Sampling strategies in computer graphics have long been divided between local approaches, which optimize sample distributions independently at each pixel, and global approaches, which use high-dimensional low-discrepancy sequences to ensure uniformity across all dimensions, including adjacent pixels. While global samplers are most-commonly used thanks to generally having better convergence rates, it comes at a cost of very limited user control over the distribution of samples on subspaces, making it difficult to handle common aliasing artifacts. We introduce a novel modular meta-sampler architecture that bridges local and global sampling, allowing integrator designers to employ specialized low-dimensional samplers while still achieving high-dimensional uniformity. Our approach leverages high-dimensional low-discrepancy sequences to orchestrate sample generation by a collection of local samplers operating over power-of-two hierarchical intervals. We demonstrate how existing local sampling techniques, including stratified, blue noise, and dyadic nets sampling, can be reformulated to be used with this framework, enabling hybrid sampling strategies that combine the benefits of both paradigms.

Low-discrepancy sequences have seen widespread adoption in computer graphics thanks to the superior rates of convergence that they provide. Because rendering integrals often are comprised of products of lower-dimensional integrals, recent work has focused on developing sequences that are also well-distributed in lower-dimensional projections. To this end, we introduce a novel construction of binary-based (0, 4)-sequences; that is, progressive fully multi-stratified sequences of 4D points, and extend the idea to higher power-of-two dimensions. We further show that not only it is possible to nest lower-dimensional sequences in higher-dimensional ones—for example, embedding a (0, 2)-sequence within our (0, 4)-sequence—but that we can ensemble two (0, 2)-sequences into a (0, 4)-sequence, four (0, 4)-sequences into a (0,16)-sequence, and so on. Such sequences can provide excellent rates of convergence when integrals include lower-dimensional integration problems in 2, 4, 16,... dimensions. Our construction is based on using 2×2 block matrices as symbols to construct larger matrices that potentially generate a sequence with the target (0, s )-sequence in base s property. We describe how to search for suitable alphabets and identify two distinct, cross-related alphabets of block symbols, which we call s and z , hence SZ for the resulting family of sequences. Given the alphabets, we construct candidate generator matrices and search for valid sets of matrices. We then infer a simple recurrence formula to construct full-resolution (64-bit) matrices. Because our generator matrices are binary, they allow highly-efficient implementation using bitwise operations and can be used as a drop-in replacement for Sobol matrices in existing applications. We compare SZ sequences to state-of-the-art low discrepancy sequences, and demonstrate mean relative squared error improvements up to 1.93× in common rendering applications.

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.

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.

Abstract Fourier analysis is gaining popularity in image synthesis as a tool for the analysis of error in Monte Carlo (MC) integration. Still, existing tools are only able to analyse convergence under simplifying assumptions (such as randomized shifts) which are not applied in practice during rendering. We reformulate the expressions for bias and variance of sampling‐based integrators to unify non‐uniform sample distributions [importance sampling (IS)] as well as correlations between samples while respecting finite sampling domains. Our unified formulation hints at fundamental limitations of Fourier‐based tools in performing variance analysis for MC integration. At the same time, it reveals that, when combined with correlated sampling, IS can impact convergence rate by introducing or inhibiting discontinuities in the integrand. We demonstrate that the convergence of multiple importance sampling (MIS) is determined by the strategy which converges slowest and propose several simple approaches to overcome this limitation. We show that smoothing light boundaries (as commonly done in production to reduce variance) can improve (M)IS convergence (at a cost of introducing a small amount of bias) since it removes C 0 discontinuities within the integration domain. We also propose practical integrand‐ and sample‐mirroring approaches which cancel the impact of boundary discontinuities on the convergence rate of estimators.

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 Modern physically based rendering techniques critically depend on approximating integrals of high dimensional functions representing radiant light energy. Monte Carlo based integrators are the choice for complex scenes and effects. These integrators work by sampling the integrand at sample point locations. The distribution of these sample points determines convergence rates and noise in the final renderings. The characteristics of such distributions can be uniquely represented in terms of correlations of sampling point locations. Hence, it is essential to study these correlations to understand and adapt sample distributions for low error in integral approximation. In this work, we aim at providing a comprehensive and accessible overview of the techniques developed over the last decades to analyze such correlations, relate them to error in integrators, and understand when and how to use existing sampling algorithms for effective rendering workflows.

Abstract Distributions of samples play a very important role in rendering, affecting variance, bias and aliasing in Monte‐Carlo and Quasi‐Monte Carlo evaluation of the rendering equation. In this paper, we propose an original sampler which inherits many important features of classical low‐discrepancy sequences (LDS): a high degree of uniformity of the achieved distribution of samples, computational efficiency and progressive sampling capability. At the same time, we purposely tailor our sampler in order to improve its spectral characteristics, which in turn play a crucial role in variance reduction, anti‐aliasing and improving visual appearance of rendering. Our sampler can efficiently generate sequences of multidimensional points, whose power spectra approach so‐called Blue‐Noise (BN) spectral property while preserving low discrepancy (LD) in certain 2‐D projections. In our tile‐based approach, we perform permutations on subsets of the original Sobol LDS. In a large space of all possible permutations, we select those which better approach the target BN property, using pair‐correlation statistics. We pre‐calculate such “good” permutations for each possible Sobol pattern, and store them in a lookup table efficiently accessible in runtime. We provide a complete and rigorous proof that such permutations preserve dyadic partitioning and thus the LDS properties of the point set in 2‐D projections. Our construction is computationally efficient, has a relatively low memory footprint and supports adaptive sampling. We validate our method by performing spectral/discrepancy/aliasing analysis of the achieved distributions, and provide variance analysis for several target integrands of theoretical and practical interest.

We present a novel technique that produces two-dimensional low-discrepancy (LD) blue noise point sets for sampling. Using one-dimensional binary van der Corput sequences, we construct two-dimensional LD point sets, and rearrange them to match a target spectral profile while preserving their low discrepancy. We store the rearrangement information in a compact lookup table that can be used to produce arbitrarily large point sets. We evaluate our technique and compare it to the state-of-the-art sampling approaches.

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 propose a new spectral analysis of the variance in Monte Carlo integration, expressed in terms of the power spectra of the sampling pattern and the integrand involved. We build our framework in the Euclidean space using Fourier tools and on the sphere using spherical harmonics. We further provide a theoretical background that explains how our spherical framework can be extended to the hemispherical domain. We use our framework to estimate the variance convergence rate of different state-of-the-art sampling patterns in both the Euclidean and spherical domains, as the number of samples increases. Furthermore, we formulate design principles for constructing sampling methods that can be tailored according to available resources. We validate our theoretical framework by performing numerical integration over several integrands sampled using different sampling patterns.

We introduce a fast tile-based method for adaptive two-dimensional sampling with user-specified spectral properties. At the core of our approach is a deterministic, hierarchical construction of self-similar, equi-area, tri-hex tiles whose centroids have a spatial distribution free of spurious spectral peaks. A lookup table of sample points, computed offline using any existing point set optimizer to shape the samples' Fourier spectrum, is then used to populate the tiles. The result is a linear-time, adaptive, and high-quality sampling of arbitrary density functions that conforms to the desired spectral distribution, achieving a speed improvement of several orders of magnitude over current spectrum-controlled sampling methods.

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.

We present a fast, scalable algorithm to generate high-quality blue noise point distributions of arbitrary density functions. At its core is a novel formulation of the recently-introduced concept of capacity-constrained Voronoi tessellation as an optimal transport problem. This insight leads to a continuous formulation able to enforce the capacity constraints exactly, unlike previous work. We exploit the variational nature of this formulation to design an efficient optimization technique of point distributions via constrained minimization in the space of power diagrams. Our mathematical, algorithmic, and practical contributions lead to high-quality blue noise point sets with improved spectral and spatial properties.

We present an original error-diffusion method which produces visually pleasant halftone images while preserving fine details and visually identifiable structures present in the original images. Our method is conceptually simple and computationally efficient. The source image is analyzed, and its local frequency content is detected. The main component of the frequency content (main frequency, orientation and contrast) serve as lookup table indices to a pre-calculated database of modifications to a standard error diffusion. The modifications comprise threshold modulation and variation of error-diffusion coefficients. The whole system is calibrated in such a way that the produced halftone images are visually close to the original images (patches of constant intensity, patches containing sinusoidal waves of different frequencies/orientations/contrasts, as well as natural images of different origins). Our system produces images of visual quality comparable to that presented in [Pang et al. 2008], but much faster. When processing typical images of linear size of several hundreds of pixels, our error-diffusion system is two to three orders of magnitude faster than [Pang et al. 2008]. Thanks to its speed combined with high visual quality, our error-diffusion algorithm can be used in many practical applications which may require digital halftoning: printing, visualization, geometry processing, various sampling techniques, etc.

We present a new general-purpose method for fast hierarchical importance sampling with blue-noise properties. Our approach is based on self-similar tiling of the plane or the surface of a sphere with rectifiable polyominoes. Sampling points are associated with polyominoes, one point per polyomino. Each polyomino is recursively subdivided until the desired local density of samples is reached. A numerical code generated during the subdivision process is used for thresholding to accept or reject the sample. The exact position of the sampling point within the polyomino is determined according to a structural index, which indicates the polyomino's local neighborhood. The variety of structural indices and associated sampling point positions are computed during the offline optimization process, and tabulated. Consequently, the sampling itself is extremely fast. The method allows both deterministic and pseudo-non-deterministic sampling. It can be successfully applied in a large variety of graphical applications, where fast sampling with good spectral and visual properties is required. The prime application is rendering.

This paper presents a novel method for efficiently generating a good sampling pattern given an importance density over a 2D domain. A Penrose tiling is hierarchically subdivided creating a sufficiently large number of sample points. These points are numbered using the Fibonacci number system, and these numbers are used to threshold the samples against the local value of the importance density. Pre-computed correction vectors, obtained using relaxation, are used to improve the spectral characteristics of the sampling pattern. The technique is deterministic and very fast; the sampling time grows linearly with the required number of samples. We illustrate our technique with importance-based environment mapping, but the technique is versatile enough to be used in a large variety of computer graphics applications, such as light transport calculations, digital halftoning, geometry processing, and various rendering techniques.

In this contribution, we introduce a new error-diffusion scheme that produces higher quality results. The algorithm is faster than the universally used Floyd-Steinberg algorithm, while maintaining its original simplicity. The efficiency of our algorithm is based on a deliberately restricted choice of the distribution coefficients. Its pleasing nearly artifact-free behavior is due to the off-line minimization process applied to the basic algorithm's parameters (distribution coefficients). This minimization brings the Fourier spectra of the selected key intensity levels as close as possible to the corresponding “blue noise” spectra. The continuity of the algorithm's behavior across the full range of intensity levels is achieved thanks to smooth interpolation between the distribution coefficients corresponding to key levels. This algorithm is applicable in a wide range of computer graphics applications, where a color quantization algorithm with good visual properties is needed.

This contribution introduces the basic techniques for digital facial engraving, which imitates traditional copperplate engraving. Inspired by traditional techniques, we first establish a set of basic rules thanks to which separate engraving layers are built on the top of the original photo. Separate layers are merged according to simple merging rules and according to range shift/scale masks specially introduced for this purpose. We illustrate the introduced technique by a set of black/white and color engravings, showing different features such as engraving-specific image enhancements, mixing different regular engraving lines with mezzotint, irregular perturbations of engraving lines etc. We introduce the notion of engraving style which comprises a set of separate engraving layers together with a set of associated range shift/scale masks. The engraving style helps to port the look and feel of one engraving to another. Once different libraries of pre-defined mappable engraving styles and an appropriate user interface are added to the basic system, producing a decent gravure starting from a simple digital photo will be a matter of seconds. The engraving technique described in this contribution opens new perspectives for digital art, adding unprecedented power and precision to the engraver's work.

A multi-color dithering algorithm is proposed, which converts a barycentric combination of color intensities into a multi-color non-overlapping surface coverage. Multi-color dithering is a generalization of standard bi-level dithering. Combined with tetrahedral color separation, multi-color dithering makes it possible to print images made of a set of non-standard inks. In contrast to most previous color halftoning methods, multi-color dithering ensures by construction that the different selected basic colors are printed side by side. Multi-color dithering is applied to generate color images whose screen dots are made of artistic shapes (letters, symbols, ornaments, etc.). Two dither matrix postprocessing techniques are developed, one for enhancing the visibility of screen motives and one for the local equilibration of large dither matrices. The dither matrix equilibration process corrects disturbing local intensity variations by taking dot gain and the human visual system transfer function into account. Thanks to the combination of the presented techniques, high quality images can be produced, which incorporate at the micro level the desired artistic screens and at the macro level the full color image. Applications include designs for advertisements and posters as well as security printing. Multi-color dithering also offers new perspectives for printing with special inks, such as fluorescent and metallic inks.

We investigate the general problem of reproducing color images on an offset press using custom inks in any combination and number.While this problem has been explored previously for the case of two inks, there are a number of new mathematical and algorithmic challenges that arise as the number of inks increases.These challenges include more complex gamut mapping strategies, more efficient ink selection strategies, and fast and numerically accurate methods for computing ink separations in situations that may be either over-or under-constrained.In addition, the demands of high-quality color printing require an accurate physical model of the colors that result from overprinting multiple inks using halftoning, including the effects of trapping, dot gain, and the interreflection of light between ink layers.In this paper, we explore these issues related to printing with multiple custom inks, and address them with new algorithms and physical models.Finally, we present some printed examples demonstrating the promise of our methods.

Dither-based methods for the halftoning of images on multi-level printing devices such as multi-level inkjet printers are presented. Due to the relatively large size of single droplets, halftoning algorithms are still needed. However, since halftoning occurs between the basic levels attainable by printing one, two or several droplets at the same position, artefacts are less visible than in equal resolution bilevel printers. When dithering algorithms are used for the halftoning task, the dither threshold tiles should have oblique orientations so as to make the halftoning artifacts less visible. They should be designed so as to break up the inherent artifacts of variable dot size printers, such as for example continuous lines made up of elongated elliptic dots. The resulting visual effects are shown by simulating the printed dots of a multilevel inkjet printer.

Artistic screening is a new image reproduction technique incorporating freely created artistic screen elements for generating halftones.Fixed predefined dot contours associated with given intensity levels determine the screen dot shape's growing behavior.Screen dot contours associated with each intensity level are obtained by interpolation between the fixed predefined dot contours.A user-defined mapping transforms screen elements from screen element definition space to screen element rendition space.This mapping can be tuned to produce various effects such as dilatations, contractions and nonlinear deformations of the screen element grid.Discrete screen elements associated with all desired intensity levels are obtained by rasterizing the interpolated screen dot shapes in the screen element rendition space.Since both the image to be reproduced and the screen shapes can be designed independently, the design freedom offered to artists is very great.The interaction between the image to be reproduced and the screen shapes enables the creation of graphic designs of high artistic quality.Artistic screening is particularly well suited for the reproduction of images on large posters.When looked at from a short distance, the poster's screening layer may deliver its own message.Furthermore, thanks to artistic screening, both full size and microscopic letters can be incorporated into the image reproduction process.In order to avoid counterfeiting, banknotes may comprise grayscale images with intensity levels produced by microletters of varying size and shape.

Rotated dispersed-dot dither is proposed as a new dither technique for digital halftoning. It is based on the discrete one-to-one rotation of a Bayer dispersed-dot dither array. Discrete rotation has the effect of rotating and splitting a significant part of the frequency impulses present in Bayer's halftone arrays into many low-amplitude distributed impulses. The halftone patterns produced by the rotated dither method therefore incorporate fewer disturbing artifacts than the horizontal and vertical components present in most of Bayer's halftone patterns. In grayscale wedges produced by rotated dither, texture changes at consecutive gray levels are much smoother than in error diffusion or in Bayer's dispersed-dot dither methods, thereby avoiding contouring effects.