VarianceAware MIS P Grittmann I Georgiev P Slusallek

Variance-Aware MIS P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek sa 2019. siggraph. org

Goal: The ultimate rendering algorithm Algorithm Technique 1 Technique 2 Technique 3 2 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Key ingredient: MIS • Multiple importance sampling (MIS) [Veach & Guibas ‘ 95] Meh. Nice! 3 Meh. Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

A use-case: bidirectional path tracing Difficult indirect illum. Simple direct illum. 4 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

A failure case: bidirectional path tracing Path tracer 5 Path tracer + Bidir. (Balance heuristic) Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Our contributions • Variance reduction effects ignored by MIS • Example: stratification • A new heuristic: MIS + variance estimates 6 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Background

Monte Carlo integration in rendering Pixel 8 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Monte Carlo integration in rendering 9 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Variance determines error Rendered image 10 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Combining estimators First technique 11 Second technique Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Combining estimators First technique 12 Second technique Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Combining estimators First technique 13 Second technique Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek Combined image

Combining estimators • Pros • Simple • Never worse than the best technique alone • Cons: • Coarse, weights whole estimators • Relies on estimates 14 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Multiple importance sampling • Per-sample weights • Balance heuristic 15 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Previous work on weighting functions • Original heuristics [Veach and Guibas 1995] • Balance, power, maximum, cut-off • All based on effective densities • Tailored solutions for special cases [Georgiev et al. 2012] • Still relies on effective densities 16 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Optimal MIS weights • [Kondapaneni et al. 2019] • Can be negative and much better! • Quite expensive… 17 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Two simple heuristics Balance heuristic MIS Variance-based 18 • Coarse • Fine grained • Relies on estimates • Uses exact values • Accounts for all effects on the variance • Ignores some variance altering effects Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

MIS ignores stratification (and other effects)

Image plane stratification One sample per pixel 20 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

No image plane stratification (e. g. BDPT) One sample per pixel 21 Sample over entire image Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Effective density with stratification One sample per pixel 22 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Effective density without stratification Sample over entire image The same! 23 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Our new heuristic

Our new heuristic

Variance and the balance heuristic • Minimized by optimal weights Minimized by balance 26 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

The residual differs between techniques • 27 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

When is the balance heuristic optimal? • When the residual is tiny: • i. e. , when variance is high 28 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Variance factor • We compute the ratio between the considered term and the whole variance: • And modify the balance heuristic with it: 29 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Behaviour of our weights • Low variance: increase weight • Where balance is known to perform poorly • High variance: balance heuristic • Where balance works well 30 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Implementation

One of many ways to use our weights • Split rendering in two stages • First stage: • Uses balance heuristic • Estimates variance factors • Second stage: • Uses our heuristic 32 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Simple variance estimation • 33 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Results: BDPT

Bidirectional path tracing • Path tracing is stratified on the image plane • Light tracing is not 35 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Results: improvement in failure cases Path tracing Balance Power 36 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek Ours

Results: no harm done otherwise Path tracing Balance Power 37 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek Ours

Results: more robust than variance alone Ours Variance-only 38 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Summary

Limitations and future work • We still use estimates (but don’t rely on them) • Limited to positive weights • How to apply to biased techniques? 40 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

A cheap and effective heuristic • Moderate overhead • Memory to store variance estimates • Trivial to implement • A few extra multiplications • Robust • Reverts to the balance heuristic with increasing variance 41 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Source code • https: //github. com/pgrit/var-aware-mis-pbrt 42 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Thank you P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek sa 2019. siggraph. org

Discussion: equal balance weights • 44 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Results: Defensive sampling

Defensive sampling • For light source selection • We combine: • Importance sampling based on unoccluded contribution • Uniform selection (defensive distribution) • Equivalent setting to [Kondapaneni et al. 2019] 46 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

We improve low-variance combinations 47 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Optimal MIS can be even better 48 Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek

Comparison to [Kondapeneni et al. 2019] Optimal weights • Optimal • With many techniques: • Complicated • Expensive • Requires exact pdfs 49 Our heuristic • Heuristic • With many techniques: • Still simple • But some memory overhead • Can handle approximated pdfs Variance-Aware Multiple Importance Sampling P. Grittmann, I. Georgiev, P. Slusallek, and J. Křivánek