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A Novel Hemispherical Basis for Accurate and Efficient
Rendering
P. Gautron J. Křivánek
S. Pattanaik K. Bouatouch
Eurographics Symposium on Rendering 2004
15th Eurographics Workshop on Rendering - 21-23 June, Norrköping, Sweden
EGSR 2004 – Norrköping, Sweden 2
Problem Statement
BRDF Incoming/Outgoing Radiance
F(, ) Sample set
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Problem Statement
Original Function Piecewise linear approximation
Need a more compact and smoothed representation
Better fitting Fast computation of integrals
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Contribution
New set of basis functionsFormula similar to Spherical Harmonics Designed for representing hemispherical functions
Several rotation methods for projected functions
Applications in lighting simulation
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Outline
Applications
BRDF representationEnvironment mappingDirectional radiance caching
Previous workBasis functionsRepresentation of hemispherical functions
Three approaches to hemispherical rotation
The new basisDefinition
EGSR 2004 – Norrköping, Sweden 6
Outline
Previous work
Three approaches to hemispherical rotation
Applications
BRDF representationEnvironment mappingDirectional radiance caching
Basis functionsRepresentation of hemispherical functions
The new basisDefinition
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Basis Functions
fi = f(x)bi(x)dx f(x) = fi bi(x)
g(x) = gi bi(x) f(x)g(x)dx = fi gi
EGSR 2004 – Norrköping, Sweden 8
Spherical Harmonics
Y lm(,) l
m()K l
mP l
m(cos )=
(0,0)
(1,-1)
(2,-2) (2,-1) (2,0) (2,1) (2,2)
(1,0) (1,1)
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Spherical HarmonicsMain Properties
Simple projection and reconstruction
Analytical rotations
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SH For Hemispherical FunctionsZero Hemisphere
Equator discontinuity
Artifacts
Original SH
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SH For Hemispherical Functions
Improve accuracy
Avoid equator discontinuity
Original
Optimizationmatrix
Even Reflection[Westin92]
Least-SquaresApproximation
[Sloan03]
Reflected Original
SH
SH
SH
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SH For Hemispherical Functions
No rotation
No dot product
R
Above equator
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SH For Hemispherical FunctionsConclusion
Do not fit the hemisphere
Specific improvements
No rotations
No dot product
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Hemispherical Basis Functions
[Koenderink96] : Zernike Polynomials
Accurate representation
No rotationsUsed in CUReT BRDF Database
[Makhotkin96] : Shifted Jacobi Polynomials
Accurate representation
No rotationsNot used previously in computer graphics
EGSR 2004 – Norrköping, Sweden 15
Outline
Previous work
Three approaches to hemispherical rotation
Applications
BRDF representationEnvironment mappingDirectional radiance caching
Basis functionsRepresentation of hemispherical functions
The new basisDefinition
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Our Novel Basis
Y lm(,) l
m()K l
mP l
m(cos )=
Spherical Harmonics
(0,0)
(1,-1)
(2,-2) (2,-1) (2,0) (2,1) (2,2)
(1,0) (1,1)
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Our Novel BasisShifting
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Our Novel Basis
H lm(,) l
m() P l
m(2cos -1)= K l
m~
(0,0)
(1,-1)
(2,-2) (2,-1) (2,0) (2,1) (2,2)
(1,0) (1,1)
Hemispherical Harmonics
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HSH Rotation
Intuitive: conversion of HSH coefficients to SH
Analytic: Comparison of SH and HSH basis functions
Brute Force: Precomputation of rotation matrices
3 Methods
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HSH RotationIntuitive
HSH SH R(SH) R(HSH)C RSH C-1
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HSH RotationIntuitive
HSH SH R(SH) R(HSH)C RSH C-1
SparseComputed Numerically
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HSH Rotation
Intuitive: conversion of HSH coefficients to SH
Analytic: Comparison of SH and HSH basis functions
Brute Force: Precomputation of rotation matrices
3 Methods
Reminders: Euler rotation angles
Hemispherical data rotation
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Euler’s Rotation Theorem
« An arbitrary rotation may be described by only three parameters »
ZYZ Angles
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HSH RotationRotation Around Vertical Axis
Y lm(,) l
m()K l
mP l
m(cos )=
H lm(,) l
m() P l
m(2cos -1)= K l
m~
EGSR 2004 – Norrköping, Sweden 25
HSH RotationRotation Around Other Axes
Y lm(,) l
m()K l
mP l
m(cos )=
H lm(,) l
m() P l
m(2cos -1)= K l
m~
EGSR 2004 – Norrköping, Sweden 26
Partial Deletion
β
Deleting vanishing part
(0,0)
C1 x
(1,-1)
C2 x
(1,0)
C3 x
(1,1)
C4 x
Deletion Matrix : projection of « cut » basis functions
computed numericallyhigh frequency dense matrix
EGSR 2004 – Norrköping, Sweden 27
HSH RotationAnalytic
Idea: Use SH rotation matrices
βSH
βHSH
HSH-projected function
SH-projected function using same coefficients
SH rotation
Impact of SH rotation onHSH projected function
βSH = arccos(2cos(βHSH)-1)
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HSH RotationBrute Force
20° 40° 60° 80°Precomputed Rotation Matrices
50° Rotation around Y Axis ?
≈50°x 0.5x 0.5
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Outline
Previous work
Three approaches to hemispherical rotation
Applications
BRDF representationEnvironment mappingDirectional radiance caching
Basis functionsRepresentation of hemispherical functions
The new basisDefinition
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Application: BRDF RepresentationPrinciple
BRDF = 4D FunctionParabolic Parameterization
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Application: BRDF Representation
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Application: BRDF Representation
SHHSH
Less Ringing
Higher Frequency
Accuracy
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Application: Environment MappingPrinciple
For each vertex
CPU
Rotation
CPU
Conversion
GPU
Environment BRDF
Additional Step
EGSR 2004 – Norrköping, Sweden 34
Application: Environment MappingPerformance
Rotation on CPU for SH and HSH
Added conversion (sparse matrix)
Accuracy overcomes computational overhead
EGSR 2004 – Norrköping, Sweden 35
Application : Radiance Caching
Goal : computation of indirect diffuse lightingIrradiance Caching Scheme
EGSR 2004 – Norrköping, Sweden 36
Application : Radiance Caching
Goal : computation of indirect diffuse lightingIrradiance Caching Scheme
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Application : Radiance Caching
Interpolation
Goal : computation of indirect diffuse lightingIrradiance Caching Scheme
EGSR 2004 – Norrköping, Sweden 38
Application : Radiance Caching
HSHHSH
Goal : computation of indirect glossy lighting
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Application : Radiance Caching
Goal : computation of indirect glossy lighting
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Application : Radiance Caching
Interpolation
Goal : computation of indirect glossy lighting
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Application : Radiance Caching
Incident Radiance BRDF
dot product
Goal : computation of indirect glossy lighting
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Application : Radiance Caching
Low frequency BRDFs
New translational gradients formulas
Rotational gradient replaced by rotation
Results
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Conclusion
New basis more accurate than SH
3 methods for computing rotations
Easy to use in SH applications : BRDF Representation, Environment Mapping, Global Illumination
More details on Radiance Caching in« Radiance Caching for Efficient Global Illumination Computation »
(J. Krivanek, P. Gautron, S. Pattanaik, K. Bouatouch)IRISA Technical Report #1623
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Perspectives
Analytic formulas for
SH HSH Conversion MatrixHSH Rotation Matrices
Improve Radiance Caching Hardware Interactive Global Illumination
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Any Questions ?
Rendered using Radiance Caching
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Papers Download
http://www.cgg.cvut.cz/~xkrivanj/papers/index.htm
A Novel Hemispherical Basis for Accurate and Efficient Rendering
Radiance Caching for Efficient Global Illumination Computation
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BRDF Representation Accuracy
Phong BRDF
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BRDF Representation Accuracy
Anisotropic Ward BRDF
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