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Fast spectral multiplication for real-time renderingWaddle, C Allen 02 May 2018 (has links)
In computer graphics, the complex phenomenon of color appearance, involving the interaction of light, matter and the human visual system, is modeled by the multiplication of RGB triplets assigned to lights and materials. This efficient heuristic produces plausible images because the triplets assigned to materials usually function as color specifications. To predict color, spectral rendering is required, but the O(n) cost of computing reflections with n-dimensional point-sampled spectra is prohibitive for real-time rendering.
Typical spectra are well approximated by m-dimensional linear models, where m << n, but computing reflections with this representation requires O(m^2) matrix-vector multiplication. A method by Drew and Finlayson [JOSA A 20, 7 (2003), 1181-1193], reduces this cost to O(m) by “sharpening” an n x m orthonormal basis with a linear transformation, so that the new basis vectors are approximately disjoint. If successful, this transformation allows approximated reflections to be computed as the products of coefficients of lights and materials. Finding the m x m change of basis matrix requires solving m eigenvector problems, each needing a choice of wavelengths in which to sharpen the corresponding basis vector. These choices, however, are themselves an optimization problem left unaddressed by the method's authors.
Instead, we pose a single problem, expressing the total approximation error incurred across all wavelengths as the sum of dm^2 squares for some number d, where, depending on the inherent dimensionality of the rendered reflectance spectra, m <= d << n, a number that is independent of the number of approximated reflections. This problem may be solved in real time, or nearly, using standard nonlinear optimization algorithms. Results using a variety of reflectance spectra and three standard illuminants yield errors at or close to the best lower bound attained by projection onto the leading m characteristic vectors of the approximated reflections. Measured as CIEDE2000 color differences, a heuristic proxy for image difference, these errors can be made small enough to be likely imperceptible using values of 4 <= m <= 9.
An examination of this problem reveals a hierarchy of simpler, more quickly solved subproblems whose solutions yield, in the typical case, increasingly inaccurate approximations. Analysis of this hierarchy explains why, in general, the lowest approximation error is not attained by simple spectral sharpening, the smallest of these subproblems, unless the spectral power distributions of all light sources in a scene are sufficiently close to constant functions. Using the methods described in this dissertation, spectra can be rendered in real time as the products of m-dimensional vectors of sharp basis coefficients at a cost that is, in a typical application, a negligible fraction above the cost of RGB rendering. / Graduate
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Géométrie et optimisation riemannienne pour la diagonalisation conjointe : application à la séparation de sources d'électroencéphalogrammes / Riemannian geometry and optimization for approximate joint diagonalization : application to source separation of electroencephalogramsBouchard, Florent 22 November 2018 (has links)
La diagonalisation conjointe approximée d’un ensemble de matrices permet de résoudre le problème de séparation aveugle de sources et trouve de nombreuses applications, notamment pour l’électroencéphalographie, une technique de mesure de l’activité cérébrale.La diagonalisation conjointe se formule comme un problème d’optimisation avec trois composantes : le choix du critère à minimiser, la contrainte de non-dégénérescence de la solution et l’algorithme de résolution.Les approches existantes considèrent principalement deux critères, les moindres carrés et la log-vraissemblance.Elles sont spécifiques à une contrainte et se restreignent à un seul type d’algorithme de résolution.Dans ce travail de thèse, nous proposons de formuler le problème de diagonalisation conjointe selon un modèle géométrique, qui généralise les travaux précédents et permet de définir des critères inédits, notamment liés à la théorie de l’information.Nous proposons également d’exploiter l’optimisation riemannienne et nousdéfinissons un ensemble d’outils qui permet de faire varier les trois composantes indépendamment, créant ainsi de nouvelles méthodes et révélant l’influence des choix de modélisation.Des expériences numériques sur des données simulées et sur des enregistrements électroencéphalographiques montrent que notre approche par optimisation riemannienne donne des résultats compétitifs par rapport aux méthodes existantes.Elles indiquent aussi que les deux critères traditionnels ne sont pas les meilleurs dans toutes les situations. / The approximate joint diagonalisation of a set of matrices allows the solution of the blind source separation problem and finds several applications, for instance in electroencephalography, a technique for measuring brain activity.The approximate joint diagonalisation is formulated as an optimization problem with three components: the choice of the criterion to be minimized, the non-degeneracy constraint on the solution and the solving algorithm.Existing approaches mainly consider two criteria, the least-squares and the log-likelihood.They are specific to a constraint and are limited to only one type of solving algorithms.In this thesis, we propose to formulate the approximate joint diagonalisation problem in a geometrical fashion, which generalizes previous works and allows the definition of new criteria, particularly those linked to information theory.We also propose to exploit Riemannian optimisation and we define tools that allow to have the three components varying independently, creating in this way new methods and revealing the influence of the choice of the model.Numerical experiments on simulated data as well as on electroencephalographic recordings show that our approach by means of Riemannian optimisation gives results that are competitive as compared to existing methods.They also indicate that the two traditional criteria do not perform best in all situations.
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Blind source separation based on joint diagonalization of matrices with applications in biomedical signal processingZiehe, Andreas January 2005 (has links)
<p>This thesis is concerned with the solution of the blind source
separation problem (BSS). The BSS problem occurs frequently in various
scientific and technical applications. In essence, it consists in
separating meaningful underlying components out of a mixture of a
multitude of superimposed signals.</p>
<P>
In the recent research literature there are two related approaches to
the BSS problem: The first is known as Independent Component Analysis (ICA),
where the goal is to transform the data such that the components
become as independent as possible. The second is based on the notion
of diagonality of certain characteristic matrices derived from the
data. Here the goal is to transform the matrices such that they become
as diagonal as possible. In this thesis we study
the latter method of approximate joint diagonalization (AJD) to
achieve a solution of the BSS problem. After an introduction to the
general setting, the thesis provides an overview on particular choices
for the set of target matrices that can be used for BSS by joint
diagonalization.</p>
<P>
As the main contribution of the thesis, new algorithms for
approximate joint diagonalization of several matrices with
non-orthogonal transformations are developed.</p>
<P>
These newly developed algorithms will be tested on synthetic
benchmark datasets and compared to other previous diagonalization
algorithms.</p>
<P>
Applications of the BSS methods to biomedical signal processing are
discussed and exemplified with real-life data sets of multi-channel
biomagnetic recordings.</p> / <p>Diese Arbeit befasst sich mit der Lösung des Problems der blinden
Signalquellentrennung (BSS). Das BSS Problem tritt häufig in vielen
wissenschaftlichen und technischen Anwendungen auf. Im Kern besteht das
Problem darin, aus einem Gemisch von überlagerten Signalen die
zugrundeliegenden Quellsignale zu extrahieren.</p>
<P>
In wissenschaftlichen Publikationen zu diesem Thema werden
hauptsächlich zwei Lösungsansätze verfolgt:</p>
<P>
Ein Ansatz ist die sogenannte "Analyse der unabhängigen
Komponenten", die zum Ziel hat, eine lineare Transformation <B>V</B> der
Daten <B>X</B> zu finden, sodass die Komponenten U<sub>n</sub> der transformierten
Daten <B>U</B> = <B> V X</B> (die sogenannten "independent components") so
unabhängig wie möglich sind.
Ein anderer Ansatz beruht auf einer simultanen Diagonalisierung
mehrerer spezieller Matrizen, die aus den Daten gebildet werden.
Diese Möglichkeit der Lösung des Problems der blinden
Signalquellentrennung bildet den Schwerpunkt dieser Arbeit.</p>
<P>
Als Hauptbeitrag der vorliegenden Arbeit präsentieren wir neue
Algorithmen zur simultanen Diagonalisierung mehrerer Matrizen mit
Hilfe einer nicht-orthogonalen Transformation.</p>
<P>
Die neu entwickelten Algorithmen werden anhand von numerischen
Simulationen getestet und mit bereits bestehenden
Diagonalisierungsalgorithmen verglichen. Es zeigt sich, dass unser
neues Verfahren sehr effizient und leistungsfähig ist. Schließlich
werden Anwendungen der BSS Methoden auf Probleme der biomedizinischen
Signalverarbeitung erläutert und anhand von realistischen
biomagnetischen Messdaten wird die Nützlichkeit in der explorativen
Datenanalyse unter Beweis gestellt.</p>
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EEG Source AnalysisCongedo, Marco 22 October 2013 (has links) (PDF)
Electroencephalographic data recorded on the human scalp can be modeled as a linear mixture of underlying dipolar source generators. The characterization of such generators is the aim of several families of signal processing methods. In this HDR we consider in several details three of such families, namely 1) EEG distributed inverse solutions, 2) diagonalization methods, including spatial filtering and blind source separation and 3) Riemannian geometry. We highlight our contributions in each of this family, we describe algorithms reporting all necessary information to make purposeful use of these methods and we give numerous examples with real data pertaining to our published studies. Traditionally only the single-subject scenario is considered; here we consider in addition the extension of some methods to the simultaneous multi-subject recording scenario. This HDR can be seen as an handbook for EEG source analysis. It will be particularly useful to students and other colleagues approaching the field.
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