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.

中文翻译：

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微分 Owen 打乱


准蒙特卡洛集成是渲染的核心。该技术通过评估所选样本位置的被积函数来估计积分的值。这些采样点旨在尽可能均匀地覆盖域，以实现比纯随机点更好的收敛速率。确定性的低差异序列已被证明优于许多竞争对手，因为它保证了由所谓的差异度量衡量的良好一致性，并且间接地通过一个整数 t 值将落入每个域层的点数与层面积相关联（t 越低越好）。为了实现随机性，加扰技术会产生多个实现，保留 t 值，使构造随机。其中，Owen 置乱是一种流行的方法，它以递归方式排列每个维度的区间。但是，依赖排列树使其与平滑优化框架不兼容。我们提出了一个可微的 Owen 置乱，它使排列正则化。我们表明，它可以有效地与自动微分工具一起使用，以优化低差异序列，以提高最佳传输均匀性、积分误差、设计功率谱或投影特性等指标，同时保持其初始 t 值，如 Owen 加扰所保证的。在某些渲染设置中，我们表明优化的序列改善了渲染错误。