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Random-walk solution of the global illumination problem

Random-walk solution of the global illumination problem. L ászló Szirmay-Kalos. Expansion solution of the global illuminaton problem. M (  n L e ) = ... R ( w 1 ,...,w n ) L e ( w 1 ,...,w n ) d w 1 ,..., d w n. y. w 2. L (x, w ). ’ 1. ’ 2. w 1. w. x. L e.

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Random-walk solution of the global illumination problem

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

  2. Expansion solution of the global illuminaton problem M ( nLe )=...R(w1,...,wn)Le(w1,...,wn) dw1,...,dwn y w2 L(x,w) ’1 ’2 w1 w x Le R(w1,w2) =1/Spfr (w1,x,)cos’1 fr (w2,y,1)cos’2

  3. pixel w 1 e w L 2 Expansion R(w1,...,wn,x) = w1 w2 …wn wi= fr cos ’ F= n...R(w1,...,wn,x)Le(w1,...,wn,x) dw1,...,dwndx  1/MiMS(w1(i),...,wn(i),x(i) )

  4. Why Monte-Carlo expansion? • Avoid the exponential core: • Monte-Carlo or quasi-Monte Carlo integration • Intuitive: R(w1,...,wn,x) · Le(w1,...,wn,x) integrand= probability · emission • Integrand is • L2 : MC finite variance • Not of finite variation, but finite length of discontinuities: QMC

  5. Importance Sampling: F=M nLe= = n...R(w1,...,wn)Le(w1,...,wn) dw1,...,dwndp R(w1,...,wn,p) = 1/Spw1 w2 …wn, wi=fr cos’ F= M (Le +(Le +(Le + …) ))= = M (Le+w1(Le + w2(Le + Le …) dw2 ) dw1)

  6. Iterative importance sampling • Pr(wi)wi (Le +…) • reflection prob x (direct lighting + indirect lighting) • BRDF sampling: Pr(wi)wi • Direct lighting: Pr(wi)Le • Global imp. sampling: Pr(wi)wi (Lin estimate) • Adaptive imp. sampling: find a process that converges to Pr(wi)wi (Le +…)

  7. BRDF sampling: Pr(win |wout) w =fr (win,wout)cos’ • Transformation of random variables: • 1. use spherical coordinates: w(win |wout)dw=w(,’)sin’d d’ • 2. normalization: p(,’) d d’ = w dw / fr (win,wout)cos’dw • scale: a (wout) =  fr (win,wout)cos’dw • 3. decomposition: p(,’) = p() p’(’) • 4. probability distributions: • P’(’)=’ p’() d, P()= p () d • 5. Generation of uniform random variables r1, r2. • 6. Find ,’ sample by transforming r1, r2 by the inverse probability distributions • ’= P’-1(r1),  = P-1(r2)

  8. Diffuse BRDF sampling • 1. use spherical coordinates: • w(win |wout)dw= kd cos’sin ’d d’ • 2. normalization: albedo= kd   • p(,’) = cos’sin ’/ • 3. decomposition: • p(,’) = 1/2 · 2 cos’sin ’= 1/2  · sin 2’ • 4. probability distributions: • P()=/2, P’(’)=sin2 ’ • 5. Generation of uniform random variables r1, r2. • 6. Find ,’ samples •  =2r2, ’= arcsin r1

  9. 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); } };

  10. dA/cos  dA rejected samples Diffuse BRDF sampling: geometric approach dw dwcos • 1. Generate a uniformly distributed point in a square • 2. Reject and generate new if not in the base circle • do { x = r1, y = r2 } while (x2+ y2 > 1) • 3. Map to the sphere to find the direction • z =  1 - x2- y2

  11. Specular BRDF sampling • 1.use spherical coordinates: • w(win |wout) dw= ks cosny cos’ siny d dy • 2. compromise: p(,’)ks cosny siny • 3. normalization:  kscosny siny d dy =2ks /(n+1) • p(,y) = (n+1)/2 cosny siny • 3. decomposition: p(,y)= 1/2 ·(n+1)cosny siny • 4. probability distributions: • P()=/2, Py(y)= 1- cosn+1y • 5. Generation of uniform random variables r1, r2. • 6. Find ,’ samples: •  =2r2, y= arccos r11/(n+1) • 7. Radiance is 0 if’(, y) > 90 degrees

  12. 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) * 2.0 - R; if (N * L < 0) return 0; return (shine+1)/2/M_PI * pow(cos_V_R, shine); } };

  13. Ideal reflection sampling • w(win |wout) is non-zero just for the reflection direction • Select the reflection direction with probability 1.

  14. Infinite dimensional integrals: Russian roulette • 1. MC integral of a bounce: • wi(Le +...) dwi = E[wi(Le +...) /p(wi)]= E[ Lrefl ] • 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[ Lrefl* ] = s E[ Lrefl/s ] + (1-s) 0 = E[ Lrefl ] • Variance is increased • D2[ Lrefl* ] = s E[ (Lrefl/s)2 ] + (1-s) 0 - E2[ Lrefl ]= • (1/s - 1) E[(Lrefl)2 ] + D2[ Lrefl ]

  15. Selection of the termination probability: albedo • 1. BRDFimportance sampling: • wi(Le +...) dwi = E[wi (wi, wi-1)/p(wi) (Le +...) ]= • E[a(wi-1) (Le +...) ] • 2. Continuation with the probability of the albedo: • if rnd() < a(wi-1) • return Le +... • else return 0

  16. Russian roulette with QMC • Using less number of samples if the integrand is smaller: w Lin dw’ • Adding an indicator (0/Const) function C(w’,r): w = C(w’,r)dr • Substitution : • w Lindw’ = r w’C(w’, r)Lin dw’ dr

  17. Combined BRDF sampling • 1.BRDF is a sum of elementary BRDFs: • fr = f1 + f2 + … + fn, w = w1+ w2+ … + wn • 2.  wLindw=  w1Lindw + ... +  wnLindw • 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  w1Lindw/ s1 + ... + sn  wnLindw/ sn +(1-..)0= •  wLindw

  18. Low variance estimator • si is the albedo of the BRDF • 1. Select reflection model: • probabilities: a1, a2, … an, • stop with (1-a1-a2-…an) • 2. Generate the direction using the reflection model • 3. Lrefl = Lin from the selected direction or 0 if stopped  wLindw =  w1Lindw+...+ wnLindw= E[w1/p1Lin ]+...+ E[wn/pnLin ] = E[a1Lin ]+...+ E[anLin ]

  19. 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); };

  20. 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 }

  21. 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); } }

  22. 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); } }

  23. Requirements of BRDF models • Local illumination and ray-tracing: • Calculation of the Lout radiance from Lin • Random walk global illumination: • Generation of a random w’ : • Computation of the albedo • diffuse, ideal reflection: yes • Phong, Ward, Schlick, ideal refraction: almost • Cook-Torrance, He-Torrance: no

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

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

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

  27. Adaptive importance sampling • Single pass methods • 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

  28. Metropolis sampling • Adaptation is guided by a Markov chain f(zi) f(zt) T zi zt acceptance probability: f(zt) / f(zi) zi+1

  29. Design of a Metropolis • “Arbitrary” mutation function T(zizt ) • Construct the acceptance a(zizt ) so that the limiting probabilities: p(z)  f(z) p(z1) T(z1z2 ) a(z1z2 ) a(z1z2 ) a(z2z1 ) f(z2) T(z2z1) f(z1) T(z1z2 ) z2 = z1 Detailed balance z3

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

  31. Convergence of the probability density p1 f(z) p2 p z0

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

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

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