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Weighted Importance Sampling Techniques for Monte Carlo Radiosity

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## Weighted Importance Sampling Techniques for Monte Carlo Radiosity

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**Weighted Importance Sampling Techniques forMonte Carlo**Radiosity Ph. Bekaert, M. Sbert, Y. Willems Department of Computer Science, K.U.Leuven I.M.A., U.d.Girona**Overview**• Weighted Importance Sampling = Biased but consistent generalization of importance sampling • Application to Form Factor Integration • Application to Stochastic Jacobi Radiosity**Uniform Sampling**• xi chosen uniformly • scores have equal weights 1/N • unbiased**Importance Sampling**• xi sampled non-uniformly (pdf q) • scores have equal weights 1/N**Importance Sampling**• xi sampled non-uniformly • scores have equal weights • better if pdf matches function more closely • pdf must be practical**Weighted Uniform Sampling**• xi sampled uniformly • non-uniform weights wi compensate for not sampling p • Also perfect if p matches f • Powell&Swann’66**Weighted Importance Sampling**• xi sampled according to source pdf q • non-uniform weights wi compensate for not sampling the target pdf p • Spanier’79**Weighted Importance Sampling**• A-posteriori correction to Importance Sampling: • Compare MSE: • Bias vanishes rapidly with N:**Global vs Local Weighting**Characteristic function • Consider sub-domain: • Global weighting: Correction factor depends on all samples in D, is the same for every sub-interval and is little effective.**Global vs Local Weighting**Characteristic function • Consider sub-domain: • Global weighting: • Local Weighting: Correction factor depends only on samples in sub-domain. normalized restriction of p and q to sub-domain**Form Factor Integration**• Sample points x on S1 uniformly: • 3 strategies to sample points y on S2: • (A) Uniform Area Sampling. • (B) Uniform Direction Sampling (Arvo’95). • (C) Cosine-distributed Direction Sampling. • Mimic (C) using (A): Low Variance in spite of uniform area sampling**Form Factor Integration: Results**Infinite variance!! About as good as or even a little bit better than uniform direction sampling**Form Factor Integration: Results**Some bias at low nr of samples**Jacobi Iterative Method**• Power equations: • Deterministic Jacobi Algorithm: (quadratic cost)**Stochastic Jacobi Iterations**A) Local Lines 1) Select patch j: 2) Select i conditional on j: 3) Score (form factor cancels!!) VARIANCE: (log-linear cost)**1) Select patch j and i:**B) Global Lines 2) Score Higher variance per sample, but cheaper samples.**Global Weighting**• Mimic local lines with cheaper global lines: Problem: same correction factor for all patches + depends on all samples.**Per-patch (local) Weighting**• Normalization of target pdf required, so can only use global line pdf as target: • When is weighting better? Compare estimates for M.S.E.: • Decide a-posteriori whether to weight or not! • Low additional cost with M.S.E. weighting M.S.E. no weighting**Weighting reduces noise**Without weighting With weighting 3250 patches about 250 rays per patch 30 seconds CPU time (195MHz R10k) Reflectivity about 70%**deep blue = 10 times better**light blue = 3 times better green = neutral yellow = 3 times worse red = 10 times worse Speedup • predicted and observed speed-up correspond well • good prediction for reasonable number of samples, except on tiny patches. Predicted (1 run of 250 rays/patch) Observed (100 runs of 250 rpp) Predicted (1 run of 100,000 rpp)**Bias is not objectionable**250 rays/patch (100 runs) 100,000 rays per patch (1 run) (Highly exaggerated) High bias on tiny patches (very low nr. of samples) Bad meshing (shadow leak)**With hierarchical refinement**• Without weighting: • High variance on small • elements created during • refinement • Shadow leak artifacts. With weighting: (solution based on the same samples!) Indirect illumination computed using stochastic Jacobi (2 min.) Direct illumination computed using stochastic ray tracing**Conclusion**• Consistent generalization of importance sampling (a-posteriori correction) • Potentially high benefit (factor >10 in SJR) at very low additional cost • and if it doesn’t help it doesn’t harm either • Future Work: • benefit estimation for very low nr of samples • other applications.