arXiv subjects

Victor Ostromoukhov

Publications and source records attributed to Victor Ostromoukhov.

7 recordsLinked to original sources

arXiv2026

Perfectly equidistributed Quasi-Monte Carlo sequences from Artin-Schreier polynomials

To numerically integrate a function, one may resort to Quasi-Monte Carlo estimators, that average integrand values at pseudo-random well-distributed uniform sampling locations. Better uniformity improves the worst-case integration-error bound. A standard measure of uniformity is given by an integer $t$ value, where $t=0$ yields the best uniformity. Producing sequences of samples with bounded $t$ values can be achieved with Sobol' recursive construction, that uses coefficients of irreducible polynomials. While $b$-dimensional sequences with $t=0$ can be obtained by taking $b$ polynomials of degree $1$ over the Galois Field $\mathrm{GF}(b)$, we show conditions that guarantee $t=0$ for specific higher degree polynomials. In particular, we relate the Sobol' construction to tensorized powers of Pascal matrices when the chosen polynomials only differ by a constant and exhibit simple conditions to guarantee $t=0$ in this case. We then focus on Artin-Schreier irreducible polynomials, in the form $p_i(x) = x^b - x + c_i$, where $i \in \{1, \dots, b-1\}$ and $b$ is prime, and we make explicit conditions that always guarantees $t=0$ in $b-1$ dimensions. Combining $b$-dimensional Sobol' of degree $1$ and our $(b-1)$-dimensional Artin-Schreier sequence of degree $b$, we provide a fast greedy procedure that optimizes the $(2b-1)$-dimensional combined $t$ value, while guaranteeing $t=0$ projection in subspaces.

cs.DM↗

arXiv2025

SZ Sequences: Binary-Based $(0, 2^q)$-Sequences

Low-discrepancy sequences have seen widespread adoption in computer graphics thanks to their superior convergence rates. Since rendering integrals often comprise 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 convergence rates when integrals include lower-dimensional integration problems in 2, 4, 16, ... dimensions. Our construction is based on using 2$\times$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 \emph{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 formula to construct full-resolution (64-bit) matrices. Our binayr generator matrices 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 $1.93\times$ in common rendering applications.

cs.GR↗

arXiv2023

Generator Matrices by Solving Integer Linear Programs

In quasi-Monte Carlo methods, generating high-dimensional low discrepancy sequences by generator matrices is a popular and efficient approach. Historically, constructing or finding such generator matrices has been a hard problem. In particular, it is challenging to take advantage of the intrinsic structure of a given numerical problem to design samplers of low discrepancy in certain subsets of dimensions. To address this issue, we devise a greedy algorithm allowing us to translate desired net properties into linear constraints on the generator matrix entries. Solving the resulting integer linear program yields generator matrices that satisfy the desired net properties. We demonstrate that our method finds generator matrices in challenging settings, offering low discrepancy sequences beyond the limitations of classic constructions.

cs.GR↗

arXiv2023

Example-Based Sampling with Diffusion Models

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.

cs.GR↗

arXiv2019

Blue-noise sampling for human retinal cone spatial distribution modeling

This paper proposes a novel method for modeling human retinal cone distribution. It is based on Blue-noise sampling algorithms that share interesting properties with the sampling performed by the mosaic formed by cone photoreceptors in the retina. Here we present the method together with a series of examples of various real retinal patches. The same samples have also been created with alternative algorithms and compared with plots of the center of the inner segments of cone photoreceptors from imaged retinas. Results are evaluated with different distance measure used in the field, like nearest-neighbor analysis and pair correlation function. The proposed method can describe features of a human retinal cone distribution with a certain degree of similarity to the available data and can be efficiently used for modeling local patches of retina.

cs.GR↗

arXiv2015

Variance Analysis for Monte Carlo Integration: A Representation-Theoretic Perspective

In this report, we revisit the work of Pilleboue et al. [2015], providing a representation-theoretic derivation of the closed-form expression for the expected value and variance in homogeneous Monte Carlo integration. We show that the results obtained for the variance estimation of Monte Carlo integration on the torus, the sphere, and Euclidean space can be formulated as specific instances of a more general theory. We review the related representation theory and show how it can be used to derive a closed-form solution.

cs.GR↗

arXiv2008

Polyomino-Based Digital Halftoning

In this work, we present a new method for generating a threshold structure. This kind of structure can be advantageously used in various halftoning algorithms such as clustered-dot or dispersed-dot dithering, error diffusion with threshold modulation, etc. The proposed method is based on rectifiable polyominoes -- a non-periodic hierarchical structure, which tiles the Euclidean plane with no gaps. Each polyomino contains a fixed number of discrete threshold values. Thanks to its inherent non-periodic nature combined with off-line optimization of threshold values, our polyomino-based threshold structure shows blue-noise spectral properties. The halftone images produced with this threshold structure have high visual quality. Although the proposed method is general, and can be applied on any polyomino tiling, we consider one particular case: tiling with G-hexominoes. We compare our polyomino-based threshold structure with the best known state-of-the-art methods for generation threshold matrices, and conclude considerable improvement achieved with our method.

cs.GR↗