CS 655 Computer Graphics

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CS 655 Computer Graphics

• Path Tracing
• (some material from University of Wisconsin)

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Path Tracing

• An extension to ray tracing
• Simulates global illumination
• Can handle all possible light bounces
• L(SD)E
• Introduced by Kajiya as a solution to the
  rendering equation
• Stochastically samples all light paths
• Handles area light sources, diffuse reflections
• Approximates the integral of illumination

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Path Tracing Algorithms

• Determine the intensity of each pixel by tracing
  light transport paths
• Paths that start at light sources and carry
  energy
• A path of length k is a sequence of vertices
• ltx0, , xk-1gt where every xi and xi1 is
  mutually
• visible and x0 is on a light

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Important Paths

• We are most interested in the important paths
• Paths that go from a light source to the eye
• Paths that carry the most energy
• The rendering equation can be written as a sum of
  integrals, each one integrating over a different
  path length

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The Rendering Equation - reordered
Light to x directly from x
Light from light source to x, then to x
Light to x via x scattered twice
Light to x via x scattered three times
etc.
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Sampling Important Paths

• We want to evaluate the integral using importance
  sampling
• i.e., attempt to sample the most important paths
• How do we find those paths?
• Several approaches have been tried

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Naïve Path Tracing (version 1)

• Start at a light
• Build a path by randomly choosing a direction at
  each bounce, send the ray in that direction, and
  add the point hit to the path vertex list
• Join the last point to the eye
• Problems? What path is achieved by this approach?

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Naïve Path Tracing (version 2)

• Start at eye
• Build a path by randomly choosing a direction at
  each bounce, follow the path in that direction,
  add the point hit to the path vertex list
• Optionally join the last point to a light
• Problems? What paths are generated?

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Path Tracing (Kajiya)

• Start at eye
• At each bounce, send a ray out in the direction
  determined according to some distribution
• At each point on the path, cast a shadow ray and
  add direct lighting contribution at that point
• Send multiple paths per pixel, average the
  contributions to get the final intensity

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Path Tracing (Kajiya) Sampling Strategies

• The way in which the direction of each bounce is
  determined makes a big difference in image
  quality.
• Stratified Sampling
• Break the possible directions into sub-regions
  and cast one sample per sub-region.
• Importance Sampling
• Sample according to the BRDF

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Path Tracing (Kajiya) - Summary

• Pros
• Focuses on objects that are visible doesnt
  waste time on objects that cant be seen
• Spends equal time on all path lengths
• ray tracing spends more time on longer paths
• Cons
• Little information gain for each ray cast
• Not easy to get good (important) samples
• Spends equal time on slow-varying diffuse
  components and fast varying specular components

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Path Tracing Algorithm

• Send a ray through a pixel
• Trace the ray to its first intersected object
• From the intersected object, send out
• One ray to each light source
• One additional ray
• a diffusely reflected ray,
• a specularly reflected ray, or
• a transmissive ray
• Trace the ray and recursively follow it, as above
• Produces a ray path not a ray tree

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Light ray
Light
Eye
Image plane
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Illuminance ray
Light
Diffuse?
Specular?
Transmission?
Eye
Image plane
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Path Tracing

• This provides a Monte Carlo approach to global
  illumination
• Stochastic samples are taken that should
  represent the actual surface properties
• The lighting distribution is sampled by tracing
  rays stochastically along all possible light
  paths
• Averaging a large number of sample rays gives an
  estimate of the integral of all light paths
  through the pixel

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Selecting the ray to trace

• How do we select which ray to trace?
• Each material has a kd, ks, and kt
• Let ktot kd ks kt
• Select a random number R in the range (0, ktot )
• if (R lt kd) then send diffuse ray
• else if (R lt kd ks) then send specular ray
• else send transmission ray

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Tracing a Diffuse Reflection

• Ideal diffuse reflection

Light is scattered equally in all directions
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Tracing a Diffuse Reflection

• Tracing a ray from the eye, the light I see could
  have come from any direction (direct from the
  light, or indirect from another object)

Randomly pick a direction and trace it
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Computing the Diffuse reflection

• We can compute a random direction as follows
• Given two random numbers x1in 0,1 and x2 in
  0,1, the randomly reflected direction wd is
  given by
• wd (q, f) (cos-1(sqrt(x1)), 2px2 )

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Tracing a Specular Reflection

• Ideal specular reflection

Light is scattered according to the reflection
direction
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Tracing a Specular Reflection

• As with diffuse lighting, tracing a ray from the
  eye, the light I see could have come from any
  direction (direct from the light, or indirect
  from another object)

Light is scattered according to the reflection
direction
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Problems

• Need to trace a lot of rays to get an accurate
  image
• Typically trace 100 1000 rays per pixel

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Problems

• Need to trace a lot of rays to get an accurate
  image
• Typically trace 100 1000 rays per pixel

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• 10,000 rays per pixel (Henrik Wann Jensen)

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Bi-directional Path Tracing

• What if we were to combine rays from both
  directions?
• Send a ray from the eye and trace it into the
  scene
• Send a ray from the light and trace it into the
  scene
• Combine them by sending a ray from the end of one
  path to the end of the other path

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Bi-directional Path Tracing

• Pros
• Each ray cast contributes to many paths
• Building from both ends can catch difficult cases
• All specular paths
• Caustics
• The idea extends to participating media
• Cons
• Still spends a lot of time in slow varying
  diffuse areas
• May not sample some of the more difficult paths

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Example Mirror
Eye ray tracing ESDL
Light ray tracing LSDE
Problematic ESDSL
Even Worse LSDSDSE
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