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Title:

Monte Carlo Path Tracing

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Today Path tracing Random walks and Markov chains Eye vs. light ray tracing Bidirectional ray tracing Next Irradiance caching Photon mapping Light Path Light ... – PowerPoint PPT presentation

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Title: Monte Carlo Path Tracing


1
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

2
Light Path
3
Light Transport
  • Integrate over all paths
  • Questions
  • How to sample space of paths

4
Path Tracing
5
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
6
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.

7
M. Fajardo Arnold Path Tracer
8
Cornell Box Path Tracing
10 rays per pixel
100 rays per pixel
From Jensen, Realistic Image Synthesis Using
Photon Maps
9
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.

10
Discrete Random Walk
11
Discrete Random Process
States
Creation
Termination
Transition
Assign probabilities to each process
12
Discrete Random Process
States
Creation
Termination
Transition
Equilibrium number of particles in each state
13
Discrete Random Walk
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
14
Monte Carlo Algorithm
  • Define a random variable on the space of paths
  • Path
  • Probability
  • Estimator
  • Expectation

15
Monte Carlo Algorithm
  • Define a random variable on the space of paths
  • Probability
  • Estimator

16
Estimator
  • Count the number of particles terminating in
    state j

17
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

18
Light Ray Tracing
19
Examples
Backward ray tracing, Arvo 1986
20
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

21
Path Tracing From Lights
  • Step 1. Choose a light ray
  • Step 2. Find ray-surface intersection
  • Step 3. Interaction
  • u rand()
  • if u lt reflectance
  • Choose new ray r BRDF
  • goto Step 2
  • else
  • terminate on surface deposit energy

22
Bidirectional Path Tracing
23
Bidirectional Ray Tracing
24
Path Pyramid
From Veach and Guibas
25
Comparison
Bidirectional path tracing
Path tracing
From Veach and Guibas
26
Generating All Paths
27
Adjoint Formulation
28
Symmetric Light Path
29
Symmetric Light Path
30
Symmetric Light Path
31
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

32
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
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