Tabular Importance Sampling Methods in Global Illumination

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Tabular Importance Sampling Methods in Global
Illumination

• David Cline
• Brigham Young University

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Global Illumination
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The Measurement Equation
Lens
Pixel
Incoming radiance
Image plane
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The Rendering Equation
Participating Medium (smoke)
Lens
Reflection
Radiance
Scatter event
Surface
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Dimensionality

• Dimensionality of the measurement equation
• Time (1D)
• Pixel Sensor (2D)
• Lens (2D)
• Distance to Scatter Event (1D)
• Scattering Direction (2D)
• 8 dimensions, just for single scattering!

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Monte Carlo (MC) Rendering

• For each pixel
• Average samples of the measurement equation (ray
  paths).

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MC Sample in World Space (Ray Path)
Point on light
Time
Scatter direction
Point on pixel
Point on lens
Distance to scatter event
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MC Sample in World Space (Ray Path)
Point on light
Time
Scatter direction
Point on pixel
Point on lens
Distance to scatter event

pixel term pixel prob
lens term lens prob
distance term distance prob
scatter term scatter prob
x
x
x
Pixel val
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MC Sample in Preimage Space, 0,1)n
x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10,
x11,
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MC Sample in Preimage Space, 0,1)n
Point on light
Distance in medium
Scatter direction
Time
x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10,
x11,
Point on lens
Point on pixel
Which light
Reflect or transmit
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How many samples?

• 50,000 samples per pixel! What can be done?

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Importance Sampling

• Vary the method that converts from preimage to
  world space.

Preimage space
Importance sampling
World space
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Tabular Importance Sampling

• Discrete probability tables to help importance
  sampling.

Less samples
More samples
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Existing Tabular Methods

• Pixel Area Sampling
• Ernst, Stamminger, Greiner
• Filter Importance Sampling (2006)
• BRDF Sampling (light scattering)
• Lawrence, Rusinkiwicz, Ramamoorthi
• Efficient BRDF importance sampling using a
  factored representation (2004)

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Existing Tabular Methods

• Environment Map Sampling (light source)
• Debevec
• A Median Cut Algorithm for Light Probe Sampling
  (2005)
• Lawrence, Rusinkiwicz, Ramamoorthi
• Adaptive Numerical Cumulative Distribution
  Functions for Efficient Importance Sampling
  (2005)
• Product Sampling (BRDF x Light Source)
• Clarberg, Jarosz, Akenine-Möller, Jensen
• Wavelet Importance Sampling (2005)

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Existing Tabular Methods

• Direction in Space Sampling
• Jensen
• Importance Driven Path Tracing using the photon
  map (1995)
• Hey, Werner
• Importance sampling with hemispherical particle
  footprints (2002)
• Steinhurst, Lastra
• Global importance sampling of glossy surfaces
  using the photon map (2006)

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Two Stage Importance Sampling

• David Cline, Parris Egbert, Justin Talbot, David
  Cardon
• Two Stage Importance Sampling for Direct
  Lighting. Eurographics Symposium on Rendering,
  2006.

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Two Stage Importance Sampling

• Simplifications
• One primary ray per pixel
• Poor anti-aliasing on the image plane
• Pinhole Camera
• Not time dependent
• No participating media
• Direct lighting only
• The light is an environment map

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Two Stage Importance Sampling

• 2D function to integrate for each pixel.

Incident Light
BRDF
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Direct Lighting Function

• Triple product
• Environment Map x BRDF x Visibility

Visibility
Environment Map
Triple Product
BRDF
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Current Solutions

• Comparison of importance functions

Multiple Importance Sampling
Environment Map
Double Product
BRDF
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Two Stage Importance Sampling

• Algorithm
• Create summed area table of environment map
• For each primary ray
• Partition environment map to approximate the
  double product
• Warp uniform samples based on the partition
• Cast rays to test visibility

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Light Source Partition

• Summed area table (cdf) of the environment map.

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Light Source Partition

• Make a kd-tree partition to approximate the BRDF
  over the domain of the environment map.

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Light Source Partition

• Make an initial partition based on the surface
  normal and peaks of the BRDF.

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Light Source Partition

• Subdivide regions likely to have high error.

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Approximate Product
Summed Area Table
Approximate Product
Light Source Partition
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Example Products
Actual Product
Actual Product
Partition
Partition
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How Can We Use the Product?
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Sample Warping
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Sample Warping
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Sample Warping
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Two Stage IS Results
Eucalyptus Grove 5 seconds
MIS (4 spp)
2 Stage (2 spp)
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Two Stage IS Results
Eucalyptus Grove 14 seconds
MIS (16 spp)
2 Stage (8 spp)
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Two Stage IS Results
Eucalyptus Grove 50 seconds
MIS (64 spp)
2 Stage (32 spp)
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What about Shadows?
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Triple Product Sampling

• David Cline, Parris Egbert, Kenric White
• Towards Triple Product Sampling in Direct
  Lighting. Poster, IEEE Symposium on Interactive
  Ray Tracing, 2006.

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Triple Product Sampling

• How can we add a visibility term to the double
  product in two stage importance sampling?

Visibility
Environment Map
Triple Product
BRDF
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Visibility Maps

• Create low resolution visibility maps at a sparse
  set of locations.

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World Space Visibility Maps
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World Space Visibility Maps
Two Stage
World Space Maps
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World Space Visibility Maps

• Problem
• Work wasted in shadowed areas.
• Observation
• Wouldnt it be better to cull shadowed samples
  BEFORE they get warped (in Preimage space)?

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Preimage Space Visibility Maps
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Preimage Space Visibility Maps
Two Stage
Preimage Space
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The Price We Pay
Bump Mapped
No Bump Map
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Preimage Space Visibility Maps

• Problems
• Bumpy surfaces
• Only direct lighting
• Little propagation of info between pixels

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What can we take from this?

• Probability maps can augment importance sampling.
  
• Preimage space is preferable.

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Sample Swarming
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Sample Swarming

• Probability maps for more things than just
  direction
• Create maps lazily
• Use Preimage space maps most of the time
• Propagate map info. by map averaging.

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Sample Swarming

• Standard Importance Sampling

Importance Sampling
Samples in World space
Uniform points in 0,1)n
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Sample Swarming

• World Space Maps

Standard Importance Sampling
Combine probabilities
Non-uniform Samples in World space
Uniform points in 0,1)n
World Space Map
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Sample Swarming

• Preimage Space Maps

Standard Importance Sampling
Preimage Space Map
Non-uniform Samples in World space
Uniform points in 0,1)n
Non-uniform Points in 0,1)n
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Importance Propagation

• Scan back and forth.
• Create current pixels maps by averaging
  neighbors.
• Once a pixel is rendered, its maps are updated
  based rendered samples.

Active Pixel Maps
Time (1D)
Image
Lens (2D)
Active rows
Participating media (1D)
Bifurcation (discrete)
Current Pixel
Light selection (discrete)
Point on light (2D)
Scatter direction (1D)
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Time Map
Helps select an instant in time
Sample Swarming (16 spp)
Path Tracing (16 spp)
Desired Image
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Lens Map
Helps select points on the lens
(same scene without blur)
Sample Swarming (32 spp)
Path Tracing (32 spp)
Desired Image
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Participating Media Map
Helps find important depths into the
participating media
Sample Swarming (32 spp)
Path Tracing (32 spp)
Desired Image
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Bifurcation Map
Helps decide between reflection and transmission
Sample Swarming (16 spp)
Path Tracing (16 spp)
Desired Image
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Light Selection Map
Helps determine which light to sample
Sample Swarming (6 spp)
Path Tracing (6 spp)
Desired Image
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Point on Light Map
Helps determine which point on a light source to
sample
Sample Swarming (32 spp)
Path Tracing (32 spp)
Desired Image
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Direction Map
Helps determine scatter direction
Sample Swarming (32 spp)
Path Tracing (32 spp)
Desired Image
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More Realistic Example
Path Tracing (256 spp)
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More Realistic Example
Sample Swarming (256 spp)
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Issues

• Each map is a 1 or 2D projection of a high
  dimensional space.
• The maps dont always cooperate.
• Map resolution and which maps are active is set
  manually.
• Information propagates down, back and forth, but
  not up.

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Thank You.