A New Mathematical Development for Radiosity Animation with Galerkin

International audienceCombining animation and global illumination constitutes, at present, a true challenge in computer graphics, especially when light sources move in a complex scene because the entire illumination has to be recomputed. This paper introduces a new algorithm, based on the Galerkin method, which can efficiently manage any moving surface -even light source- to compute animation sequences. For each new frame of a sequence, we take into account the continuous property of the moves to determine the necessary energy differences between the previous global illumination solution and the new one. Based on a mathematical development of the form factor, this new approach leads to an efficient and simple algorithm, similar to the classical progressive refinement algorithm, and which computes animated sequence about three times faster

A constructive theory of sampling for image synthesis using reproducing kernel bases

Sampling a scene by tracing rays and reconstructing an image from such pointwise samples is fundamental to computer graphics. To improve the efficacy of these computations, we propose an alternative theory of sampling. In contrast to traditional formulations for image synthesis, which appeal to nonconstructive Dirac deltas, our theory employs constructive reproducing kernels for the correspondence between continuous functions and pointwise samples. Conceptually, this allows us to obtain a common mathematical formulation of almost all existing numerical techniques for image synthesis. Practically, it enables novel sampling based numerical techniques designed for light transport that provide considerably improved performance per sample. We exemplify the practical benefits of our formulation with three applications: pointwise transport of color spectra, projection of the light energy density into spherical harmonics, and approximation of the shading equation from a photon map. Experimental results verify the utility of our sampling formulation, with lower numerical error rates and enhanced visual quality compared to existing techniques

A Theoretical Analysis of Compactness of the Light Transport Operator

International audienceRendering photorealistic visuals of virtual scenes requires tractable models for the simulation of light. The rendering equation describes one such model using an integral equation, the crux of which is a continuous integral operator. A majority of rendering algorithms aim to approximate the effect of this light transport operator via discretization (using rays, particles, patches, etc.). Research spanning four decades has uncovered interesting properties and intuition surrounding this operator. In this paper we analyze compactness, a key property that is independent of its discretization and which characterizes the ability to approximate the operator uniformly by a sequence of finite rank operators. We conclusively prove lingering suspicions that this operator is not compact and therefore that any discretization that relies on a finite-rank or nonadaptive finite-bases is susceptible to unbounded error over arbitrary light distributions. Our result justifies the expectation for rendering algorithms to be evaluated using a variety of scenes and illumination conditions. We also discover that its lower dimensional counterpart (over purely diffuse scenes) is not compact except in special cases, and uncover connections with it being noninvertible and acting as a low-pass filter. We explain the relevance of our results in the context of previous work. We believe that our theoretical results will inform future rendering algorithms regarding practical choices.Le rendu d'images photoréalistes de scènes virtuelles nécessite la simulation du transport lumineux. L'équation du rendu décrit un tel modèle à l'aide d'une équation intégrale, ou intervient un opérateur intégral continu. Une part significative des d'algorithmes de rendu visent à approximer l'effet de cet opérateur via une discrétisation (à l'aide de rayons, de particules, de patchs, etc.). Quatre décennies de recherches ont mis à jour des propriétés et une intuition entourant cet opérateur. Dans cet article, nous analysons sa compacité, une propriété clé qui est indépendante de la discrétisation et qui caractérise la possibilité d'approcher uniformément l'opérateur par une suite d'opérateurs de rang fini. Nous justifions les soupçons persistants que cet opérateur n'est pas compact et donc que toute discrétisation qui repose sur un rang fini ou des bases finies non adaptatives n'apporte pas de guarantie d'erreur sur des distributions de lumière arbitraires. Notre résultat justifie le besoin d'évaluer chaque méthode en utilisant une variété de scènes et de conditions d'éclairage. Nous montrons également que son homologue de dimension inférieure (sur des scènes purement diffuses) n'est pas compact sauf dans des cas particuliers, et établissons un lien avec le fait qu'il est non inversible et agit comme un filtre passe-bas. Nous expliquons la pertinence de nos résultats dans le contexte de travaux antérieurs. Nous pensons que nos résultats théoriques éclaireront les futurs algorithmes de rendu concernant les choix pratiques

Novel illumination algorithms for off-line and real-time rendering

This thesis presents new and efficient illumination algorithms for off-line and real-time rendering. The realistic rendering of arbitrary indirect illumination is a difficult task. Assuming ray optics model of light, the rendering equation describes the propagation of light in the scene with high accuracy. However, the computation is expensive, and thus even in off-line rendering, i.e., in prerendered animations, indirect illumination is often approximated as it would otherwise constitute a bottleneck in the production pipeline. Indirect illumination can be computed using Monte Carlo integration, but when restrained to a reasonable amount of computation time, the result is often corrupted by noise. This thesis includes a method that effectively reduces the noise by applying a spatially varying filter to the noisy illumination. For real-time performance, some components of indirect illumination can be precomputed. Irradiance volume and many variations of it precompute reflections and shadowing of a static scene into a volumetric data structure. This data is then used to shade dynamic objects in real-time. The practical usage of the method is limited due to aliasing artifacts. This thesis shows that with a suitable super-sampling approach, a significant quality improvement can be obtained. Another direction is to precompute how light propagates in the scene and use the precomputed data during run-time to solve both direct and indirect illumination based on the known incident lighting. To keep the memory and precomputation costs tractable, these methods are typically restricted to infinitely distant lighting. Those that are not, require a very long precomputation time. This thesis presents an algorithm that adopts a wavelet-based hierarchical finite element method for the precomputation. A significant performance improvement over the existing techniques is obtained. When full global illumination cannot be afforded, ambient occlusion is an attractive alternative. This thesis includes two methods for real-time rendering of ambient occlusion in dynamic scenes. The first method models the shadowing of ambient light between rigid moving bodies. The second method gives a data-oriented solution for rendering approximate ambient occlusion for animated characters in real-time. Both methods achieve unprecedented efficiency.reviewe

Development and Verification of Enclosure Radiation Capabilities in the CHarring Ablator Response (CHAR) Code

With the recent development of multi-dimensional thermal protection system (TPS) material response codes, the capability to account for surface-to-surface radiation exchange in complex geometries is critical. This paper presents recent efforts to implement such capabilities in the CHarring Ablator Response (CHAR) code developed at NASA's Johnson Space Center. This work also describes the different numerical methods implemented in the code to compute geometric view factors for radiation problems involving multiple surfaces. Verification of the code's radiation capabilities and results of a code-to-code comparison are presented. Finally, a demonstration case of a two-dimensional ablating cavity with enclosure radiation accounting for a changing geometry is shown

Foundations of realistic rendering : a mathematical approach

Die vorliegende Dissertation ist keine gewöhnliche Abhandlung, sondern sie ist als Lehrbuch zum realistischen Rendering für Studenten im zweiten Studienabschnitt, sowie Forscher und am Thema Interessierte konzipiert. Aus mathematischer Sicht versteht man unter realistischem Rendering das Lösen der stationären Lichttransportgleichung, einer komplizierten Fredholm Integralgleichung der 2tenArt, deren exakte Lösung, wenn überhaupt berechenbar, nur in einem unendlich- dimensionalen Funktionenraum existiert. Während in den existierenden Büchern, die sich mit globaler Beleuchtungstheorie beschäftigen, vorwiegend die praktische Implementierung von Lösungsansätzen im Vordergrund steht, sind wir eher daran interessiert, den Leser mit den mathematischen Hilfsmitteln vertraut zu machen, mit welchen das globale Beleuchtungsproblem streng mathematisch formuliert und letzendlich auch gelöst werden kann. Neue, effzientere und elegantere Algorithmen zur Berechnung zumindest approxima- tiver Lösungen der Lichttransportgleichung und ihrer unterschiedlichen Varianten können nur im Kontext mit einem vertieften Verständnis der Lichttransportgleichung entwickelt werden. Da die Probleme des realistischen Renderings tief in verschiedenen mathematis- chen Disziplinen verwurzelt sind, setzt das vollständige Verständnis des globalen Beleuch- tungsproblems Kenntnisse aus verschiedenen Bereichen der Mathematik voraus. Als zen- trale Konzepte kristallisieren sich dabei Prinzipien der Funktionalanalysis, der Theorie der Integralgleichungen, der Maß- und Integrationstheorie sowie der Wahrscheinlichkeitstheo- rie heraus. Wir haben uns zum Ziel gesetzt, dieses Knäuel an mathematischen Konzepten zu entflechten, sie für Studenten verständlich darzustellen und ihnen bei Bedarf und je nach speziellem Interesse erschöpfend Auskunft zu geben.The available doctoral thesis is not a usual paper but it is conceived as a text book for realistic rendering, made for students in upper courses, as well as for researchers and interested people. From mathematical point of view, realistic rendering means solving the stationary light transport equation, a complicated Fredholm Integral equation of 2nd kind. Its exact solution exists|if possible at all|in an infinite dimensional functional space. Whereas practical implementation of approaches for solving problems are in the center of attentionin the existing textbooks that treat global illumination theory, we are more interested in familiarizing our reader with the mathematical tools which permit them to formulate the global illumination problem in accordance with strong mathematical principles and last but not least to solve it. New, more eficient and more elegant algorithms to calculate approximate solutions for the light transport equation and their different variants must be developed in the context of deep and complete understanding of the light transport equation. As the problems of realistic rendering are deeply rooted in different mathematical disciplines, there must precede the complete comprehension of all those areas. There are evolving principles of functional analysis, theory of integral equations, measure and integration theory as well as probability theory. We have set ourselves the target to remerge this bundle of fluff of mathematical concepts and principles, to represent them to the students in an understandable manner, and to give them, if required, exhaustive information

A custom designed density estimation method for light transport

We present a new Monte Carlo method for solving the global illumination problem in environments with general geometry descriptions and light emission and scattering properties. Current Monte Carlo global illumination algorithms are based on generic density estimation techniques that do not take into account any knowledge about the nature of the data points --- light and potential particle hit points --- from which a global illumination solution is to be reconstructed. We propose a novel estimator, especially designed for solving linear integral equations such as the rendering equation. The resulting single-pass global illumination algorithm promises to combine the flexibility and robustness of bi-directional path tracing with the efficiency of algorithms such as photon mapping

Development, Verification and Validation of Enclosure Radiation Capabilities in the CHarring Ablator Response (CHAR) Code

With the recent development of multi-dimensional thermal protection system (TPS) material response codes including the capabilities to account for radiative heating is a requirement. This paper presents the recent efforts to implement such capabilities in the CHarring Ablator Response (CHAR) code developed at NASA's Johnson Space Center. This work also describes the different numerical methods implemented in the code to compute view factors for radiation problems involving multiple surfaces. Furthermore, verification and validation of the code's radiation capabilities are demonstrated by comparing solutions to analytical results, to other codes, and to radiant test data

A 'Surface-based' Geometrical Acoustic formulation within a Galerkin Boundary Element framework

As sound propagates within a room, it experiences high-order reflection, diffraction, and scattering. This causes the reflection density to increase over time, such that the sound field becomes diffuse and chaotic. Under these conditions, there is little benefit in running a computationally costly full wave solver – if it is even feasible – so methods based on Geometrical models of acoustic propagation prevail. Raytracing is currently the de-facto method for this late-time, high-frequency regime in room acoustic modelling. It samples the propagating distributions of acoustic intensity by launching a set of rays and individually tracing their trajectories. This is computationally efficient when only specular reflections are present, but accurate inclusion of scattering or diffraction requires ‘ray-splitting’ to be introduced, causing an exponential increase in computational cost with reflection order, crippling the algorithm. Hence only crude Monte Carlo implantations of these processes are tractable with Raytracing.An emerging solution for modelling late-reflections is “Surface-Based” Geometrical Acoustics. These formulations map a distribution of rays arriving at a boundary onto a pre- defined ‘approximation space’ of basis functions spanning position and angle, so the sound field is represented by a vector of boundary coefficients. Re-radiation of subsequent reflections is thus reduced to a matrix multiplication, with the steady-state solvable via a Neumann series. As rays only propagate one reflection order before being collected, the multiple ‘child’ rays that would be produced by scattering or diffraction of a ‘parent’ ray at the boundary are absorbed into the ‘approximation’ space at each reflection order. This maintains a fixed number of degrees of freedom and a linear computational cost with reflection order. This thesis presents a Surface-Based Geometrical Acoustic formulation cast in a Galerkin Boundary Element framework.This thesis presents and implements the formulation in two dimensions and validates it against an Image Source Model for a rectangular room. The Galerkin Boundary Element scheme expediates comparison of different approximation schemes and their effect on convergence and accuracy can be easily studied. Examination of the resulting power distributions on the boundary for early reflections show power being smudged over a range of reflection angles, indicating approximation in the scheme. But this is perceptually appropriate for late-time diffuse fields as individual reflections will no longer be distinguishable, and late time energy decay rates are shown to be correct. Receiver responses for early reflections show very good agreement also, so long as angular resolution is set sufficiently high. The formulation is shown to converge with the number of angular degrees of freedom as well as smaller element sizes. The results show a high degree of accuracy and identical convergence trends when using continuous orthogonal polynomials, such as Legendre, Chebyshev or Lobatto, as angular basis. In contrast, other functions, such as continuous piecewise-linear, or discontinuous piecewise- constant, exhibit a significant degree of approximation for higher interpolation orders due to their discontinuous or non-smooth nature. Solutions in Geometrical Acoustics can be discontinuous. The ultimate ambition in formulating the model presented in this thesis is to include diffraction, and solutions when it is included will be continuous. This capability still remains as work for the future, but the choices made in this thesis were informed by that end goal