Quasi-Monte Carlo integration is at the core of rendering. This technique estimates the value of an integral by evaluating the integrand at well-chosen sample locations. These sample points are designed to cover the domain as uniformly as possible to achieve better convergence rates than purely random points. Deterministic low-discrepancy sequences have been shown to outperform many competitors by guaranteeing good uniformity as measured by the so-called discrepancy metric, and, indirectly, by an integer t value relating the number of points falling into each domain stratum with the stratum area (lower t is better). To achieve randomness, scrambling techniques produce multiple realizations preserving the t value, making the construction stochastic. Among them, Owen scrambling is a popular approach that recursively permutes intervals for each dimension. However, relying on permutation trees makes it incompatible with smooth optimization frameworks. We present a differentiable Owen scrambling that regularizes permutations. We show that it can effectively be used with automatic differentiation tools for optimizing low-discrepancy sequences to improve metrics such as optimal transport uniformity, integration error, designed power spectra or projective properties, while maintaining their initial t -value as guaranteed by Owen scrambling. In some rendering settings, we show that our optimized sequences improve the rendering error.

Much effort has been put into developing samplers with specific properties, such as producing blue noise, low-discrepancy, lattice or Poisson disk samples. These samplers can be slow if they rely on optimization processes, may rely on a wide range of numerical methods, are not always differentiable. The success of recent diffusion models for image generation suggests that these models could be appropriate for learning how to generate point sets from examples. However, their convolutional nature makes these methods impractical for dealing with scattered data such as point sets. We propose a generic way to produce 2-d point sets imitating existing samplers from observed point sets using a diffusion model. We address the problem of convolutional layers by leveraging neighborhood information from an optimal transport matching to a uniform grid, that allows us to benefit from fast convolutions on grids, and to support the example-based learning of non-uniform sampling patterns. We demonstrate how the differentiability of our approach can be used to optimize point sets to enforce properties.