A Novel Hemispherical Basis for Accurate and Efficient Rendering P. Gautron J. Křivánek S. Pattanaik K. Bouatouch Eurographics Symposium on Rendering 2004.

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


Problem Statement

Original Function Piecewise linear approximation

Need a more compact and smoothed representation

Better fitting Fast computation of integrals


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


Outline

Applications

BRDF representationEnvironment mappingDirectional radiance caching

Previous workBasis functionsRepresentation of hemispherical functions

Three approaches to hemispherical rotation

The new basisDefinition


Outline

Previous work


Applications


Basis functionsRepresentation of hemispherical functions



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


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)


Spherical HarmonicsMain Properties

Simple projection and reconstruction

Analytical rotations


SH For Hemispherical FunctionsZero Hemisphere

Equator discontinuity

Artifacts

Original SH


SH For Hemispherical Functions

Improve accuracy

Avoid equator discontinuity

Original

Optimizationmatrix

Even Reflection[Westin92]

Least-SquaresApproximation

[Sloan03]

Reflected Original

SH

SH

SH


SH For Hemispherical Functions

No rotation

No dot product

R

Above equator


SH For Hemispherical FunctionsConclusion

Do not fit the hemisphere

Specific improvements

No rotations

No dot product


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


Outline

Previous work


Applications





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)


Our Novel BasisShifting


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


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


HSH RotationIntuitive

HSH SH R(SH) R(HSH)C RSH C-1


HSH RotationIntuitive

HSH SH R(SH) R(HSH)C RSH C-1

SparseComputed Numerically


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


Euler’s Rotation Theorem

« An arbitrary rotation may be described by only three parameters »

ZYZ Angles


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~


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~


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


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)


HSH RotationBrute Force

20° 40° 60° 80°Precomputed Rotation Matrices

50° Rotation around Y Axis ?

≈50°x 0.5x 0.5


Outline

Previous work


Applications





Application: BRDF RepresentationPrinciple

BRDF = 4D FunctionParabolic Parameterization


Application: BRDF Representation


Application: BRDF Representation

SHHSH

Less Ringing

Higher Frequency

Accuracy


Application: Environment MappingPrinciple

For each vertex

CPU

Rotation

CPU

Conversion

GPU

Environment BRDF

Additional Step


Application: Environment MappingPerformance

Rotation on CPU for SH and HSH

Added conversion (sparse matrix)

Accuracy overcomes computational overhead


Application : Radiance Caching

Goal : computation of indirect diffuse lightingIrradiance Caching Scheme






Interpolation




HSHHSH

Goal : computation of indirect glossy lighting






Interpolation




Incident Radiance BRDF

dot product




Low frequency BRDFs

New translational gradients formulas

Rotational gradient replaced by rotation

Results


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


Perspectives

Analytic formulas for

SH HSH Conversion MatrixHSH Rotation Matrices

Improve Radiance Caching Hardware Interactive Global Illumination


Any Questions ?

Rendered using Radiance Caching


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


BRDF Representation Accuracy

Phong BRDF


BRDF Representation Accuracy

Anisotropic Ward BRDF

A Novel Hemispherical Basis for Accurate and Efficient Rendering P. Gautron J. Křivánek S. Pattanaik K. Bouatouch Eurographics Symposium on Rendering 2004.

Documents

sweden slide

hsh rotation rotation

equator slide

rotation methods

arbitrary rotation

new basis definition

hsh rotation brute force

hsh rotation analytic