Monte Carlo Path Tracing
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1 Page 1 Monte Carlo Path Tracing Today Path tracing Random wals and Marov chains Eye vs. light ray tracing Bidirectional ray tracing Next Irradiance caching Photon mapping Light Path Sx (, x) 0 1 f ( x, x, x ) r x 0 Gx (, x) 0 1 Gx ( 1, x2) x 1 x 2 f ( x, x, x ) r L ( S x, 0 x,, ) 1 x2 x3 L ( x, x, x, x ) = S( x, x) G( x, x) f ( x, x, x ) G( x, x ) f ( x, x, x ) S o 1 r r 1 2 3
2 Page 2 Light Transport Integrate over all paths Lx (, x) 1 = = 1 M M L ( x,, x, x, x ) da( x ) da( x ) 2 2 S Questions How to sample space of paths Path Tracing
3 Page 3 Penumbra: Trees vs. Paths 4 eye rays per pixel 16 shadow rays per eye ray 64 eye rays per pixel 1 shadow ray per eye ray Path Tracing: From Camera Step 1. Choose a camera ray r given the (x,y,u,v,t) sample weight = 1; Step 2. Find ray-surface intersection Step 3. if light return weight * Le(); else weight *= reflectance(r) Choose new ray r ~ BRDF pdf(r) Go to Step 2.
4 Page 4 M. Fajardo Arnold Path Tracer Cornell Box: Path Tracing 10 rays per pixel 100 rays per pixel From Jensen, Realistic Image Synthesis Using Photon Maps
5 Page 5 Path Tracing: Include Direct Lighting Step 1. Choose a camera ray r given the (x,y,u,v,t) sample weight = 1; Step 2. Find ray-surface intersection Step 3. weight += Lr(light sources) Choose new ray r ~ BRDF pdf(r) Go to Step 2. Discrete Random Wal
6 Page 6 Discrete Random Process Creation States Termination Transition Assign probabilities to each process p 0 i : probability of creation in state pij, : probability of transition from state i j p * i : probability of termination in state i i p * i = 1 p j i, j Discrete Random Process Creation States Termination Transition Equilibrium number of particles in each state P = p P + p 0 i i, j j i j 0 P= MP+ p M = p i, j i, j
7 Page 7 Discrete Random Wal Creation States Termination Transition 1. Generate random particles from sources. 2. Undertae a discrete random wal. 3. Count how many terminate in state i [von Neumann and Ulam; Forsythe and Leibler; 1950s] Monte Carlo Algorithm Define a random variable on the space of paths Path: Probability: Estimator: α = ( i, i,, i ) P( α ) W( α ) 1 2 Expectation: EW [ ] P( α ) W( α ) = = 1 α
8 Page 8 Monte Carlo Algorithm Define a random variable on the space of paths Probability: Estimator: P( α ) = p p p p 0 * i i, i i, i i δ Wj( α ) = * p i, j i Estimator Count the number of particles terminating in state j δ EW [ ] ( p p p p ) 0 * i, j j = i1 i1, i 2 i 1, i i * = 1 i i pj p Mp M p j j j = + +
9 Page 9 Equilibrium Distribution of States Total probability of being in states P ( ) P = I + M+ M + p 2 0 Note that this is the solution of the equation ( I M) P = p 0 Thus, the discrete random wal is an unbiased estimate of the equilibrium number of particles in each state Light Ray Tracing
10 Page 10 Examples Bacward ray tracing, Arvo 1986 Path Tracing: From Lights Step 1. Choose a light ray Choose a light source according to the light source power distribution function. Choose a ray from the light source radiance (area) or intensity (point) distribution function w = 1; Step 2. Trace ray to find surface intersect Step 3. Interaction
11 Page 11 Path Tracing: From Lights Step 1. Choose a light ray Step 2. Find ray-surface intersection Step 3. Interaction u = rand() if u < reflectance Choose new ray r ~ BRDF goto Step 2 else terminate on surface; deposit energy Bidirectional Path Tracing
12 Page 12 Bidirectional Ray Tracing = l+ e l = 0, e= 3 l = 1, e= 2 l = 2, e= 1 l = 3, e= 0 = 3 Path Pyramid =3 ( l = 2, e = 1) = 4 =5 =6 ( l = 5, e = 1) l From Veach and Guibas e
13 Page 13 Comparison Bidirectional path tracing Path tracing From Veach and Guibas Generating All Paths
14 Page 14 Adjoint Formulation Symmetric Light Path S( x, x ) 0 1 f ( x, x, x ) r x 0 Gx (, x) 0 1 Gx ( 1, x2) x 1 x 2 Gx (, x) 2 3 x 3 f ( x, x, x ) r R( x, x ) 2 3 M = S( x, x) G( x, x) f ( x, x, x ) G( x, x ) f ( x, x, x ) G( x, x ) R( x, x ) 0 1 o 1 r r
15 Page 15 Symmetric Light Path S( x, x ) 0 1 f ( x, x, x ) = f ( x, x, x ) r r x 0 Gx (, x) 0 1 Gx ( 1, x2) x 1 x 2 Gx (, x) = Gx (, x) x 3 f ( x, x, x ) r R( x, x ) 2 3 = r r M S( x, x) G( x, x) f ( x, x, x ) G( x, x ) f ( x, x, x ) G( x, x ) R( x, x ) Symmetric Light Path Sx ( 1, x0) f ( x, x, x ) r x 0 Gx ( 1, x0) Gx (, x) x x 2 Gx (, x) x f ( x, x, x ) r R( x, x ) 3 2 = r r M R( x, x ) G( x, x ) f ( x, x, x) G( x, x) f ( x, x, x ) G( x, x ) S( x, x )
16 Page 16 Three Consequences 1. Forward estimate equal bacward estimate - May use forward or bacward ray tracing 2. Adjoint solution - Importance sampling paths 3. Solve for small subset of the answer Example: Linear Equations Solve a linear system Mx = b Solve for a single x i? Solve the adjoint equation Source Estimator x i ( 2 i i i ), < x + Mx + M x + b> More efficient than solving for all the unnowns [von Neumann and Ulam]
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