The numerical evaluation of integrals is a core computer graphics research problem, notably for rendering realistic images of 3D scenes. Many physically-based rendering techniques rely on the random sampling of an integrand, a process called Monte Carlo integration. For path tracing, it typically consists of following paths from the camera to light sources by randomly bouncing rays in the scene. In many cases, obtaining noise-free images requires computing hundreds to thousands of such paths. Improving the convergence rate of this integral estimator can be acheived by replacing these random values by samples that are particularly well distributed over the integration domain in a highly uniform fashion. Intuitively, correlating samples to avoid holes and clusters on the domain makes the estimate more e cient. This can be formally assessed by various uniformity measures and associated variance reduction theorems, such as the discrepancy and the Koksma-Hlawka theorem [Hlawka 1961].

Several techniques exist to obtain low discrepancy samples, either relying on sequences of values uniformly covering the domain by construction [Lemieux 2009] or nely optimizing point sets of xed cardinality by minimizing well chosen energies [Keller 2013]. Among these options, the Sobol’sequence has gained signi cant popularity for rendering since it is fast and produces very well distributed samples that e ectively reduce noise in rendered images. However, it has been suggested that uniformity over the integration domain is not su cient, and that uniformity over the twodimensional projections used for sampling re ectance, light sources, camera lenses or sensors, is also important to improve image quality [Ahmed and Wonka 2020; Paulin et al. 2020; Perrier et al. 2018; Reinert et al. 2016]. In this context, we show that the popular Sobol’sequence does not always satisfy this requirement, and that consecutive pairs of dimensions can produce very poor distributions in 2-dimensional projections. They may exhibit dominant structures that may result in aliasing artifacts (see Figs. 9 and 10). We propose to alleviate this problem by introducing a new sampler based on consecutive calls to Sobol’functions (a construction we call cascaded Sobol’sampling), which we prove to provide well distributed low discrepancy point sets in consecutive pairs of dimensions. For now, this comes at a cost: we cannot preserve the sequential aspect of the original Sobol’algorithm, and hence need to x the number of samples in advance. Still, in addition to uniformity over 2-dimensional projections, we preserve uniformity over the high-dimensional integration domain by optimizing initialization tables over a range of sample cardinalities and dimensions useful for computer graphics applications [Joe and Kuo 2008]. We also propose a technique that improves uniformity, and hence the convergence rate of Monte Carlo rendering. While the deterministic Sobol’sequence is often accompanied with a randomization strategy–for example, Owen scrambling [Owen 1998], see Section 3.2–we show that increasing the bit depth of this randomization technique allows one to more uniformly distribute samples in the domain of integration and to optimize the generation. Aside from the predetermined number of samples, our approach can act as a drop-in replacement of Sobol’samplers in existing rendering engines to e ectively obtain faster convergence. Our