure

Photon shooting + Importance generation   2 4 Photon shooting Looking the photon impact from a patch Dicretising the hemisphere ,, Dicrete pdf 2/6 4/6 ,, 2/6 1 Dicrete CDF Random variable

Adaptive importance sampling l Single pass methods l Use previous walks to guide the importance of the future walks –explicit storage of importance: VEGAS sampling: approximate high-D importance functions by the product of 1D functions –implicit: use a process that converges to the desired probability density: Metropolis

Metropolis sampling zizi ztzt f(zi)f(zi) z i+1 f(zt)f(zt) acceptance probability: f(z t ) / f(z i ) T Adaptation is guided by a Markov chain

Design of a Metropolis l “Arbitrary” mutation function T(z i  z t ) l Construct the acceptance a(z i  z t ) so that the limiting probabilities: p(z)  f(z) z1z1 z2z2 z3z3 p(z 1 ) T(z 1  z 2 ) a(z 1  z 2 ) a(z 1  z 2 ) a(z 2  z 1 ) f (z 2 ) T(z 2  z 1 ) f (z 1 ) T(z 1  z 2 ) = Detailed balance

Metropolis algorithm FOR i=1 TO M DO Using z i choose another random, tentative point z t a(z i  z t ) = (f (z t ) T(z t  z i )) /((f (z i ) T(z i  z t )) IF a(z i  z t ) > 1 THEN accept z i+1 = z t ELSE // accept with probability a(z i  z t ) Generate random number r in [0,1] IF r < a(z i  z t ) THEN z i+1 = z t ELSE z i+1 = z i ENDIF ENDFOR

Convergence of the probability density z0z0 f(z)f(z) p1p1 p2p2 pp

Definition of a Metropolis algorithm l Perturbation (mutation) strategy: –how to find a new tentative sample in the neightborhood of the actual samples. –Req: any state can be reached l Benefits: adaptive, exact importance sampling l Drawbacks: adaptation phase: start-up bias

Metropolis method for the rendering equation L e Mutations: perturbing the directions, adding and deleting steps pixel