Sliced Optimal Transport Sampling

Abstract

In this paper, we introduce a numerical technique to generate sample dis- tributions in arbitrary dimension for improved accuracy of Monte Carlo integration. We point out that optimal transport offers theoretical bounds on Monte Carlo integration error, and that the recently-introduced numerical framework of sliced optimal transport (SOT) allows us to formulate a novel and efficient approach to generating well-distributed high-dimensional pointsets. The resulting sliced optimal transport sampling, solely involving repeated 1D solves, is particularly simple and efficient for the common case of a uniform density over a d-dimensional ball. We also construct a volume- preserving map from a d-ball to a d-cube (generalizing the Shirley-Chiu mapping to arbitrary dimensions) to offer fast SOT sampling over d-cubes. We provide ample numerical evidence of the improvement in Monte Carlo integration accuracy that SOT sampling brings compared to existing QMC techniques, and derive a projective variant for rendering which rivals, and at times outperforms, current sampling strategies using low-discrepancy sequences or optimized samples.

Caption: Sliced Optimal Transport Sampling. Global illumination of a scene (top left, San Miguel) requires integrating radiance over a high-dimensional space of light paths. The projective variant of our sliced optimal transport (SOT) sampling technique, leveraging the particular nature of integral evaluation in rendering and further combined with a micro-Cranley-Patterson rotation per pixel, outperforms standard Monte Carlo and Quasi-Monte Carlo techniques, exhibiting less noise and no structured artifact (top right, 32spp) while offering a better spatial distribution of error (bottom right, errors from blue (small) to red (large)). Moreover, our projective SOT sampling produces better convergence of the mean absolute error for the central 7×7 zone of the highlighted reference window as a function of the number of samples per pixel (from 4spp to 4096spp, bottom-left graph) in the case of indirect lighting with one bounce.

@article{paulin2020,
      author = {Loïs Paulin and Nicolas Bonneel and David Coeurjolly and Jean-Claude Iehl and Antoine Webanck and Mathieu Desbrun and Victor Ostromoukhov},
      journal = {ACM Transactions on Graphics (Proceedings of SIGGRAPH)},
      month = {July},
      number = {4},
      title = {Sliced Optimal Transport Sampling},
      volume = {39},
      year = {2020}
}