22 research outputs found
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
How Different Materials are Carried on the Routes of India in Supply Chain.
In this research , it was presented that how there are different kind of materials which can be transported by commercial vehicle, various category of heavy vehicles, weight segments for carrying distinct product with safety, emission standard for vehicles and numerous challenges faced while transporting the goods. This study focusses on five category material, commercial vehicles, weight, new emission norms and challenges. For analysing this paper descriptive coding was used so that data can easily be divided into these categories. The result shows that for loading and unloading of materials documents, workforce, equipment and safety kit are needed, different type of commercial vehicle is needed for carrying distinct material like liquid in tankers, parcel in container and generic product in trailers, safety measures like plastic covers and straps for open body vehicles, SS tank for chemical products and
wooden or cardboard sheets in containers, the new norms has improved the engines, vehicles produce less pollution, better safety equipment, only improvement is done but no major changes has been noticed. Finally, the challenges faced in transit are theft, less margin, low infrastructure, harassment and corruption
Perceptual error optimization for Monte Carlo rendering
Realistic image synthesis involves computing high-dimensional light transport
integrals which in practice are numerically estimated using Monte Carlo
integration. The error of this estimation manifests itself in the image as
visually displeasing aliasing or noise. To ameliorate this, we develop a
theoretical framework for optimizing screen-space error distribution. Our model
is flexible and works for arbitrary target error power spectra. We focus on
perceptual error optimization by leveraging models of the human visual system's
(HVS) point spread function (PSF) from halftoning literature. This results in a
specific optimization problem whose solution distributes the error as visually
pleasing blue noise in image space. We develop a set of algorithms that provide
a trade-off between quality and speed, showing substantial improvements over
prior state of the art. We perform evaluations using both quantitative and
perceptual error metrics to support our analysis, and provide extensive
supplemental material to help evaluate the perceptual improvements achieved by
our methods
Scalable multi-class sampling via filtered sliced optimal transport
We propose a multi-class point optimization formulation based on continuous
Wasserstein barycenters. Our formulation is designed to handle hundreds to
thousands of optimization objectives and comes with a practical optimization
scheme. We demonstrate the effectiveness of our framework on various sampling
applications like stippling, object placement, and Monte-Carlo integration. We
a derive multi-class error bound for perceptual rendering error which can be
minimized using our optimization. We provide source code at
https://github.com/iribis/filtered-sliced-optimal-transport.Comment: 15 pages, 17 figures, ACM Trans. Graph., Vol. 41, No. 6, Article 261.
Publication date: December 202
End-to-end Sampling Patterns
Sample patterns have many uses in Computer Graphics, ranging from procedural
object placement over Monte Carlo image synthesis to non-photorealistic
depiction. Their properties such as discrepancy, spectra, anisotropy, or
progressiveness have been analyzed extensively. However, designing methods to
produce sampling patterns with certain properties can require substantial
hand-crafting effort, both in coding, mathematical derivation and compute time.
In particular, there is no systematic way to derive the best sampling algorithm
for a specific end-task.
Tackling this issue, we suggest another level of abstraction: a toolkit to
end-to-end optimize over all sampling methods to find the one producing
user-prescribed properties such as discrepancy or a spectrum that best fit the
end-task. A user simply implements the forward losses and the sampling method
is found automatically -- without coding or mathematical derivation -- by
making use of back-propagation abilities of modern deep learning frameworks.
While this optimization takes long, at deployment time the sampling method is
quick to execute as iterated unstructured non-linear filtering using radial
basis functions (RBFs) to represent high-dimensional kernels. Several important
previous methods are special cases of this approach, which we compare to
previous work and demonstrate its usefulness in several typical Computer
Graphics applications. Finally, we propose sampling patterns with properties
not shown before, such as high-dimensional blue noise with projective
properties
Patternshop: Editing Point Patterns by Image Manipulation
Point patterns are characterized by their density and correlation. While
spatial variation of density is well-understood, analysis and synthesis of
spatially-varying correlation is an open challenge. No tools are available to
intuitively edit such point patterns, primarily due to the lack of a compact
representation for spatially varying correlation. We propose a low-dimensional
perceptual embedding for point correlations. This embedding can map point
patterns to common three-channel raster images, enabling manipulation with
off-the-shelf image editing software. To synthesize back point patterns, we
propose a novel edge-aware objective that carefully handles sharp variations in
density and correlation. The resulting framework allows intuitive and
backward-compatible manipulation of point patterns, such as recoloring,
relighting to even texture synthesis that have not been available to 2D point
pattern design before. Effectiveness of our approach is tested in several user
experiments.Comment: 14 pages, 16 figure
Analysis of Sample Correlations for Monte Carlo Rendering
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.publishe
Monte Carlo Convergence Analysis for Anisotropic Sampling Power Spectra
Traditional Monte Carlo (MC) integration methods use point samples to numerically approximate the underlying integral. This approximation introduces variance in the integrated result, and this error can depend critically on the sampling patterns used during integration. Most of the well known samplers used for MC integration in graphics, e.g. jitter, Latin hypercube (n-rooks), multi-jitter, are anisotropic in nature. However, there are currently no tools available to analyze the impact of such anisotropic samplers on the variance convergence behavior of Monte Carlo integration. In this work, we propose a mathematical tool in the Fourier domain that allows analyzing the variance, and subsequently the convergence rate, of Monte Carlo integration using any arbitrary (anisotropic) sampling power spectrum. We apply our analysis to common anisotropic point sampling strategies in Monte Carlo integration, and extend our analysis to recent Monte Carlo approaches relying on line samples which have inherently anisotropic power spectra. We validate our theoretical results with several experiments using both point and line samples
Analyse de variance et échantillonnage pour l'intégration Monte Carlo sur la sphère
This dissertation introduces a theoretical framework to study different sampling patterns in the spherical domain and their effects in the evaluation of global illumination integrals. Evaluating illumination (light transport) is one of the most essential aspect in image synthesis to achieve realism which involves solving multi-dimensional space integrals. Monte Carlo based numerical integration schemes are heavily employed to solve these high dimensional integrals. One of the most important aspect of any numerical integration method is sampling. The way samples are distributed on an integration domain can greatly affect the final result. For example, in images, the effects of various sampling patterns appear in the form of either structural artifacts or completely unstructured noise. In many cases, we may get completely false (biased) results due to the sampling pattern used in integration. The distribution of sampling patterns can be characterized using their Fourier power spectra. It is also possible to use the Fourier power spectrum as input, to generate the corresponding sample distribution. This further allows spectral control over the sample distributions. Since this spectral control allows tailoring new sampling patterns directly from the input Fourier power spectrum, it can be used to improve error in integration. However, a direct relation between the error in Monte Carlo integration and the sampling power spectrum is missing. In this work, we propose a variance formulation, that establishes a direct link between the variance in Monte Carlo integration and the power spectra of both the sampling pattern and the integrand involved. To derive our closed-form variance formulation, we use the notion of homogeneous sample distributions that allows expression of error in Monte Carlo integration, only in the form of variance. Based on our variance formulation, we develop an analysis tool that can be used to derive theoretical variance convergence rates of various state-of-the-art sampling patterns. Our analysis gives insights to design principles that can be used to tailor new sampling patterns based on the integrandCette thèse introduit un cadre théorique pour l'étude de différents schémas d'échantillonnage dans un domaine sphérique, et de leurs effets sur le calcul d'intégrales pour l'illumination globale. Le calcul de l'illumination (du transport lumineux) est un composant majeur de la synthèse d'images réalistes, qui se traduit par l'évaluation d'intégrales multidimensionnelles. Les schémas d'intégration numériques de type Monte-Carlo sont utilisés intensivement pour le calcul de telles intégrales. L'un des aspects majeurs de tout schéma d'intégration numérique est l'échantillonnage. En effet, la façon dont les échantillons sont distribués dans le domaine d'intégration peut fortement affecter le résultat final. Par exemple, pour la synthèse d'images, les effets liés aux différents schémas d'échantillonnage apparaissent sous la forme d'artéfacts structurés ou, au contrire, de bruit non structuré. Dans de nombreuses situations, des résultats complètement faux (biaisés) peuvent être obtenus à cause du schéma d'échantillonnage utilisé pour réaliser l'intégration. La distribution d'un échantillonnage peut être caractérisée à l'aide de son spectre de Fourier. Des schémas d'échantillonnage peuvent être générés à partir d'un spectre de puissance dans le domaine de Fourier. Cette technique peut être utilisée pour améliorer l'erreur d'intégration, car un tel contrôle spectral permet d'adapter le schéma d'échantillonnage au spectre de Fourier de l'intégrande. Il n'existe cependant pas de relation directe entre l'erreur dans l'intégration par méthode de Monte-Carlo et le spectre de puissance de la distribution des échantillons. Dans ces travaux, nous proposons une formulation de la variance qui établit un lien direct entre la variance d'une méthode de Monte-Carlo, les spectres de puissance du schéma d'échantillonnage ainsi que de l'intégrande. Pour obtenir notre formulation de la variance, nous utilisons la notion d'homogénéité de la distribution des échantillons qui permet d'exprimer l'erreur de l'intégration par une méthode de Monte-Carlo uniquement sous forme de variance. À partir de cette formulation de la variance, nous développons un outil d'analyse pouvant être utilisé pour déterminer le taux de convergence théorique de la variance de différents schémas d'échantillonnage proposés dans la littérature. Notre analyse fournit un éclairage sur les bonnes pratiques à mettre en œuvre dans la définition de nouveaux schémas d'échantillonnage basés sur l'intégrand