Importance Sampling from Product of the BRDF and the Illumination using Spherical Radial Basis Funct

1
Importance Sampling from Product of the BRDF and
the Illumination using Spherical Radial Basis
Functions

• Student Qing-Zhen Jiang
• Advisor Prof. Zen-Chung Shih
• Institute of Multimedia EngineeringNational
  Chiao Tung University

2
Outline

• Introduction
• Related Work
• Background of SRBFs
• System Overview
• Off-Line SRBF Fitting Process
• Run-Time Rendering Process
• Results
• Conclusions and Future Works

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Introduction
4
Introduction

• The goal of global illumination is to solve the
  rendering equation Kajiya 1986
• Monte Carlo Technique
• To increase efficiency, importance sampling is a
  powerful technique.

5
Introduction
Uniform Sampling
6
Introduction
Importance Sampling
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Introduction
X
Environment map importance sampling
BRDF importance sampling
From Clarberg et al. 2005
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Complex Models
measured BRDF data
HDR environment map
Matusik et al. 2003
St. Peters Basilica
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Related Work
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Related Work BRDF Importance Sampling

• Lafortune Model (Multiple cosine lobes)
• Lafortune et al. SIGGRAPH 1997
• Non-linear fitting
• Matrix factorization
• Lawrence et al. SIGGRAPH 2004
• Non-negative Matrix Factorization (NMF)
• Wavelet
• Matusik et al. SIGGRAPH 2003
• Lalonde PhD thesis 1997
• Spherical Radial Basis Functions (SRBFs)
• Weng and Shih Master thesis 2006
• Non-uniform and non-negative fitting

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Related Work Environment Map Importance
Sampling

• According the energy distribution
• Cohen and Debevec 2001, Agarwal et al. 2003,
  Kollig and Keller 2003, Ostromoukhov 2004.
• Based on clustering algorithm or hierarchical
  tiling scheme
• Spherical Harmonics
• Ramamoorthi and Hanrahan 2002
• Wavelet
• Ng R. et al. 2003

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Related Work Sampling from Product
Distributions

• Bidirectional sampling
• Bruke et al. Eurographics 2005
• Rejection sampling
• Sampling-importance resampling

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Related Work Sampling from Product
Distributions

• Wavelet Important Sampling
• Clarberg et al. SIGGRAPH 2005

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Related Work Sampling from Product
Distributions

• Wavelet Important Sampling
• Clarberg et al. SIGGRAPH 2005

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Background of SRBFs
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Background of SRBFs
Center

• General Formula

Bandwidth
Coefficient
Legendre Polynomial
?
?
?
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Background of SRBFs

• Spherical singular integral
• Gaussian SRBF
• There is a simple mathematical for the
  convolution of two Gaussian SRBF kernels
• where

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System Overview
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System Overview Off-Line SRBF Fitting Process
Lighting Direction
Viewing Direction

Measured BRDF Data
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System Overview Off-Line SRBF Fitting Process
Environment map
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System Overview Run-Time Rendering Process
Environment map
BRDF
Viewing Direction
Product
Generate Samples from each SRBF
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Off-Line SRBF Fitting Process
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Non-uniform and Non-negative SRBF Fitting
Algorithm HDR environment map

Initial Guess
optimize centers
optimize bandwidths
L-BRFS-B solver
No
optimize coefficients
SE lt t Iters gt n
Yes
terminate
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Initial Guess
Accept!!
Reject!!
Coverage-Weighted Square of Intensity
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Non-uniform and Non-negative SRBF Fitting
Algorithm HDR environment map
St. Peters Basilica
Reconstructed from 300 SRBFs
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Non-uniform and Non-negative SRBF Fitting
Algorithm HDR environment map
Uffizi Gallery
Reconstructed from 300 SRBFs
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Non-uniform and Non-negative SRBF Fitting
Algorithm Measured BRDF Data

• Weng and Shih Master thesis 2006

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Run-Time Rendering Process
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Product of BRDF and illumination

• Product of Gaussian SRBF
• where , ,
• We prune smaller efficient terms to reduce
  computation cost.

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Importance sampling

• Monte Carlo estimator
• Choose a density function p that is similar to
  the integrand f
• Multiple Importance Sampling
• Veach and Guibas SIGGRAPH 95

??
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Multiple Importance Sampling

• Combining estimators
• Weighted-average of all estimators
• Weights depend on the sampling positions

Product of BRDF and illumination
PDF computed from SRBF
??
Weight
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Allocate Samples for each SRBF

• The density of each SRBF kernel can be estimated
  by its integral
• Integral can be calculated easily with spherical
  singular integral property.
• The number of samples that should be taken from
  each SRBF is determined by
• Calculate a 1D CDF for these probabilities.
• Allocating samples by generating uniform
  variables in 0, 1

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Generate Samples for each SRBF

• Choosing azimuth angle f
• Uniform sample in 0, 2p
• Choosing elevation angle ?
• Metropolis Random Walk Algorithm
• Use random mutations to produce a set of samples
  with a desired density

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Results
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Results
10 samples
40 samples
60 samples
Environment map Grace cathedral
100 samples
100 Samples (BRDF Only)
80 samples
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Results
Environment mapSt. Peters Basilica
60 samples
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Conclusions and Future Works
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Conclusions and Future Works

• Conclusions
• A new method for sampling the products of complex
  functions.
• SRBF importance sampling can render noise-free
  images using 60-100 samples per pixel.
• Future Works
• The major computation cost for ray tracing is the
  visibility testing.
• Generate samples smarter based on some heuristic
  approach.