Monte Carlo Path Tracing
Today Path tracing Random walks and Markov chains Eye vs. light ray tracing Bidirectional ray tracing Next Irradiance caching Photon mapping University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Light Path Should we show the directional form of this equation? University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Light Transport Integrate over all paths Questions
How to sample space of paths University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Path Tracing Penumbra: Trees vs. Paths
Path tracing vs. recursive ray tracing using trees Use less samples for this Superimpose paths and trees over these images 4 eye rays per pixel 16 shadow rays per eye ray 64 eye rays per pixel 1 shadow ray per eye ray University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell 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. Should illustrate this algorithm using the Cornell Box Confusion about reflectance The BRDF is not normalized to be a pdf The directional hemispherical reflectance is the correct normalization factor How to choose between Le and Lr University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell M. Fajardo Arnold Path Tracer
Return to this when discussing the adjoint formulation University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Cornell Box: Path Tracing
Check that I am referring to these images in the correct way 10 rays per pixel 100 rays per pixel From Jensen, Realistic Image Synthesis Using Photon Maps University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell 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. Should illustrate this algorithm using the Cornell Box Confusion about reflectance The BRDF is not normalized to be a pdf The directional hemispherical reflectance is the correct normalization factor How to choose between Le and Lr University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Discrete Random Walk Discrete Random Process
States Creation Termination Transition Assign probabilities to each process University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Discrete Random Process
States Creation Termination Transition Equilibrium number of particles in each state University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Discrete Random Walk Generate random particles from sources.
States Generate random particles from sources. Undertake a discrete random walk. Count how many terminate in state i [von Neumann and Ulam; Forsythe and Leibler; 1950s] Creation Termination Transition University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Monte Carlo Algorithm Define a random variable on the space of paths
Probability: Estimator: Expectation: University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Monte Carlo Algorithm Define a random variable on the space of paths
Probability: Estimator: University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Estimator Count the number of particles terminating in state j
Do I have I,j reversed in Mi,j? University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Equilibrium Distribution of States
Total probability of being in states P Note that this is the solution of the equation Thus, the discrete random walk is an unbiased estimate of the equilibrium number of particles in each state University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Light Ray Tracing Backward ray tracing, Arvo 1986
Examples Backward ray tracing, Arvo 1986 University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell 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 University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell 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 University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Bidirectional Path Tracing Bidirectional Ray Tracing
Need to make notation clear K is the number of vertices on a path K-2 is the number of bounces on the path Should probably show an example with k=4, not k=3 University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Path Pyramid From Veach and Guibas
25 samples per pixel Combined using the power heuristic Middle image in the k=4 row Rightmost image in the bottom row 5 light-path vertices 5 steps from the floor-lamp to the table Everyone really liked this picture Understand the phenomenology behind each subimage Is there a version of this picture that contains the weights for each pixel Review multiple importance sampling: need to keep the weights and values If there are multiple samples for each pixel, Kurt suggests premultiplying by the weights and keeping a running sum of the weights. From Veach and Guibas University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Comparison Bidirectional path tracing Path tracing
From Veach and Guibas University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Generating All Paths University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Adjoint Formulation Symmetric Light Path All this adjoint stuff is confusing What are the key points Can always formulate adjoint version of the problem Rendering operators are self-adjoint Solution of the adjoint equation can be used as an importance function Clear explanation with examples of why you want to reverse sources and receivers University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Symmetric Light Path University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Symmetric Light Path University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Three Consequences Forward estimate equal backward estimate
- May use forward or backward ray tracing Adjoint solution - Importance sampling paths Solve for small subset of the answer University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell Example: Linear Equations
Solve a linear system Solve for a single xi? Solve the adjoint equation Source Estimator More efficient than solving for all the unknowns [von Neumann and Ulam] University of Texas at AustinCS395T -Advanced Image Synthesis Spring Don Fussell