light transport guiding product importance sampling forsungeui/icg/students... · [eurographics...
TRANSCRIPT
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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
tracingpath
tracing
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Previous work• Jensen [1995] photon
tracingpath
tracing
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Previous work• Jensen [1995] photon
tracingpath
tracing
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k-NN 14
Previous work• Jensen [1995]: reconstruction
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Previous work• Jensen [1995]: reconstruction
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Previous work• Peter and Pietrek [1998]
pathtracing
photontracing
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Previous work• Peter and Pietrek [1998]
pathtracing
photontracing
importontracing
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Previous work• Peter and Pietrek [1998] PT
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Previous work• Peter and Pietrek [1998] PT
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Previous work• Peter and Pietrek [1998] PT
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Previous work• Peter and Pietrek [1998] PT
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Previous work• Peter and Pietrek [1998] PT
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Previous work• Peter and Pietrek [1998] PT
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Previous work• Peter and Pietrek [1998] PT
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Limitations of previous work
25
• 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 workPT
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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 constraint1st pass
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Overcoming the memory constraint1st pass
GMM
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Overcoming the memory constraint1st pass
k-NN34
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Overcoming the memory constraint1st pass
GMM
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Overcoming the memory constraint1st pass
GMM
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Overcoming the memory constraint1st pass
GMM
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Overcoming the memory constraint1st pass 2nd pass
GMM
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Overcoming the memory constraint1st pass
GMM
2nd pass
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Overcoming the memory constraint1st pass 2nd pass 3rd pass …
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GMM
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Overcoming the memory constraint1st 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 Overview1. Preprocessing
2. Training
3. Rendering
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Process Overview1. Preprocessing
2. Training
3. Rendering
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Process Overview1. Preprocessing
2. Training
3. Rendering
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Process Overview1. 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
on-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 original 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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ResultMultiple 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