L'imagerie ultrasonore (US) permet de réaliser des examens médicaux non invasifs avec des méthodes d'acquisition rapides à des coûts modérés. L'imagerie cardiaque, abdominale, fœtale, ou mammaire sont quelques-unes des applications où elle est largement utilisée comme outil de diagnostic. En imagerie US classique, des ondes acoustiques sont transmises à une région d'intérêt du corps humain. Les signaux d'écho rétrodiffusés, sont ensuite formés pour créer des lignes radiofréquences. La formation de voies (FV) joue un rôle clé dans l'obtention des images US, car elle influence la résolution et le contraste de l'image finale. L'objectif de ce travail est de modéliser la formation de voies comme un problème inverse liant les données brutes aux signaux RF. Le modèle de formation de voies proposé ici améliore le contraste et la résolution spatiale des images échographiques par rapport aux techniques de FV existants. Dans un premier temps, nous nous sommes concentrés sur des méthodes de FV en imagerie US. Nous avons brièvement passé en revue les techniques de formation de voies les plus courantes, en commencent par la méthode par retard et somme standard puis en utilisant les techniques de formation de voies adaptatives. Ensuite, nous avons étudié l'utilisation de signaux qui exploitent une représentation parcimonieuse de l'image US dans le cadre de la formation de voies. Les approches proposées détectent les réflecteurs forts du milieu sur la base de critères bayésiens. Nous avons finalement développé une nouvelle façon d'aborder la formation de voies en imagerie US, en la formulant comme un problème inverse linéaire liant les échos réfléchis au signal final. L'intérêt majeur de notre approche est la flexibilité dans le choix des hypothèses statistiques sur le signal avant la formation de voies et sa robustesse dans à un nombre réduit d'émissions. Finalement, nous présentons une nouvelle méthode de formation de voies pour l'imagerie US basée sur l'utilisation de caractéristique statistique des signaux supposée alpha-stable. / Ultrasound (US) allows non-invasive and ultra-high frame rate imaging procedures at reduced costs. Cardiac, abdominal, fetal, and breast imaging are some of the applications where it is extensively used as diagnostic tool. In a classical US scanning process, short acoustic pulses are transmitted through the region-of-interest of the human body. The backscattered echo signals are then beamformed for creating radiofrequency(RF) lines. Beamforming (BF) plays a key role in US image formation, influencing the resolution and the contrast of final image. The objective of this thesis is to model BF as an inverse problem, relating the raw channel data to the signals to be recovered. The proposed BF framework improves the contrast and the spatial resolution of the US images, compared with the existing BF methods. To begin with, we investigated the existing BF methods in medical US imaging. We briefly review the most common BF techniques, starting with the standard delay-and-sum BF method and emerging to the most known adaptive BF techniques, such as minimum variance BF. Afterwards, we investigated the use of sparse priors in creating original two-dimensional beamforming methods for ultrasound imaging. The proposed approaches detect the strong reflectors from the scanned medium based on the well-known Bayesian Information Criteria used in statistical modeling. Furthermore, we propose a new way of addressing the BF in US imaging, by formulating it as a linear inverse problem relating the reflected echoes to the signal to be recovered. Our approach offers flexibility in the choice of statistical assumptions on the signal to be beamformed and it is robust to a reduced number of pulse emissions. At the end of this research, we investigated the use of the non-Gaussianity properties of the RF signals in the BF process, by assuming alpha-stable statistics of US images.

abstract: Inverse problems model real world phenomena from data, where the data are often noisy and models contain errors. This leads to instabilities, multiple solution vectors and thus ill-posedness. To solve ill-posed inverse problems, regularization is typically used as a penalty function to induce stability and allow for the incorporation of a priori information about the desired solution. In this thesis, high order regularization techniques are developed for image and function reconstruction from noisy or misleading data. Specifically the incorporation of the Polynomial Annihilation operator allows for the accurate exploitation of the sparse representation of each function in the edge domain. This dissertation tackles three main problems through the development of novel reconstruction techniques: (i) reconstructing one and two dimensional functions from multiple measurement vectors using variance based joint sparsity when a subset of the measurements contain false and/or misleading information, (ii) approximating discontinuous solutions to hyperbolic partial differential equations by enhancing typical solvers with l1 regularization, and (iii) reducing model assumptions in synthetic aperture radar image formation, specifically for the purpose of speckle reduction and phase error correction. While the common thread tying these problems together is the use of high order regularization, the defining characteristics of each of these problems create unique challenges. Fast and robust numerical algorithms are also developed so that these problems can be solved efficiently without requiring fine tuning of parameters. Indeed, the numerical experiments presented in this dissertation strongly suggest that the new methodology provides more accurate and robust solutions to a variety of ill-posed inverse problems. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2018

Técnicas de reconstrução térmica inversa são muito usadas em diferentes aplicações tais como a determinação de propriedades térmicas de novos materiais, controle da produção de calor, temperatura em processos de manufatura, etc. Apesar da ampla aplicabilidade, o problema inverso é intrinsecamente mal condicionado e tem sido tema de trabalhos de vários pesquisadores. A solução de um problema térmico inverso tridimensional é significantemente complexa, e, assim requer uma formulação que não contenha condições experimentais não realistas tais como confinamento bidimensional e estabilidade do campo térmico com relação a mudanças em parâmetros internos. Uma das abordagens adotada é baseada na formulação variacional sobre a forma do erro quadrático para reconstrução da distribuição de condução de calor interna e coeficiente de condução de calor parietal para um problema tridimensional. Dentro desta estrutura, a natureza mal condicionada do problema se manifesta na superfície de otimização por produzir topologias problemáticas tais como, vários mínimos locais, pontos de sela, vales e platôs ao redor da solução etc. Para viabilizar a abordagem escolhida, um modelo numérico foi escrito baseado na discretização por diferenças finitas da equação diferencial governante e condições de contorno. O erro funcional foi definido pela comparação entre medidas experimentais e numéricas de temperatura. O objetivo foi realizar simulações numéricas a fim de mapear a superfície de otimização correspondente e identificar a estrutura problemática associada ou patologia, chegando assim à reconstrução do coeficiente de convecção h. / Inverse thermal reconstruction techniques are widely used in different applications such as the determination of thermal properties of new materials, control of heat generation, temperature in manufacturing processes, etc. Despite the broad range of applicability, an inverse problem is intrinsically ill conditioned and has been the subject of the work of several researchers. The solution of an inverse 3-dimesional thermal problem is significantly complex, and, thus, requires a formulation that do not contain unrealistic experimental conditions such as 2-dimensional confinement and steadiness of the thermal field with respect to changes in internal parameters. One of the most adopted is the variational formulation based on quadratic error forms for the reconstruction of the internal heat conduction distribution and convection coefficient for a 3-dimensional problem. Within this framework, the ill conditioned nature of the problem manifests itself on the optimization surface by producing problematic topologies such as contour and multiple local minima, saddle points, plateaux around the solution pit and so on. To be able to apply th method a numerical model was written based on a finite difference discretization of the governing differential equation and boundary conditions. An error functional was defined by comparing experimental and numerical measurement temperatures. Numerical simulations aiming at mapping the corresponding optimization surfaces andatidentifing the associated problematic structures or pathologies, resulting in the reconstruction of convection coefficient.

The initial question addressed in this thesis is how to take into account the multi-objective aspect of decision problems in inverse optimization. The most straightforward extension consists of finding a minimal adjustment of the objective functions coefficients such that a given feasible solution becomes efficient. However, there is not only a single question raised by inverse multi-objective optimization, because there is usually not a single efficient solution. The way we define inverse multi-objective<p>optimization takes into account this important aspect. This gives rise to many questions which are identified by a precise notation that highlights a large collection of inverse problems that could be investigated. In this thesis, a selection of inverse problems are presented and solved. This selection is motivated by their possible applications and the interesting theoretical questions they can rise in practice. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

Les problèmes inverses liés à des processus physiques sont d'une grande importance dans la plupart des domaines liés au traitement du signal, tels que la tomographie, l'acoustique, les communications sans fil, le radar, l'imagerie médicale, pour n'en nommer que quelques uns. Dans le même temps, beaucoup de ces problèmes soulèvent des défis en raison de leur nature mal posée. Par ailleurs, les signaux émanant de phénomènes physiques sont souvent gouvernées par des lois s'exprimant sous la forme d'équations aux dérivées partielles (EDP) linéaires, ou, de manière équivalente, par des équations intégrales et leurs fonctions de Green associées. De plus, ces phénomènes sont habituellement induits par des singularités, apparaissant comme des sources ou des puits d'un champ vectoriel. Dans cette thèse, nous étudions en premier lieu le couplage entre de telles lois physiques et une hypothèse initiale de parcimonie des origines du phénomène physique. Ceci donne naissance à un concept de dualité des régularisations, formulées soit comme un problème d'analyse coparcimonieuse (menant à la représentation en EDP), soit comme une parcimonie à la synthèse équivalente à la précédente (lorsqu'on fait plutôt usage des fonctions de Green). Nous dédions une part significative de notre travail à la comparaison entre les approches de synthèse et d'analyse. Nous défendons l'idée qu'en dépit de leur équivalence formelle, leurs propriétés computationnelles sont très différentes. En effet, en raison de la parcimonie héritée par la version discrétisée de l'EDP (incarnée par l'opérateur d'analyse), l'approche coparcimonieuse passe bien plus favorablement à l'échelle que le problème équivalent régularisé par parcimonie à la synthèse. Nos constatations sont illustrées dans le cadre de deux applications : la localisation de sources acoustiques, et la localisation de sources de crises épileptiques à partir de signaux électro-encéphalographiques. Dans les deux cas, nous vérifions que l'approche coparcimonieuse démontre de meilleures capacités de passage à l'échelle, au point qu'elle permet même une interpolation complète du champ de pression dans le temps et en trois dimensions. De plus, dans le cas des sources acoustiques, l'optimisation fondée sur le modèle d'analyse \emph{bénéficie} d'une augmentation du nombre de données observées, ce qui débouche sur une accélération du temps de traitement, plus rapide que l'approche de synthèse dans des proportions de plusieurs ordres de grandeur. Nos simulations numériques montrent que les méthodes développées pour les deux applications sont compétitives face à des algorithmes de localisation constituant l'état de l'art. Pour finir, nous présentons deux méthodes fondées sur la parcimonie à l'analyse pour l'estimation aveugle de la célérité du son et de l'impédance acoustique, simultanément à l'interpolation du champ sonore. Ceci constitue une étape importante en direction de la mise en œuvre de nos méthodes en en situation réelle. / Inverse problems related to physical processes are of great importance in practically every field related to signal processing, such as tomography, acoustics, wireless communications, medical and radar imaging, to name only a few. At the same time, many of these problems are quite challenging due to their ill-posed nature. On the other hand, signals originating from physical phenomena are often governed by laws expressible through linear Partial Differential Equations (PDE), or equivalently, integral equations and the associated Green’s functions. In addition, these phenomena are usually induced by sparse singularities, appearing as sources or sinks of a vector field. In this thesis we primarily investigate the coupling of such physical laws with a prior assumption on the sparse origin of a physical process. This gives rise to a “dual” regularization concept, formulated either as sparse analysis (cosparse), yielded by a PDE representation, or equivalent sparse synthesis regularization, if the Green’s functions are used instead. We devote a significant part of the thesis to the comparison of these two approaches. We argue that, despite nominal equivalence, their computational properties are very different. Indeed, due to the inherited sparsity of the discretized PDE (embodied in the analysis operator), the analysis approach scales much more favorably than the equivalent problem regularized by the synthesis approach. Our findings are demonstrated on two applications: acoustic source localization and epileptic source localization in electroencephalography. In both cases, we verify that cosparse approach exhibits superior scalability, even allowing for full (time domain) wavefield interpolation in three spatial dimensions. Moreover, in the acoustic setting, the analysis-based optimization benefits from the increased amount of observation data, resulting in a speedup in processing time that is orders of magnitude faster than the synthesis approach. Numerical simulations show that the developed methods in both applications are competitive to state-of-the-art localization algorithms in their corresponding areas. Finally, we present two sparse analysis methods for blind estimation of the speed of sound and acoustic impedance, simultaneously with wavefield interpolation. This is an important step toward practical implementation, where most physical parameters are unknown beforehand. The versatility of the approach is demonstrated on the “hearing behind walls” scenario, in which the traditional localization methods necessarily fail. Additionally, by means of a novel algorithmic framework, we challenge the audio declipping problemregularized by sparsity or cosparsity. Our method is highly competitive against stateof-the-art, and, in the cosparse setting, allows for an efficient (even real-time) implementation.

[pt] Esta dissertação apresenta uma metodologia para medição visual de campos de deformações (2D) em materiais, por meio da aplicação da técnica SIFT (Scale Invariant Feature Transform). A análise de imagens capturadas é feita por uma câmera digital em estágios diferentes durante o processo de deformação de um material quando este é submetido a esforços mecânicos. SIFT é uma das técnicas modernas de visão computacional e um eficiente filtro para extração e descrição de pontos de características relevantes em imagens, invariantes a transformações em escala, iluminação e rotação. A metodologia é baseada no cálculo do gradiente de funções que representam o campo de deformações em um material durante um ensaio mecânico sob diferentes condições de contorno. As funções são calibradas com a aplicação da análise inversa sobre o conjunto de pontos homólogos de duas imagens extraídos pelo algoritmo SIFT. A formulação da solução ao problema inverso combina os dados experimentais fornecidos pelo SIFT e o método linear de mínimos quadrados para estimação dos parâmetros de deformação. Os modelos propostos para diferentes corpos de prova são avaliados experimentalmente com a ajuda de extensômetros para medição direta das deformações. O campo de deformações identificado pelo sistema de visão computacional é comparado com os valores obtidos pelos extensômetros e por simulações feitas no programa de Elementos Finitos ANSYS. Os resultados obtidos mostram que o campo de deformações pode ser medido utilizando a técnica SIFT, gerando uma nova ferramenta visual de medição para ensaios mecânicos que não se baseia nas técnicas tradicionais de correlação de imagens. / [en] This thesis presents a methodology for measurement of strain fields in materials by applying the SIFT technique (Scale Invariant Feature Transform). The images analyzed are captured by a digital camera at different stages during the deformation process of a material when it is subjected to mechanical stress. SIFT is one of the modern computer vision techniques and an efficient filter for extraction and description of relevant feature points in images. These interest points are largely invariant to changes in scale, illumination and rotation. The methodology is based on the calculation of the gradient of the functions that represents the corresponding strain field in the material during a mechanical test under different boundary conditions. The functions are calibrated with the application of inverse analysis on the set of homologous points of two images extracted by the SIFT algorithm. The formulation of the solution to the inverse problem combines the experimental data processed by the SIFT and linear least squares method for the estimation of strain parameters. The proposed models for different specimens are evaluated experimentally with strain gauges for direct measurement of the deformations. The strain field identified by the computer vision system is compared with values obtained by strain gauges and simulations with the ANSYS finite element program. The proposed models for different types of measurements are experimentally evaluated with strain gages, including the estimation of mechanical properties. The results show that the strain field can be measured using the SIFT technique, developing a new visual tool for measurement of mechanical tests that are not based on traditional techniques of image correlation.

A recording of the defense presentation for this dissertation is available at: http://hdl.handle.net/10754/625197 / Steady-state elliptic partial differential equations (PDEs) are frequently used to model a diverse range of physical phenomena. The source and boundary data estimation problems for such PDE systems are of prime interest in various engineering disciplines including biomedical engineering, mechanics of materials and earth sciences. Almost all existing solution strategies for such problems can be broadly classified as optimization-based techniques, which are computationally heavy especially when the problems are formulated on higher dimensional space domains. However, in this dissertation, feedback based state estimation algorithms, known as state observers, are developed to solve such steady-state problems using one of the space variables as time-like. In this regard, first, an iterative observer algorithm is developed that sweeps over regular-shaped domains and solves boundary estimation problems for steady-state Laplace equation. It is well-known that source and boundary estimation problems for the elliptic PDEs are highly sensitive to noise in the data. For this, an optimal iterative observer algorithm, which is a robust counterpart of the iterative observer, is presented to tackle the ill-posedness due to noise. The iterative observer algorithm and the optimal iterative algorithm are then used to solve source localization and estimation problems for Poisson equation for noise-free and noisy data cases respectively. Next, a divide and conquer approach is developed for three-dimensional domains with two congruent parallel surfaces to solve the boundary and the source data estimation problems for the steady-state Laplace and Poisson kind of systems respectively. Theoretical results are shown using a functional analysis framework, and consistent numerical simulation results are presented for several test cases using finite difference discretization schemes.

This thesis discusses the inverse problem associated with the propagation of field autocorrelation of light through a highly scattering object like tissue. In the first part of the thesis we consider the mathematical issues involved in inverting boundary measurements made from diffuse propagation of light through highly scattering objects for their optical and mechanical properties. We present the convergence analysis of the Gauss-Newton algorithm for the recovery of object properties applicable for both diffuse correlation tomography (DCT) and diffuse optical tomography (DOT). En route to this, we establish the existence of solution and Frechet differentiability of the forward propagation equation. The two cases of the delta source and the Gaussian source illuminations are considered separately and the smoothness of solution of the forward equation in these cases is established. Considering DCT as an example, we establish the feasibility of recovering the particle diffusion coefficient (DB ) through minimizing the data-model mismatch of the field autocorrelation at the boundary using the Gauss-Newton algorithm. Some numerical examples validating the theoretical results are also presented. In the second part of the thesis, we reconstruct optical absorption coefficient, µa, and particle diffusion coefficient, DB , from simulated measurements which are integrals of a quantity computed from the measured intensity and intensity autocorrelation g2(τ ) at the boundary. We also recovered the mean square displacement (MSD) distribution of particles in an inhomogeneous object from the sampled g2(τ ) measured on the boundary. From the MSD, we compute the storage and loss moduli distributions in the object. We have devised computationally easy methods to construct the sensitivity matrices which are used in the iterative reconstruction algorithms for recovering these parameters from these measurements. The results of reconstruction of inhomogeneities in µa, DB , MSD and the visco-elastic parameters, which are presented, show forth reasonably good positional and quantitative accuracy. Finally we introduce a self regularized pseudo-dynamic scheme to solve the above inverse problem, which has certain advantages over the usual minimization method employing a variant of the Newton algorithm. The computational difficulties involved in the inversion of ill-conditioned matrices arising in the nonlinear inverse DCT problem are avoided by introducing artificial dynamics and considering the solution to be the steady-state response (if it exists) of the artificially evolving dynamical system, represented by ordinary differential equations (ODE) in pseudo-time. We show that the asymptotic solution obtained through the pseudo-time marching converges to the optimal solution which minimizes a mean-square error functional, provided the Hessian of the forward equation is positive definite in the neighborhood of this optimal solution. The superior noise tolerance and regularization-insensitive nature of pseudo-dynamic strategy are proven through numerical simulations in the context of DCT.

Dans la première partie, on démontre des inégalités de Carleman pour des systèmes paraboliques. Au chapitre 1, on démontre des inégalités de stabilité pour un système parabolique 2 x 2 en utilisant des inégalités de Carleman avec une seule observation. Il s'agit d'un problème inverse pour l'identification des coefficients et les conditions initiales du système. Le chapitre2 est consacré aux inégalités de Carleman pour des systèmes paraboliques dont les coefficients de diffusion sont de classe C1 par morceaux ou à variations bornées. A la fin, on donne quelques applications à la contrôlabilité à zéro. La seconde partie est consacrée à l'étude d'un problème de diffraction d'ondes acoustiques dans un demi-plan homogène. Il s'agit d'un problème aux limites associé à l'équation de Helmholtz dans le demi-plan supérieur avec une donnée de Neumann non homogène au bord. On apporte des éléments de réponse sur la question d'unicité et d'existence des solutions pour certaines classes de la donnée au bord. / In the first part, we prove Carleman estimates for parabolic systems. In chapter1, we prove stability inequalities for 2 x 2 parabolic system using Carleman estimates with one observation. It is concerns to the identification of the coefficients and initial conditions of the system. The chapter2 is devoted to th Carleman estimates of parabolic systems for which the diffusion coefficients are assumed to be ofclass piecewise C1 or with bounded variations. In the end, we give some applications to the null controllability. The second part is devoted to the study of the scattering problem of acoustics waves in a homogeneous half-plane. It is about a boundary value problem associated to the Helmholtz equation in theupper half-plane with a nonhomogeneous Neumann boundary data. We provide some answers to the question of uniqueness and existence of solutions for some classes of the boundary data.

Typical aero-engine assemblies have at least two nested rotors mounted within a flexible casing via squeeze-film damper (SFD) bearings. As a result, the flexible casing structures become highly sensitive to the vibration excitation arising from the High and Low pressure rotors. Lowering vibrations at the aircraft engine casing can reduce harmful effects on the aircraft engine. Inverse problem techniques provide a means toward solving the unbalance identification problem for a rotordynamic system supported by nonlinear SFD bearings, requiring prior knowledge of the structure and measurements of vibrations at the casing. This thesis presents two inverse solution techniques for the nonlinear rotordynamic inverse problem, which are focused on applications where the rotor is inaccessible under operating conditions, e.g. high pressure rotors. Numerical and experimental validations under hitherto unconsidered conditions have been conducted to test the robustness of each technique. The main contributions of this thesis are:• The development of a non-invasive inverse procedure for unbalance identification and balancing of a nonlinear SFD rotordynamic system. This method requires at least a linear connection to ensure a well-conditioned explicit relationship between the casing vibration and the rotor unbalance via frequency response functions. The method makes no simplifying assumptions made in previous research e.g. neglect of gyroscopic effects; assumption of structural isotropy; restriction to one SFD; circular centred orbits (CCOs) of the SFD. • The identification and validation of the inverse dynamic model of the nonlinear SFD element, based on recurrent neural networks (RNNs) that are trained to reproduce the Cartesian displacements of the journal relative to the bearing housing, when presented with given input time histories of the Cartesian SFD bearing forces.• The empirical validation of an entirely novel approach towards the solution of a nonlinear inverse rotor-bearing problem, one involving an identified empirical inverse SFD bearing model. This method is suitable for applications where there is no adequate linear connection between rotor and casing. Both inverse solutions are formulated using the Receptance Harmonic Balance Method (RHBM) as the underpinning theory. The first inverse solution uses the RHBM to generate the backwards operator, where a linear connection is required to guarantee an explicit inverse solution. A least-squares solution yields the equivalent unbalance distribution in prescribed planes of the rotor, which is consequently used to balance it. This method is successfully validated on distinct rotordynamic systems, using simulated data considering different practical scenarios of error sources, such as noisy data, model uncertainty and balancing errors. Focus is then shifted to the second inverse solution, which is experimentally-based. In contrast to the explicit inverse solution, the second alternative uses the inverse SFD model as an implicit inverse solution. Details of the SFD test rig and its set up for empirical identification are presented. The empirical RNN training process for the inverse function of an SFD is presented and validated as a part of a nonlinear inverse problem. Finally, it is proved that the RNN could thus serve as reliable virtual instrumentation for use within an inverse rotor-bearing problem.