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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

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EGSR 2004 – Norrköping, Sweden 2 Problem Statement BRDFIncoming/Outgoing Radiance F( , ) Sample set

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EGSR 2004 – Norrköping, Sweden 3 Problem Statement Original FunctionPiecewise linear approximation Need a more compact and smoothed representation Better fitting Fast computation of integrals

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EGSR 2004 – Norrköping, Sweden 4 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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EGSR 2004 – Norrköping, Sweden 5 Outline Applications BRDF representation Environment mapping Directional radiance caching Previous work Basis functions Representation of hemispherical functions Three approaches to hemispherical rotation The new basis Definition

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EGSR 2004 – Norrköping, Sweden 6 Outline Previous work Three approaches to hemispherical rotation Applications BRDF representation Environment mapping Directional radiance caching Basis functions Representation of hemispherical functions The new basis Definition

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EGSR 2004 – Norrköping, Sweden 7 Basis Functions f i = f(x)b i (x)dx f(x) = fifi b i (x) g(x) = gigi b i (x) f(x)g(x)dx = fifi gigi

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EGSR 2004 – Norrköping, Sweden 8 Spherical Harmonics Y l m (,)(,) l m ()() K l m P 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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EGSR 2004 – Norrköping, Sweden 9 Spherical Harmonics Main Properties Simple projection and reconstruction Analytical rotations

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EGSR 2004 – Norrköping, Sweden 10 SH For Hemispherical Functions Zero Hemisphere Equator discontinuity Artifacts Original SH

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EGSR 2004 – Norrköping, Sweden 11 SH For Hemispherical Functions Improve accuracy Avoid equator discontinuity Original Optimization matrix Even Reflection [Westin92] Least-Squares Approximation [Sloan03] Reflected Original SH

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EGSR 2004 – Norrköping, Sweden 12 SH For Hemispherical Functions No rotation No dot product R Above equator

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EGSR 2004 – Norrköping, Sweden 13 SH For Hemispherical Functions Conclusion Do not fit the hemisphere Specific improvements No rotations No dot product

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EGSR 2004 – Norrköping, Sweden 14 Hemispherical Basis Functions [Koenderink96] : Zernike Polynomials Accurate representation No rotations Used in CUReT BRDF Database [Makhotkin96] : Shifted Jacobi Polynomials Accurate representation No rotations Not used previously in computer graphics

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EGSR 2004 – Norrköping, Sweden 15 Outline Previous work Three approaches to hemispherical rotation Applications BRDF representation Environment mapping Directional radiance caching Basis functions Representation of hemispherical functions The new basis Definition

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EGSR 2004 – Norrköping, Sweden 16 Our Novel Basis Y l m (,)(,) l m ()() K l m P 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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EGSR 2004 – Norrköping, Sweden 17 Our Novel Basis Shifting

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EGSR 2004 – Norrköping, Sweden 18 Our Novel Basis H l m (,)(,) 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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EGSR 2004 – Norrköping, Sweden 19 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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EGSR 2004 – Norrköping, Sweden 20 HSH Rotation Intuitive HSH SHR(SH)R(HSH) C R SH C -1

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EGSR 2004 – Norrköping, Sweden 21 HSH Rotation Intuitive HSH SHR(SH)R(HSH) C R SH C -1 Sparse Computed Numerically

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EGSR 2004 – Norrköping, Sweden 22 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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EGSR 2004 – Norrköping, Sweden 23 Euler’s Rotation Theorem « An arbitrary rotation may be described by only three parameters » ZYZ Angles

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EGSR 2004 – Norrköping, Sweden 24 HSH Rotation Rotation Around Vertical Axis Y l m (,)(,) l m ()() K l m P l m (cos ) = H l m (,)(,) l m ()() P l m (2cos -1) = K l m ~

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EGSR 2004 – Norrköping, Sweden 25 HSH Rotation Rotation Around Other Axes Y l m (,)(,) l m ()() K l m P l m (cos ) = H l m (,)(,) l m ()() P l m (2cos -1) = K l m ~

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EGSR 2004 – Norrköping, Sweden 26 Partial Deletion β Deleting vanishing part (0,0) C 1 x (1,-1) C 2 x (1,0) C 3 x (1,1) C 4 x Deletion Matrix : projection of « cut » basis functions computed numerically high frequency dense matrix

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EGSR 2004 – Norrköping, Sweden 27 HSH Rotation Analytic Idea: Use SH rotation matrices β SH β HSH HSH-projected function SH-projected function using same coefficients SH rotation Impact of SH rotation on HSH projected function β SH = arccos(2cos(β HSH )-1)

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EGSR 2004 – Norrköping, Sweden 28 HSH Rotation Brute Force 20° 40° 60° 80° Precomputed Rotation Matrices 50° Rotation around Y Axis ? ≈50° x 0.5

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EGSR 2004 – Norrköping, Sweden 29 Outline Previous work Three approaches to hemispherical rotation Applications BRDF representation Environment mapping Directional radiance caching Basis functions Representation of hemispherical functions The new basis Definition

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EGSR 2004 – Norrköping, Sweden 30 Application: BRDF Representation Principle BRDF = 4D Function Parabolic Parameterization

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EGSR 2004 – Norrköping, Sweden 31 Application: BRDF Representation

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EGSR 2004 – Norrköping, Sweden 32 Application: BRDF Representation SH HSH Less Ringing Higher Frequency Accuracy

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EGSR 2004 – Norrköping, Sweden 33 Application: Environment Mapping Principle For each vertex CPU Rotation CPU Conversion GPU Environment BRDF Additional Step

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EGSR 2004 – Norrköping, Sweden 34 Application: Environment Mapping Performance Rotation on CPU for SH and HSH Added conversion (sparse matrix) Accuracy overcomes computational overhead

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EGSR 2004 – Norrköping, Sweden 35 Application : Radiance Caching Goal : computation of indirect diffuse lighting Irradiance Caching Scheme

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EGSR 2004 – Norrköping, Sweden 36 Application : Radiance Caching Goal : computation of indirect diffuse lighting Irradiance Caching Scheme

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EGSR 2004 – Norrköping, Sweden 37 Application : Radiance Caching Interpolation Goal : computation of indirect diffuse lighting Irradiance Caching Scheme

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EGSR 2004 – Norrköping, Sweden 38 Application : Radiance Caching HSH Goal : computation of indirect glossy lighting

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EGSR 2004 – Norrköping, Sweden 39 Application : Radiance Caching Goal : computation of indirect glossy lighting

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EGSR 2004 – Norrköping, Sweden 40 Application : Radiance Caching Interpolation Goal : computation of indirect glossy lighting

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EGSR 2004 – Norrköping, Sweden 41 Application : Radiance Caching Incident RadianceBRDF dot product Goal : computation of indirect glossy lighting

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EGSR 2004 – Norrköping, Sweden 42 Application : Radiance Caching Low frequency BRDFs New translational gradients formulas Rotational gradient replaced by rotation Results

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EGSR 2004 – Norrköping, Sweden 43 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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EGSR 2004 – Norrköping, Sweden 44 Perspectives Analytic formulas for SH HSH Conversion Matrix HSH Rotation Matrices Improve Radiance Caching Hardware Interactive Global Illumination

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EGSR 2004 – Norrköping, Sweden 45 Any Questions ? Rendered using Radiance Caching

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EGSR 2004 – Norrköping, Sweden 46 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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EGSR 2004 – Norrköping, Sweden 47 BRDF Representation Accuracy Phong BRDF

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EGSR 2004 – Norrköping, Sweden 48 BRDF Representation Accuracy Anisotropic Ward BRDF

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