A Novel Hemispherical Basis for Accurate and Efficient Rendering

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A Novel Hemispherical Basis for Accurate and
Efficient Rendering
Eurographics Symposium on Rendering 2004
15th Eurographics Workshop on Rendering - 21-23
June, Norrköping, Sweden

• P. Gautron J. Krivánek
• S. Pattanaik K. Bouatouch

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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 functions Formula similar to
Spherical Harmonics Designed for representing
hemispherical functions
Several rotation methods for projected functions
Applications in lighting simulation
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Outline
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Outline
Previous work
Basis functions
Representation of hemispherical functions
The new basis
Definition
Three approaches to hemispherical rotation
Applications
BRDF representation
Environment mapping
Directional radiance caching
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Basis Functions
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Spherical Harmonics

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Spherical Harmonics
Main Properties
Simple projection and reconstruction
Analytical rotations
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SH For Hemispherical Functions
Zero Hemisphere
Original
SH
Equator discontinuity
Artifacts
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SH For Hemispherical Functions
Even Reflection Westin92
Least-Squares Approximation Sloan03
Original
Avoid equator discontinuity
Improve accuracy
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SH For Hemispherical Functions
No rotation
No dot product
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SH For Hemispherical Functions
Conclusion
Do not fit the hemisphere
Specific improvements
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Hemispherical Basis Functions
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Outline
Previous work
Basis functions
Representation of hemispherical functions
The new basis
Definition
Three approaches to hemispherical rotation
Applications
BRDF representation
Environment mapping
Directional radiance caching
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Our Novel Basis
Spherical Harmonics
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Our Novel Basis
Shifting
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Our Novel Basis
Hemispherical Harmonics
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HSH Rotation
3 Methods
Intuitive conversion of HSH coefficients to SH
Analytic Comparison of SH and HSH basis functions
Brute Force Precomputation of rotation matrices
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HSH Rotation
Intuitive
HSH
SH
R(SH)
R(HSH)
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HSH Rotation
Intuitive
RSH
C
C-1
HSH
SH
R(SH)
R(HSH)
Sparse Computed Numerically
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HSH Rotation
3 Methods
Intuitive conversion of HSH coefficients to SH
Analytic Comparison of SH and HSH basis functions
Brute Force Precomputation of rotation matrices
Reminders
Euler rotation angles
Hemispherical data rotation
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Eulers Rotation Theorem
 An arbitrary rotation may be described by only
three parameters 
ZYZ Angles
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HSH Rotation
Rotation Around Vertical Axis
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HSH Rotation
Rotation Around Other Axes
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Partial Deletion
Deleting vanishing part
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HSH Rotation
Analytic
Idea Use SH rotation matrices
HSH-projected function
SH-projected function using same coefficients
SH rotation
Impact of SH rotation on HSH projected function
ßSH arccos(2cos(ßHSH)-1)
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HSH Rotation
Brute Force
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Precomputed Rotation Matrices
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Outline
Previous work
Basis functions
Representation of hemispherical functions
The new basis
Definition
Three approaches to hemispherical rotation
Applications
BRDF representation
Environment mapping
Directional radiance caching
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Application BRDF Representation
Principle
BRDF 4D Function
Parabolic Parameterization
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Application BRDF Representation
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Application BRDF Representation
Accuracy
Less Ringing
Higher Frequency
SH
HSH
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Application Environment Mapping
Principle For each vertex
Additional Step
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Application Environment Mapping
Performance
Rotation on CPU for SH and HSH
Added conversion (sparse matrix)
Accuracy overcomes computational overhead
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Application Radiance Caching
Irradiance Caching Scheme
Goal computation of indirect diffuse lighting
?
?
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Application Radiance Caching
Irradiance Caching Scheme
Goal computation of indirect diffuse lighting
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Application Radiance Caching
Irradiance Caching Scheme
Goal computation of indirect diffuse lighting
Interpolation
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Application Radiance Caching
Goal computation of indirect glossy lighting
HSH
HSH
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Application Radiance Caching
Goal computation of indirect glossy lighting
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Application Radiance Caching
Goal computation of indirect glossy lighting
Interpolation
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Application Radiance Caching
Goal computation of indirect glossy lighting
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Incident Radiance
BRDF
? dot product
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Application Radiance Caching
Results
Low frequency BRDFs
Rotational gradient replaced by rotation
New translational gradients formulas
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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
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Any Questions ?
Rendered using Radiance Caching
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Papers Download
A Novel Hemispherical Basis for Accurate and
Efficient Rendering
Radiance Caching for Efficient Global
Illumination Computation
http//www.cgg.cvut.cz/xkrivanj/papers/index.htm
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BRDF Representation Accuracy
Phong BRDF
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BRDF Representation Accuracy
Anisotropic Ward BRDF