A novel image reconstruction algorithm from edges (image gradients) follows from the Sokhostki-Plemelj Theorem of complex analysis, an elaboration of the standard Cauchy (Singular) Integral. This algorithm demonstrates the use of Singular Integral Equation methods to image processing, extending the more common use of Partial Differential Equations (e.g. based on variants of the Diffusion or Poisson equations). The Cauchy Integral approach has a deep connection to and sheds light on the (linear and non-linear) diffusion equation, the retinex algorithm and energy-based image regularization. It extends the commonly understood local definition of an edge to a global, complex analytic structure - the analytic edge - the contrast weighted kernel of the Cauchy Integral. Superposition of the set of analytic edges provides a "filled-in" image which is the piece-wise analytic image corresponding to the edge (gradient data) supplied. This is a fully parallel operation which avoids the time penalty associated with iterative solutions and thus is compatible with the short time (about 150 milliseconds) that is biologically available for the brain to construct a perceptual image from edge data. Although this algorithm produces an exact reconstruction of a filled-in image from the gradients of that image, slight modifications of it produce images which correspond to perceptual reports of human observers when presented with a wide range of "visual contrast illusion" images.

This paper is an examination of the theory of discrete complex analysis that arises from the framework of a planar cell complex. Construction of this theory is largely integration-based. A combination of two cell complexes, the double and its associated diamond complex, allows for the development of a discrete Cauchy Integral Formula. / Master of Science

The paper is devoted to the foundation of approximation methods for integral equations of the form (aI+SbI+K)f=g, where S is the Cauchy singular integral operator on (-1,1) and K is a weakly singular integral operator. Here a,b,g are given functions on (-1,1) and the unknown function f on (-1,1) is looked for. It is assumed that a and b are real-valued and Hölder continuous functions on [-1,1] without common zeros and that g belongs to some weighted space of Hölder continuous functions. In particular, g may have a finite number of singularities. Based on known spectral properties of Cauchy singular integral operators approximation methods for the numerical solution of the above equation are constructed, where both aspects the theoretical convergence and the numerical practicability are taken into account. The weighted uniform convergence of these methods is studied using a general approach based on the theory of approximation spaces. With the help of this approach it is possible to prove simultaneously the stability, the convergence and results on the order of convergence of the approximation methods under consideration.

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The paper is devoted to the foundation of approximation methods for integral equations of the form (aI+SbI+K)f=g, where S is the Cauchy singular integral operator on (-1,1) and K is a weakly singular integral operator. Here a,b,g are given functions on (-1,1) and the unknown function f on (-1,1) is looked for. It is assumed that a and b are real-valued and Hölder continuous functions on [-1,1] without common zeros and that g belongs to some weighted space of Hölder continuous functions. In particular, g may have a finite number of singularities. Based on known spectral properties of Cauchy singular integral operators approximation methods for the numerical solution of the above equation are constructed, where both aspects the theoretical convergence and the numerical practicability are taken into account. The weighted uniform convergence of these methods is studied using a general approach based on the theory of approximation spaces. With the help of this approach it is possible to prove simultaneously the stability, the convergence and results on the order of convergence of the approximation methods under consideration.

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The aim of this thesis is to give a mathematical framework for scattering of electromagnetic waves by rough surfaces. We prove that the Maxwell transmission problem with a weakly Lipschitz interface,in finite energy norms, is well posed in Fredholm sense for real frequencies. Furthermore, we give precise conditions on the material constants ε, μ and σ and the frequency ω when this transmission problem is well posed. To solve the Maxwell transmission problem, we embed Maxwell’s equations in an elliptic Dirac equation. We develop a new boundary integral method to solve the Dirac transmission problem. This method uses a boundary integral operator, the rotation operator, which factorises the double layer potential operator. We prove spectral estimates for this rotation operator in finite energy norms using Hodge decompositions on weakly Lipschitz domains. To ensure that solutions to the Dirac transmission problem indeed solve Maxwell’s equations, we introduce an exterior/interior derivative operator acting in the trace space. By showing that this operator commutes with the two basic reflection operators, we are able to prove that the Maxwell transmission problem is well posed. We also prove well-posedness for a class of oblique Dirac transmission problems with a strongly Lipschitz interface, in the L_2 space on the interface. This is shown by employing the Rellich technique, which gives angular spectral estimates on the rotation operator.

Dans cette thèse, nous établissons un modèle à deux échelles à la fois pour desmatrices de cantilevers unidimensionnels et bidimensionnels en régime de fonctionnementélastodynamique avec des applications possibles aux réseaux de microscopesà force atomique (AFM). Son élaboration est basée sur une analyseasymptotique pour les structures minces élastiques, une approximation à deuxéchelles et une mise à l’échelle utilisée pour l’homogénéisation des milieux fortementhétérogènes. Nous complétons la théorie de l’approximation à deux échellespour les problèmes aux limites du quatrième ordre posés dans des domaines mincespériodiques connexes seulement dans certaines directions. Notre modèle reproduitla dynamique globale du support ainsi que les mouvements locaux des cantilevers.Pour simplifier la suite du travail, nous concentrons nos travaux à l’étude de matricesde leviers constituées de lignes découplées en régime dynamique. Comme lesupport des leviers est élastique, l’effet du couplage entre levier est pris en compte.La vérification du modèle est soigneusement réalisée. Nous montrons que chaquemode propre peut être décomposé en produits d’un mode de base avec un modede levier. Nous présentons une méthode de discrétisation du modèle et effectuonssa vérification numérique en la comparant avec des résultats de simulation paréléments finis du problème d’élasticité tridimensionnel. Par ailleurs, nous avonsélaboré de nouveaux outils d’aide à la conception de réseaux d’AFM. Une boîte àoutils d’optimisation robuste est interfacée avec le modèle permettant d’optimiserun design avant micro-Fabrication. Un algorithme d’estimation de l’état statiquecombinant la mesure de déplacements mécaniques par interférométrie et le modèlea été introduit. Nous avons également synthétisé un régulateur quadratiquelinéaire (LQR) pour un réseau de cantilevers en mode dynamique comprenant actionneurset capteurs régulièrement espacées. Dans le but de mettre en oeuvre lecontrôle en temps réel, nous proposons une approximation semi-Décentralisée quipeut être réalisé par un circuit électronique distribué analogique. Plus précisément,notre processeur analogique peut être réalisé par un réseau périodique derésistances (PNR). La méthode d’approximation de commande est basée sur deuxconcepts généraux, à savoir sur un calcul fonctionnel (c’est-À-Dire des fonctionsd’opérateurs) et sur la formule de représentation d’une fonction d’opérateur deDunford-Schwartz. Cette méthode d’approximation est étendue pour la résolutiond’un problème de filtrage optimal robuste de type H∞ de la dynamique d’un réseaude leviers couplés avec sources aléatoires de bruit. / In this dissertation, we establish a two-Scale model both for one-Dimensionaland two-Dimensional Cantilever Arrays in elastodynamic operating regime withpossible applications to Atomic Force Microscope (AFM) Arrays. Its derivationis based on an asymptotic analysis for thin elastic structures, a two-Scale approximationand a scaling used for strongly heterogeneous media homogenization. Wecomplete the theory of two-Scale approximation for fourth order boundary valueproblems posed in thin periodic domains connected in some directions only. Ourmodel reproduces the global dynamics as well as each of the cantilever motion. Forthe sake of simplicity, we present a simplified model of mechanical behavior of largecantilever arrays with decoupled rows in the dynamic operating regime. Since thesupporting bases are assumed to be elastic, cross-Talk effect between cantileversis taken into account. The verification of the model is carefully conducted. Weexplain not only how each eigenmode is decomposed into products of a base modewith a cantilever mode but also the method used for its discretization, and reportresults of its numerical validation with full three-Dimensional Finite Element simulations.We show new tools developed for Arrays of Microsystems and especiallyfor AFM array design. A robust optimization toolbox is interfaced to aid for designbefore the microfabrication process. A model based algorithm of static stateestimation using measurement of mechanical displacements by interferometry ispresented. We also synthesize a controller based on Linear Quadratic Regulator(LQR) methodology for a one-Dimensional cantilever array with regularly spacedactuators and sensors. With the purpose of implementing the control in real time,we propose a semi-Decentralized approximation that may be realized by an analogdistributed electronic circuit. More precisely, our analog processor is made by PeriodicNetwork of Resistances (PNR). The control approximation method is basedon two general concepts, namely on functions of operators and on the Dunford-Schwartz representation formula. This approximation method is extended to solvea robust H∞ filtering problem of the coupled cantilevers for time-Invariant systemwith random noise effects.