Product Importance Sampling for Light Transport Guiding Herholtz et - - PowerPoint PPT Presentation

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Product Importance Sampling for Light Transport Guiding Herholtz et al. 2016 presenter: Eunhyouk Shin It is all about convergence Contents - Review on importance sampling - Light transport guiding techniques - Gaussian Mixture Model &

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Product Importance Sampling for Light Transport Guiding

presenter: Eunhyouk Shin

Herholtz et al. 2016

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It is all about convergence

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Contents

  • Review on importance sampling
  • Light transport guiding techniques
  • Gaussian Mixture Model & EM
  • Process overview
  • Results & Discussion
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Source Materials

[EUROGRAPHICS 2016] [SIGGRAPH 2014]

  • Main paper for this presentation
  • Baseline technology
  • Useful presentation slides from the authors
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Importance Sampling

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Rendering Equation

  • Direct analytic integration is virtually impossible
  • Recursive, due to the radiance term in the integrand
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Monte Carlo Ray Tracing

  • Random sample direction from hemisphere to

cast ray recursively

  • Unbiased, even if sampling is not uniform
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Importance Sampling

  • Lower variance when PDF is close to integrand distribution
  • i.e. make more path that contributes more to radiance (light transport guiding)
  • How can we make a good estimate for the integrand distribution?
  • BRDF (given)
  • Illumination (unknown)

∝

Better to be...

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Light Transport Guiding Techniques

(slides from Vorba et al.)

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Previous work

  • Jensen [1995]

photon tracing

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Previous work

  • Jensen [1995]

photon tracing path tracing

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Previous work

  • Jensen [1995]

photon tracing path tracing

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Previous work

  • Jensen [1995]

photon tracing path tracing

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k-NN

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Previous work

  • Jensen [1995]: reconstruction
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Previous work

  • Jensen [1995]: reconstruction
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Previous work

  • Peter and Pietrek [1998]

path tracing photon tracing

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Previous work

  • Peter and Pietrek [1998]

path tracing photon tracing importon tracing

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Previous work

  • Peter and Pietrek [1998]

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Previous work

  • Peter and Pietrek [1998]

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Previous work

  • Peter and Pietrek [1998]

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Previous work

  • Peter and Pietrek [1998]

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Previous work

  • Peter and Pietrek [1998]

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Previous work

  • Peter and Pietrek [1998]

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Previous work

  • Peter and Pietrek [1998]

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Limitations of previous work

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  • Bad approximation of in complex scenes
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Limitations of previous work

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Limitations of previous work

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Limitations of previous work

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Limitations of previous work

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Not enough memory!

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Solution: On-line Learning of Parametric Model

  • Shoot a batch of photons, then summarize into a parametric model
  • GMM (Gaussian Mixture Model) is used
  • Parametric model use less memory
  • Forget previous photon batch and shoot new batch
  • Keep updating parameters of the model: On-line learning
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Overcoming the memory constraint

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Overcoming the memory constraint

1st pass

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Overcoming the memory constraint

1st pass GMM

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Overcoming the memory constraint

1st pass

k-NN

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Overcoming the memory constraint

1st pass GMM

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Overcoming the memory constraint

1st pass GMM

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Overcoming the memory constraint

1st pass GMM

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Overcoming the memory constraint

1st pass 2nd pass GMM

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Overcoming the memory constraint

1st pass GMM 2nd pass

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Overcoming the memory constraint

1st pass 2nd pass 3rd pass

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GMM

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Overcoming the memory constraint

1st pass 2nd pass 3rd pass

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GMM

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Gaussian Mixture Model

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Gaussian Distribution (Normal Distribution)

Compact: just 6 float numbers for 2D

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Gaussian Mixture Model (GMM)

Used to approximate PDF Convex combination of Gaussians:

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Expectation Maximization (EM) Algorithm

  • Popular algorithm that can be used for fitting GMM to scattered data points
  • Consists of 2 steps: E-step (expectation) and M-step (maximization)
  • Converge to local maximum of likelihood
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EM: How It Works

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EM: Expectation Step

Soft assignment using Bayes’ rule

  • For each sample, compute

soft assignment weight to clusters

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EM: Maximization Step

  • Update each cluster

parameters (mean, variance, weight) to fit the data assigned to it

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EM example

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EM example

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EM example

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On-line learning: Weighted Stepwise EM

  • Use one sample for each step and extend to infinite stream of samples
  • Use weighted samples (can be viewed as repeated samples)
  • Fit to density of finite set of samples, compute sufficient statistics at once

Weighted stepwise EM: (variant used for this paper) Original EM:

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Process Overview

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Process Overview

1. Preprocessing 2. Training 3. Rendering

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Process Overview

1. Preprocessing 2. Training 3. Rendering

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Process Overview

1. Preprocessing 2. Training 3. Rendering

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Process Overview

1. Preprocessing 2. Training 3. Rendering

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  • 1. Preprocessing
  • BRDF is approximated by GMM
  • Cache GMM for each material, for each

(viewing) direction BRDF:Given

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  • 2. Training
  • Photon, importons guide each other in

alternating fashion

  • On-line learning with weighted step-wise EM
  • Cache the learnt illumination GMMs

Illumination: not known in advance

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  • 2. Training
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  • 3. Rendering
  • For intersection point, query the cached

BRDF, radiance GMM

  • Product distribution is calculated
  • n-the-fly
  • Sampling based on product distribution

How can we calculate efficiently?

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Gaussian x Gaussian = Gaussian

  • Extends to multi-dimensional Gaussian

X =

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( + … + ) x ( + … + )

GMM x GMM = GMM

BRDF: GMM of N components Illumination: GMM of M components

= ( + + … + + )

Product distribution: GMM of M*N components

  • Parameters for product GMM can be computed directly from
  • riginal parameters
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Reduction of GMM components

  • For the sake of efficiency, merge similar components
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Results & Discussion

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Evaluation: 1 hour rendering

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Result

Multiple importance sampling instead of product dist. No GMM reduction

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Result

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Discussion

  • No memory issue indeed
  • < 10MB for GMM cache in typical scene
  • Fast convergence for complex glossy-glossy reflection scene
  • Where product sampling is important
  • Not efficient for spatially varying BRDF
  • GMM is cached per material
  • Possible extension using SVBRDF parameters
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Summary

  • In order to perform importance sampling, we estimate illumination based on particles
  • In complex scenes, we need more particles for better estimation
  • On-line learning of GMM by weighted stepwise EM, enables to generate particles

without causing memory issues.

  • BRDF is also approximated as GMM so that we can use the product GMM as direct

approximation for the integrand of the rendering equation

  • Fast convergence for complex, glossy scenes