Low-discrepancy sequences, and more particularly Sobol' sequences are gold standard for drawing highly uniform samples for quasi-Monte Carlo applications. They produce so-called ( t,s )-sequences, that is, sequences of s -dimensional samples whose uniformity is controlled by a non-negative integer quality factor t. The Monte Carlo integral estimator has a convergence rate that improves as t decreases. Sobol' construction in base 2 also provides extremely fast sampling point generation using efficient xor-based arithmetic. Computer graphics applications, such as rendering, often require high uniformity in consecutive 2D projections and in higher-dimensional projections at the same time. However, it can be shown that, in the classical Sobol' construction, only a single 2D sequence of points (up to scrambling), constructed using irreducible polynomials x and x + 1, achieves the ideal t = 0 property. Reusing this sequence in projections necessarily loses high dimensional uniformity. We prove the existence and construct many 2D Sobol' sequences having t = 1 using irreducible polynomials p and p 2 + p + 1. They can be readily combined to produce higher-dimensional low discrepancy sequences with a high-quality t = 1, guaranteed in consecutive pairs of dimensions. We provide the initialization table that can be directly used with any existing Sobol' implementation, along with the corresponding generator matrices, for an optimized 692-dimensional Sobol' construction. In addition to guaranteeing the (1, 2)-sequence property for all consecutive pairs, we ensure that t ≤ 4 for consecutive 4D projections up to 2 15 points.

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.

Abstract Rendering photo‐realistic images using Monte Carlo path tracing often requires sampling a large number of paths to reach acceptable levels of noise. This is particularly the case when rendering participating media, that complexify light paths with multiple scattering events. Our goal is to accelerate the rendering of heterogeneous participating media by exploiting redundancy across views, for instance when rendering animated camera paths, motion blur in consecutive frames or multi‐view images such as lenticular or light‐field images. This poses a challenge as existing methods for sharing light paths across views cannot handle heterogeneous participating media and classical estimators are not optimal in this context. We address these issues by proposing three key ideas. First, we propose new volume shift mappings to transform light paths from one view to another within the recently introduced null‐scattering framework, taking into account changes in density along the transformed path. Second, we generate a shared path suffix that best contributes to a subset of views, thus effectively reducing variance. Third, we introduce the multiple weighted importance sampling estimator that benefits from multiple importance sampling for combining sampling strategies, and from weighted importance sampling for reducing the variance due to non contributing strategies. We observed significant reuse when views largely overlap, with no visible bias and reduced variance compared to regular path tracing at equal time. Our method further readily integrates into existing volumetric path tracing pipelines.

Many applications ranging from quasi-Monte Carlo integration over optimal control to neural networks benefit from high-dimensional, highly uniform samples. In the case of computer graphics, and more particularly in rendering, despite the need for uniformity, several sub-problems expose a low-dimensional structure. In this context, mastering sampling uniformity over projections while preserving high-dimensional uniformity has been intrinsically challenging. This difficulty may explain the relatively small number of mathematical constructions for such samplers. We propose a novel approach by showing that uniformity constraints can be expressed as an integer linear program that results in a sampler with the desired properties. As it turns out, complex constraints are easy to describe by means of stratification and sequence properties of digital nets. Formalized using generator matrix determinants, our new MatBuilder software solves the set of constraints by iterating the linear integer program solver in a greedy fashion to compute a problem-specific set of generator matrices that can be used as a drop-in replacement in the popular digital net samplers. The samplers created by MatBuilder achieve the uniformity of classic low discrepancy sequences. More importantly, we demonstrate the benefit of the unprecedented versatility of our constraint approach with respect to low-dimensional problem structure for several applications.

Rendering quality is largely influenced by the samplers used in Monte Carlo integration. Important factors include sample uniformity (e.g., low discrepancy) in the high-dimensional integration domain, sample uniformity in lower-dimensional projections, and lack of dominant structures that could result in aliasing artifacts. A widely used and successful construction is the Sobol' sequence that guarantees good high-dimensional uniformity and consequently results in faster convergence of quasi-Monte Carlo integration. We show that this sequence exhibits low uniformity and dominant structures in low-dimensional projections. These structures impair quality in the context of rendering, as they precisely occur in the 2-dimensional projections used for sampling light sources, reflectance functions, or the camera lens or sensor. We propose a new cascaded construction, which, despite dropping the sequential aspect of Sobol' samples, produces point sets exhibiting provably perfect dyadic partitioning (and therefore, excellent uniformity) in consecutive 2-dimensional projections, while preserving good high-dimensional uniformity. By optimizing the initialization parameters and performing Owen scrambling at finer levels of binary representations, we further improve over Sobol's integration convergence rate. Our method does not incur any overhead as compared to the generation of the Sobol' sequence, is compatible with Owen scrambling and can be used in rendering applications.

In this paper, we introduce a numerical technique to generate sample distributions 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.

Abstract We introduce a novel fitting procedure that takes as input an arbitrary material, possibly anisotropic, and automatically converts it to a microfacet BRDF. Our algorithm is based on the property that the distribution of microfacets may be retrieved by solving an eigenvector problem that is built solely from backscattering samples. We show that the eigenvector associated to the largest eigenvalue is always the only solution to this problem, and compute it using the power iteration method. This approach is straightforward to implement, much faster to compute, and considerably more robust than solutions based on nonlinear optimizations. In addition, we provide simple conversion procedures of our fits into both Beckmann and GGX roughness parameters, and discuss the advantages of microfacet slope space to make our fits editable. We apply our method to measured materials from two large databases that include anisotropic materials, and demonstrate the benefits of spatially varying roughness on texture mapped geometric models.

We present Linear Efficient Antialiased Displacement and Reflectance (LEADR) mapping, a reflectance filtering technique for displacement mapped surfaces. Similarly to LEAN mapping, it employs two mipmapped texture maps, which store the first two moments of the displacement gradients. During rendering, the projection of this data over a pixel is used to compute a noncentered anisotropic Beckmann distribution using only simple, linear filtering operations. The distribution is then injected in a new, physically based, rough surface microfacet BRDF model, that includes masking and shadowing effects for both diffuse and specular reflection under directional, point, and environment lighting. Furthermore, our method is compatible with animation and deformation, making it extremely general and flexible. Combined with an adaptive meshing scheme, LEADR mapping provides the very first seamless and hardware-accelerated multi-resolution representation for surfaces. In order to demonstrate its effectiveness, we render highly detailed production models in real time on a commodity GPU, with quality matching supersampled ground-truth images.

Abstract We present Metropolis Instant Radiosity (MIR), an unbiased algorithm to solve the Light Transport problem. MIR is a hybrid technique which consists in representing the incoming radiance field by a set of Virtual Point Lights (V PLs) and in computing the response of all sensors in the scene (i.e. camera captors) by accumulating their contributions. In contrast to other similar approaches, we propose to sample the VPLs with an innovative Multiple‐try Metropolis‐Hastings (MTMH) Algorithm: the goal is to build an efficient, aggressive, and unconditionally robust variance reduction method that works well regardless of the scene layout. Finally, we present a fast ray tracing implementation using MIR and show how our complete rendering pipeline can produce high‐quality and high‐resolution pictures in a few seconds.

This paper presents a new sampling strategy to achieve interactive global illumination on one commodity computer. The goal is to propose an efficient numerical stochastic scheme which can be well adapted to a fast rendering algorithm. As we want to provide an efficient sampling strategy to handle difficult settings without sacrificing performance in common cases, we developed an extension of Instant Radiosity [Kel97] in the same way bidirectional path tracing is an extension of path or light tracing. Our idea is to build several estimators and to efficiently combine them to find a set of virtual point light sources which are relevant for the areas of the scene seen by the camera. The resulting algorithm is faster than classical solutions to global illumination. Using today graphics hardware, an interactive frame rate and the convergence of the scheme can be easily obtained in scenes with many light sources, glossy materials or difficult visibility problems.

In this paper, we present a spectral rendering method based on a ray tracing algorithm and guided by a perceptual control of the error made. An adaptive representation of spectral data for light sources and materials is used and induces, for each pixel, the evaluation of an algebraic expression. For each visible wavelength, the computations of complex spread in refractive and dispersive materials, based on several photon maps, are locally restricted by an adaptive evaluation of the expression. This method allows to simulate high quality physically‐based pictures and, in particular, some specific phenomena like dispersion in transparent objects, scattered caustics,. . .