MONTE-CARLO PATH TRACING

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1 CS580: Computer Graphics KAIST School of Computing Chapter 4 & 5 MOTE-CARLO PATH TRACIG 2

2 Shadow Rays by Hemisphere Sampling Directions Ψ i are generated over the hemisphere Ω, after which the nearest intersection point r(, Ψ i is found. If r(, Ψ i belongs to a light source, a contribution to the direct illumination estimator is recorded. 3 Shadow Rays by Global Area Sampling Since the self-emitted radiance equals 0 over surfaces that are not a light source, it is possible to write the integral as an integration over all surfaces in the scene. L direct ( Θ = Asources!"!!"! L e (y y fr (,Θ y G(, yv(, yday, 4 2

Environment Map Illumination Galileo s Tomb http://www.pauldebevec.

surface point due to an environment map: ò Ldirect ( Q = Lmap ( Y fr (, Q«Y

W Self-shadow could occlude the incoming light toward : ò Ldirect ( Q =

3 Environment Map Illumination Galileo s Tomb 5 Sampling Environment Maps The direct illumination of a surface point due to an environment map: ò Ldirect ( Q = Lmap ( Y fr (, Q«Y cos( Y, dw Y. W Self-shadow could occlude the incoming light toward : ò Ldirect ( Q = Lmap ( Y fr (, Q«Y V (, Y cos ( Y, dw Y. W An estimator using Monte Carlo integration: Lmap ( Yi fr (, Q«Yi V(, Yi cos( Yi, Ldirect ( Q = å = p( Y i i 6 3

4 Environment Map Lighting Issues: The integration domain of the direct illumination equation has a large etent, usually increasing variance. Tetured light source. The incoming radiance contains high frequencies or discontinuities, increasing variance and stochastic noise in the final image. Product of environment map and BRDF. The product of the high-frequency illumination and glossy and specular BRDF will contain very sharp peaks. 7 Importance Sampling for Direct Illumination Direct illumination map can be sampled using importance sampling, e.g., computing the cumulative distribution function and subsequently inverting it, storing it as a 2D LUT. BDRF sampling can be conducted after the sample directions have been chosen Sampling the product of both the illumination map and the BRDF can be conducted, e.g., Wavelet importance sampling. 8 4

5 Chapter 4 & 5 IDIRECT ILLUMIATIO 9 Direct + Indirect Illumination Let s see reflection in the rendering equation L ( Q = L( Q + L ( Q ; ò e L ( Q = L ( Y f (, Q«Ycos( Y, dw, r W r ò r L ( Q = Lr ((, Y -Y f (, Q«Ycos( Y, dw. r W r Rewrite L(r(,Ψ Ψ as a sum of self-emitted (direct and reflected (indirect radiance r(,ψ at : L( Q = L( r (, Y -Y f(, Q «Ycos( Y, dw + r W e r ò W L( r (, Y -Y f(, Q «Ycos( Y, dw = L ( Q + L ( Q direct ò r r indirect 0 Y Y Y Y 5

6 Indirect Illumination The indirect illumination contribution to is epressed as: L( Θ ò Lindire ct ( Q = Lr ((, ry -Y fr (, Q«Ycos( Y, dw Y. W Estimator for indirect illumination: L indirect Lr ((, ry i -Yi fr(, Q«Yicos( Yi, ( Q = å. = p( Y i We need to evaluate the reflected radiance L r (r(,ψ i Ψ i at the closest intersection point r(,ψ i i Indirect Illumination Russian roulette can be used for stopping the recursive evaluation. local hemispherical reflectance is used as an appropriate absorption probability. 2 6

7 Indirect Illumination Pseudo code: // direct illumination from a single light source // for a surface point, direction theta indirectillumination (, theta estimateradiance = 0; if (no absorption for all indirect paths sample direction psi on hemisphere; y = trace(, psi; estimatedradiance += computeradiance(y, -psi * BRDF * cos(, psi / pdf(psi; estimatedradiance = estimatedradiance / #paths; return(estimatedradiance/(-absorption; computeradiance(,dir estimatedradiance = Le(, dir; estimatedradiance += directillumination(, dir; estimatedradiance += indirectillumination(, dir; return(transfer; 3 Importance Sampling for Indirect Illum. Simplest choice for p(ψ is a uniform PDF: p(ψ = 2π However, uniform sampling over the hemisphere does not take into account any knowledge, we might have about the integrand in the indirect illumination integral. The cosine factor: cos(ψ i, The BRDF: f r (,Θ Ψ i The incident radiance field: L r (r(,ψ i Ψ i A combination of any of the above 4 7

8 Recap: Cosine sampling A cosine weighting factor arises in the rendering equation; therefore, it is often useful to sample the hemisphere to compute radiance using a cosine PDF. The hemisphere can be sampled such that the samples are weighted by the cosine term. The PDF is cos( q p( qf, = p 5 Recap: Cosine sampling Its CDF is computed as: F = cos qdw ; p ò f q F( qf, = cos q'sin q' dq' df' p òò 0 0 f q = df' cos q'sin q' dq' p ò0 ò0 f 2 q = (-cos q '/ 2 p 0 f 2 = (- cos q 2p 6 8

by u 2, since the uniform random variables lie in the domain [0,.

9 Recap: Cosine sampling The CDF, with respect to f and q functions, is separable: f 2 Ff =, Fq = - cos q p Therefore, we compute two uniformly distributed samples u and u 2 : - fi = 2 pu, qi = cos u2 Where -u is replaced by u 2, since the uniform random variables lie in the domain [0,. These f i and q i values are distributed, according to the cosine PDF. 7 Recap: Phong vs. Blinn-Phong BRDF Phong model uses V and R, instead of and H f r = ρ π + k (V Rα s 8 9

otation mapping Original Phong model (in the previous slides f r = ρ π + k (V Rα s θ Modified Phong BRDF (in the net slides: f r (,Θ Ψ = k d + k s cos n (Ψ,Θ s Cos angles are same: V, R = Ψ,Θ s Θ s θ

10 otation mapping Original Phong model (in the previous slides f r = ρ π + k (V Rα s θ Modified Phong BRDF (in the net slides: f r (,Θ Ψ = k d + k s cos n (Ψ,Θ s Cos angles are same: V, R = Ψ,Θ s Θ s θ 9 Modified Phong BRDF Sampling Diffuse BRDF estimator: ò Lindirect ( Q = L( Y fr (, Y Q cos(, Y dw Y W Modified Phong BRDF: n fr(, Q«Y = kd + k s cos ( YQ, s where Θ s is the perfect specular direction of Θ, relative to. 20 0

11 Modified Phong BRDF Sampling Modified Phong BRDF: f r (,Ψ Θ = f r,d (,Ψ Θ + f r,s (,Ψ Θ = k d + k s cos n (Θ s,ψ = ρ d π + ρ (n + 2 s 2π cos n (Θ s,ψ 2 Modified Phong BRDF Sampling Modified Phong BRDF: L ( Q = L ( Q + L ( Q indirect diffuse specular L ( Q = L( r (, Y -Y k cos( Y, dw indirect W r d L L diffuse specular ò n + L((, ry -Y k cos ( YQ, cos( Y, dw W ( Q = ( Q = r ò å i= å i= s s L ( Yi kd cos(, Yi p ( Y Random direction on unit hemisphere, proportional to cosine lobe around normal i n L ( Yi kscos ( YQ, s cos( Yi, p ( Y 2 i Y Y 22

12 Modified Phong BRDF Sampling Diffuse reflection estimator: L indirect ( Θ = i= cosq cos(, Y PDF = p( Y = = p p rd BRDF = fr(, Y«Q = kd = p L( Ψ i f r (,Ψ Θcos(,Ψ i p(ψ i 23 Modified Phong BRDF Sampling Diffuse reflection estimator: L indirect L ( Yi fr(, Y Qcos(, Yi ( Q = å i= p( Yi ærd ö L ( Yi cos(, i ç Y p = è ø å cos(, i= Yi p rd = å L ( Yi i= 24 2

13 Modified Phong BRDF Sampling Specular reflection estimator: L specular n L ( Yi ks cos ( Y, Qscos( Yi, ( Q = å = p ( Y i 2 i with k s = ρ s n + 2 2π ; PDF = p 2 (Ψ i = n + 2π cosn (Ψ,Θ s 25 Modified Phong BRDF Sampling Specular reflection estimator: L specular ( Q = å i= n L ( Yi ks cos ( Y, Qs cos( Yi, p ( Y æ n + 2 ö n L ( Y cos (, cos(, i ç rs Y Qs Yi 2p = è ø å i= æn + ö cos n ç ( YQ, s è 2p ø æn + 2 ö = åç rsl ( Yicos( Yi, i= è n + ø æn + 2 ö = rs ç å L ( Yicos( Yi, è n + ø i= n + 2 with k PDF = p 2 (Ψ i = n + s = ρ s 2π cosn (Ψ,Θ s 2π ; 2 i 26 3

14 Modified Phong BRDF Sampling The final estimator for a sampled direction Ψ i is then equal to: L indirect ( Θ = L r ( Ψ i k d cos(ψ i, q p (Ψ i L r ( Ψ i k s cos n (Ψ i,θ s cos(ψ i, q 2 p 2 (Ψ i if event (diffuse if event 2 (specular 0 if event 3 (absoption Consider the direction as part of a single distribution. L indirect ( Θ = L r (r(,ψ i Ψ i (k d + k s cos n (Ψ i,θ s cos(ψ i, i= q p (Ψ i + q 2 p 2 (Ψ i with q = πk d, q 2 = 2π n + 2 k s 27 Modified Phong BRDF Sampling Reflectance only estimator: ρ( Θ = f r (,Ψ Θcos(,Ψdω Ψ Ω f r (,Ψ Θ = f r,d (,Ψ Θ + f r,s (,Ψ Θ = k d + k s cos n (Θ s,ψ = ρ d π + ρ (n + 2 s cos n (Θ 2π s,ψ Energy convervation: ρ d + ρ s ρ d = πk d, ρ s = 2π n + 2 k s remember f r (,Ψ Θ = dl( Θ de( Ψ = dl( Θ L( Ψcos(,Ψdω Ψ 28 4

15 Modified Phong BRDF Sampling Reflectance only estimator: ò r( Q = f (, Y Qcos(, Y dw W r ærd rs( n + 2 cos n ö = ò ( s, cos(, dw W ç + Q Y Y Y è p 2p ø ærd ö ærs( n + 2 n ö = ò cos(, dw cos ( s, cos(, dw W ç Y Y + p òw ç Q Y Y Y è ø è 2p ø rd æn + 2 ö n = cos(, dwy rs cos ( s, cos(, dwy p ò Y + W ç Q Y Y 2p òw è ø = r + r ( Q d s Y 29 Modified Phong BRDF Sampling If a PDF is the Phong lobe, r( Q = r + r ( Q d s æn + 2 ö n = rd + rsç cos ( Qs, Ycos(, Y dwy 2p òw è ø n æn + 2ö cos ( Qs, Ycos(, Y = rd + rsç å è 2 p ø i= p ( PDF(Θ = n + 2 cosn (Θ s,ψ p( = n + 2π cosn (Θ s,ψ n = n 2 cos ( s, cos(, r + æ d rs ç + ö 2p Q Y Y å è ø n + i= n cos ( Qs, Y 2p æn + 2öæ 2p ö = r + r ç ç è 2p øèn+ ø cos(, Y d s i= æn + 2ö = rd + rsç åcos(, Y è n+ ø i= å * Reading Lafortune and Willem s Using the Modified Phong Reflectance Model for Physicallybased Rendering will help. 30 5

16 Eample: Glossy Rendering 3 Eample: Glossy Rendering 32 6

Eample: Glossy Rendering 33 Area Sampling Indirect illumination as an integral over all surfaces in the scene L indirect ( Θ = Ascene Estimator using a PDF p(y L indirect ( Θ =!"!

17 Eample: Glossy Rendering 33 Area Sampling Indirect illumination as an integral over all surfaces in the scene L indirect ( Θ = Ascene Estimator using a PDF p(y L indirect ( Θ =!"!!"! L r (y y fr (,Θ y G(, yv(, yday. i=!"!!"! L r (y y fr (,Θ y G(, yv(, y p(y i Sampling area needs to evaluate the visibility function V( as part of the estimator. Putting the visibility in the estimator increases the variance for an equal number of samples. 34 7

18 Issues Direct illumination The total number of shadow rays d cast from each point. How a single light source is selected from among all the available light sources for each shadow ray. The distribution of the shadow ray over the area of a single light source. Indirect illumination umber of indirect illumination rays i distributed over the hemisphere Ω Eact distribution of these rays over the hemisphere (uniform, cosine, Absorption probabilities for Russian roulette in order to stop the recursion. 35 Pseudo codes // global illumination algorithm // stochastic ray tracing computeimage(eye for each piel radiance = 0; H = integral(h(p; for each sample // p viewing rays pick sample point p within support of h; construct ray at eye, direction p-eye; radiance = radiance + rad(ray * h(p; radiance = radiance / (#samples * H; ray(ray find closest intersection point of ray with scene; return(le(,eye- + computeradiance(, eye-; computeradiance(, dir estimateradiance += directillumination(, dir; estimateradiance += indirectillumination(, dir; return(estimatedradiance; 36 8

19 Pseudo codes directillumination(, theta estimatedradiance = 0; for all shadow rays // d shadow rays select light source k; sample point y on light source k; estimated radiance += Le * BRDF * radiancetransfer(,y/(pdf(kpdf(y k; estimatedradiance = estimatedradiance / #paths; return(estimatedradiance; indirectillumination(,theta estimatedradiance = 0; if (no absorption // Russian roulette for all indirect paths // i indirect rays sample direction psi on hemisphere; y = trace(, psi; estimatedradiance += compute_radiance(y, -psi * BRDF * cos(,psi/pdf(psi; estimatedradiance = estimatedradiance / #paths; return(estimatedradiance/(-absorption; 37 Area Light Sampling Indirect illumination as: L indirect ( Θ =!"!!"! L r (y y fr (,Θ y G(, yv(, yday. Ascene Estimator using a PDF p(y: L indirect ( Θ = G(, y i = cosα cosα y i r y 2 i=!!"!!" L r (y i y i fr (,Θ y i G(, yi V(, y i p(y i = cos(,! y!! "!!" i cos( y,y i. 2 r y α i r α j da j da i 38 9

20 Spherical Light Sampling Samples the sphere over the solid angle as seen from a point Find direction toward sphere in polar coordinates: θ ' φ ' arccos ξ + ξ 2 = 2πξ 2 r c 2 c + 2r cos(2πξ 2 ξ ( ξ where ' = c y + 2r sin(2πξ 2 ξ ( ξ c z + r( 2ξ Reference: Section 3.2 Sampling Spherical Luminaries in "Monte Carlo Techniques for Direct Lighting Calculations," ACM Transactions on Graphics, Spherical Light Sampling Transform local to world coordinates with U, V,. Find intersection points ( : PDF(' = ( ˆω ' ˆn'!!!! 2! " 2π ' r - c 2 Reference: Section 3.2 Sampling Spherical Luminaries in "Monte Carlo Techniques for Direct Lighting Calculations," ACM Transactions on Graphics,

21 Polygon Light Sampling Sampling rectangular luminaries The uniform random sampled are given by:!! ' = 0 + ξ v + ξ PDF(' = / 2 v2 v! v! 2 Sampling triangular luminaries Use barycentric coordinates of triangles The uniform random sampled are given by:!!!!!!! "!!!!!!! " ' = p 0 + ξ 2 ξ (p p 0 + ( ξ (p 2 p 0 Reference: Section 3.2 Sampling Spherical Luminaries in "Monte Carlo Techniques for Direct Lighting Calculations," ACM Transactions on Graphics, 996 PDF(' = 2 /! v! v 2 4 Pseudo codes radiancetransfer(,y transfer = G(,y*V(,y; return(transfer; 42 2

22 Results 43 Results 44 22

23 Results 45 23

The Rendering Equation Philip Dutré. Course 4. State of the Art in Monte Carlo Global Illumination Sunday, Full Day, 8:30 am - 5:30 pm

The Rendering Equation Philip Dutré Course 4. State of the Art in Monte Carlo Global Illumination Sunday, Full Day, 8:30 am - 5:30 pm 1 Overview Rendering Equation Path tracing Path Formulation Various