Electromagnetic (EM) imaging schemes are inherently non-linear and ill-posed. Albeit there exist remedies to these fundamental problems, more efficient solutions are still being sought. To this end, in this thesis, the non-linearity is tackled in- corporating a multitude of techniques (ranging from Born approximation (linear), inexact Newton (linearized) to complete nonlinear iterative Landweber schemes) that can account for weak to strong scattering problems. The ill-posedness of the EM inverse scattering problem is circumvented by formulating the above methods into a minimization problem with a sparsity constraint. More specifically, four novel in- verse scattering schemes are formulated and implemented. (i) A greedy algorithm is used together with a simple artificial neural network (ANN) for efficient and accu- rate EM imaging of weak scatterers. The ANN is used to predict the sparsity level of the investigation domain which is then used as the L0 - constraint parameter for the greedy algorithm. (ii) An inexact Newton scheme that enforces the sparsity con- straint on the derivative of the unknown material properties (not necessarily sparse) is proposed. The inverse scattering problem is formulated as a nonlinear function of the derivative of the material properties. This approach results in significant spar- sification where any sparsity regularization method could be efficiently applied. (iii) A sparsity regularized nonlinear contrast source (CS) framework is developed to di- rectly solve the nonlinear minimization problem using Landweber iterations where the convergence is accelerated using a self-adaptive projected accelerated steepest descent algorithm. (iv) A 2.5D finite difference frequency domain (FDFD) based in- verse scattering scheme is developed for imaging scatterers embedded in lossy and inhomogeneous media. The FDFD based inversion algorithm does not require the Green’s function of the background medium and appears a promising technique for biomedical and subsurface imaging with a reasonable computational time. Numerical experiments, which are carried out using synthetically generated mea- surements, show that the images recovered by these sparsity-regularized methods are sharper and more accurate than those produced by existing methods. The methods developed in this work have potential application areas ranging from oil/gas reservoir engineering to biological imaging where sparse domains naturally exist.

京都大学 / 0048 / 新制・課程博士 / 博士(農学) / 甲第22514号 / 農博第2418号 / 新制||農||1078(附属図書館) / 学位論文||R2||N5294(農学部図書室) / 京都大学大学院農学研究科地域環境科学専攻 / (主査)教授 村上 章, 教授 藤原 正幸, 教授 磯 祐介 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DGAM

This project concerns the computational solution of inverse problems formulated as partial differential equation (PDE)-constrained optimization problems with interior data. The areas addressed are twofold. First, we present a novel software architecture designed to solve inverse problems constrained by an elliptic system of PDEs. These generally require the solution of forward and adjoint problems, evaluation of the objective function, and computation of its gradient, all of which are approximated numerically using finite elements. The creation of specialized "layered"' elements to perform these tasks leads to a modular software structure that improves code maintainability and promotes functional interoperability between different software components. Second, we address issues related to forward model definition in the presence of boundary condition (BC) uncertainty. We propose two variational formulations to accommodate that uncertainty: (a) a Bayesian formulation that assumes Gaussian measurement noise and a minimum strain energy prior, and (b) a Lagrangian formulation that is completely free of displacement and traction BCs. This work is motivated by applications in the field of biomechanical imaging, where the mechanical properties within soft tissues are inferred from observations of tissue motion. In this context, the constraint PDE is well accepted, but considerable uncertainty exists in the BCs. The approaches developed here are demonstrated on a variety of applications, including simulated and experimental data. We present modulus reconstructions of individual cells, tissue-mimicking phantoms, and breast tumors.

No description available.

Les techniques de mesure de diagramme de rayonnement d’antenne actuelles partagent un paradigme commun qui stipule que l’information utile est exclusivement portée par le signal de test généré. Cela implique un effort mécanique fastidieux en faisant tourner l'antenne sous test ou en déplaçant le système de sondes afin de couvrir des angles de mesure différents jusqu'à ce qu'une caractérisation complète soit effectuée ; une limitation qui est généralement surmontée en utilisant des systèmes multisondes coûteux. En outre, toute réflexion provenant du site de mesure et des équipements de test est considérée comme parasite perturbant le signal de test et est ainsi minimisée.Dans cette thèse, un concept de mesure du diagramme de rayonnement d'antenne remettant en cause ce paradigme commun est présenté comme un moyen d'accélérer le processus de caractérisation en utilisant des systèmes économiques. Le paradigme proposé consiste en la génération d'un ensemble d'échos contrôlées, en utilisant des configurations impliquant des plaques réfléchissantes, qui contribueraient directement à la mesure en couvrant différents angles, et récupérer les informations portées par l'ensemble des signaux générés simultanément. Une diversité fréquentielle est introduite afin de générer un système d'équations équilibré où le vecteur inconnu contenant les valeurs du diagramme de rayonnement est récupéré en inversant un problème matriciel. Par conséquent, une attention considérable est accordée au conditionnement du modèle mathématique afin d'assurer la stabilité et la robustesse du systèmeTrois configurations de différents niveaux de complexité en termes d'échos contrôlés sont étudiées, en mettant l'accent sur la configuration la plus simple impliquant un seul écho contrôlé. Des modèles ont été mis au point, avec des contraintes de conception des configurations proposées en termes de dimensionnement et de bandes passante de fonctionnement, mettant en évidence la viabilité mathématique du concept. Les aspects pratiques ont également été évalués en étudiant la tolérance des modèles développés vis-à-vis des erreurs systématiques, ainsi qu’à l'impact de l’application d'un ensemble d’hypothèses simplificatrices. La faisabilité du concept ainsi que son utilité pour accélérer le processus de caractérisation par rapport aux techniques classiques ont été mises en évidence par des simulations numériques. Ce travail ouvre la porte à l'exploitation des échos, généralement considérés comme perturbateurs, dans un contexte de mesure d’antennes. / Current antenna radiation pattern measurement techniques share a common paradigm which states that useful information is exclusively carried by the generated test signal. This implies an excessive, time consuming, mechanical effort by rotating the antenna under test or displacing the probe system in order to cover different measurement angles until a complete scan is performed; a limitation that is typically overcome using costly multi-probe systems. Moreover, any reflection from the measurement site and test equipment is considered spurious as it perturbs the test signal and thus is minimized.In this thesis, an antenna radiation pattern measurement concept challenging this common paradigm is introduced as a mean of accelerating the characterization process using cost-efficient systems. The proposed paradigm consists in the generation of a set of controlled echoes, using set-ups involving highly-reflective plates, which would directly contribute to the measurement alongside the line-of-sight signal by covering different measurement angles, and retrieving the ARP information carried by the set of all generated signals concurrently. Frequency diversity is used in order to generate a balanced system of equations where the unknown ARP vector is retrieved by inverting a matrix problem. Consequently, a considerable attention is paid into the conditioning of the mathematical model in order to ensure the system stability and accuracy.Three configurations of different complexity levels in terms of controlled echoes are studied, with focus on the simplest configuration involving a single controlled echo. Models have been developed with design guidelines for the proposed configurations in terms of set-up dimensions and operating frequency bandwidth highlighting the mathematical viability of the concept. Practical issues were also assessed by studying the tolerance of developed models to systematic practical errors, as well as to the impact of an applied set of simplifying assumptions. The feasibility of the concept as well as its usefulness in accelerating the measurement process with respect to classical techniques were highlighted via numerical simulations. This thesis opens the door for exploiting echoes, generally regarded as a nuisance, in an antenna measurements context.

Abstract We are interested in studying Gaussian Markov random fields as correlation priors for Bayesian inversion. We construct the correlation priors to be discretisation-invariant, which means, loosely speaking, that the discrete priors converge to continuous priors at the discretisation limit. We construct the priors with stochastic partial differential equations, which guarantees computational efficiency via sparse matrix approximations. The stationary correlation priors have a clear statistical interpretation through the autocorrelation function. We also consider how to make structural model of an unknown object with anisotropic and inhomogeneous Gaussian Markov random fields. Finally we consider these fields on unstructured meshes, which are needed on complex domains. The publications in this thesis contain fundamental mathematical and computational results of correlation priors. We have considered one application in this thesis, the electrical impedance tomography. These fundamental results and application provide a platform for engineers and researchers to use correlation priors in other inverse problem applications.

<div> <p>Thermal processing is a critical step in shelf-stable food manufacturing to the ensure safety of the food products. To accurately model and establish the thermal processes, temperature-dependent thermal properties are needed. Existing methods for measuring the temperature-dependent thermal diffusivity (α), thermal conductivity (k) and volumetric heat capacity (C) are time consuming, tend to have high errors, and cannot provide results in a single experiment, especially at temperatures above 100°C. A novel bench scale device, named Thermal Properties Cell (TPCell), was custom made to rapidly estimate the temperature-dependent thermal parameters of food products. </p> <p> </p> <p>The TPCell used thin film heaters as the heating elements. The first study focused on estimating the thermal properties of a thin film heater. Using mathematical modeling and sequential parameter estimation, the effective thermal diffusivity of the thin film heater was found at different temperatures. The estimated thermal properties of the thin film heater were used for the second study.</p> <p> </p> <p>The objective of the second study was to design optimal complementary experiments using TPCell. Complementary experiments are a combination of experiments that enable estimation of multiple thermal parameters from the experimental temperature data, based on sensitivity analysis. Sensitivity coefficients indicate the extent of change in a measured variable due to a change in value of an input parameter. Designs of experiments were simulated and their impact on sensitivity and optimality criteria was analyzed. Results from the simulated profiles were validated using sweet potato puree. </p> <p> </p> <p>Learnings from this work can be directly applied for the optimization of all types of food thermal processes, including retort and aseptic processing. Optimally designed processes increase preservation of the heat labile nutrients, color, flavor, and taste compounds, thereby enhancing the quality of food products.</p> </div> <br>

The goal of this study is the recovery of functions and finite parametric distributions from their spherical means over spheres and designing a general formula or algorithm for the reconstruction of a function f via its spherical mean transform. The theoretical study is and supported with a numerical implementation based on radar data. In this study, we approach the reconstruction problem in two different way. The first one is to show how the reconstruction problem could be converted to a Prony-type system of equations. After solving this Prony-type system of equations, one can extract the parameters that describe the corresponding functions or distributions efficiently. The second way is to solve this problem via a backprojection procedure.

Convolution and, as a special case, autoconvolution of functions are important in many branches of mathematics and have found lots of applications, such as in physics, statistics, image processing and others. While it is a relatively easy task to determine the autoconvolution of a function (at least from the numerical point of view), the inverse problem, which consists in reconstructing a function from its autoconvolution is an ill-posed problem. Hence there is no possibility to solve such an inverse autoconvolution problem with a simple algebraic operation. Instead the problem has to be regularized, which means that it is replaced by a well-posed problem, which is close to the original problem in a certain sense. The outline of this thesis is as follows: In the first chapter we give an introduction to the type of inverse problems we consider, including some basic definitions and some important examples of regularization methods for these problems. At the end of the introduction we shortly present some general results about the convergence theory of Tikhonov-regularization. The second chapter is concerned with the autoconvolution of square integrable functions defined on the interval [0, 1]. This will lead us to the classical autoconvolution problems, where the term “classical” means that no kernel function is involved in the autoconvolution operator. For the data situation we distinguish two cases, namely data on [0, 1] and data on [0, 2]. We present some well-known properties of the classical autoconvolution operators. Moreover, we investigate nonlinearity conditions, which are required to show applicability of certain regularization approaches or which lead convergence rates for the Tikhonov regularization. For the inverse autoconvolution problem with data on the interval [0, 1] we show that a convergence rate cannot be shown using the standard convergence rate theory. If the data are given on the interval [0, 2], we can show a convergence rate for Tikhonov regularization if the exact solution satisfies a sparsity assumption. After these theoretical investigations we present various approaches to solve inverse autoconvolution problems. Here we focus on a discretized Lavrentiev regularization approach, for which even a convergence rate can be shown. Finally, we present numerical examples for the regularization methods we presented. In the third chapter we describe a physical measurement technique, the so-called SD-Spider, which leads to an inverse problem of autoconvolution type. The SD-Spider method is an approach to measure ultrashort laser pulses (laser pulses with time duration in the range of femtoseconds). Therefor we first present some very basic concepts of nonlinear optics and after that we describe the method in detail. Then we show how this approach, starting from the wave equation, leads to a kernel-based equation of autoconvolution type. The aim of chapter four is to investigate the equation and the corresponding problem, which we derived in chapter three. As a generalization of the classical autoconvolution we define the kernel-based autoconvolution operator and show that many properties of the classical autoconvolution operator can also be shown in this new situation. Moreover, we will consider inverse problems with kernel-based autoconvolution operator, which reflect the data situation of the physical problem. It turns out that these inverse problems may be locally well-posed, if all possible data are taken into account and they are locally ill-posed if one special part of the data is not available. Finally, we introduce reconstruction approaches for solving these inverse problems numerically and test them on real and artificial data.

L'évaluation et le contrôle non destructifs (E.C.N.D.) des matériaux et des structures sont une problématique industrielle très importante dans les domaines du transport, de l'aéronautique et du spatial, et dans le milieu médical. La thermographie infrarouge active est une technique d'E.C.N.D qui consiste à apporter une excitation extérieure afin d'entraîner une élévation de température dans le matériau, puis à évaluer le champ de température résultant à la surface. Cependant, les excitateurs thermiques utilisés (lampes flash, halogènes, lasers) agissent uniquement sur la surface du matériau. Plusieurs systèmes de conversion d'énergie peuvent en revanche mener à l'apparition de sources volumiques : on peut citer en particulier les phénomènes de thermo-acoustique, de thermo-induction, de thermomécanique ou de thermochimie. Par exemple, une excitation par ondes ultrasonores peut entraîner des sources thermiques volumiques si le matériau est viscoélastique ou s'il y a présence de défaut. La reconstruction de ces sources est donc la première étape permettant de remonter aux paramètres responsables de l'échauffement. Caractériser une source thermique consiste à reconstruire sa géométrie et la puissance qu'elle génère. Cependant, l'identification de sources thermiques volumiques par la mesure des champs de température de surface est un problème mathématiquement mal posé. Le caractère diffusif de la température en est le principal responsable. Dans ce travail, la reconstruction 3D des sources volumiques à partir du champ de température résultant à la surface, mesuré par InfraRouge, est étudié. Tout d'abord, une analyse du problème physique permet de spécifier les limites de la reconstruction. En particulier, un critère sur la résolution spatiale atteignable est défini et une limitation de reconstruction pour les sources en profondeur est mise en lumière. Ensuite, une méthode de reconstruction par approche probabiliste est proposée et comparée aux méthodes d'inversions existantes. Le temps d'exécution et la sensibilité au bruit de mesure sont étudiés pour chacune de ces méthodes. Des applications numériques et expérimentales seront enfin présentées pour illustrer les résultats. / Non Destructive Testing (N.D.T.) of materials and structures is a very important industrial issue in the fields of transport, aeronautics and space and in the medical domain. Active infrared thermography is a N.D.T. method that consists in providing an external excitation to cause an elevation of temperature field in the material and then to evaluate the resulting temperature field at the surface. However, thermal exciters used (flash lamps, halogen, lasers) act only on the surface of the sample. Several energy conversion systems can on the other hand lead to the generation of volumetric sources: the phenomena of thermo-acoustic, thermo-induction, thermomechanic or thermochemistry can be cited. For example, ultrasonic waves can generate volumetric heat sources if the material is viscoelastic or if there is a defect. The reconstruction of these sources is the first step for the quantification of parameters responsible of the heating. Characterizing a heat source means reconstructing its geometry and the power it generates. For example, a defect in a structure and / or the viscoelasticity of a material can be detected and quantified by this technique if it acts directly on temperature field. However, identification of volumetric heat sources from surface temperature fields is a mathematical ill-posed problem. The diffusive nature of the temperature is the main cause. In this work, the 3D reconstruction of the volumetric heat sources from the resulting surface temperature field, measured by InfraRed, is studied. First, an analysis of the physical problem enables to specify the limits of the reconstruction. In particular, a criterion on achievable spatial resolution is defined and a reconstruction limitation for in-depth sources is highlighted. Then, a probabilistic approach for the reconstruction is proposed and compared to existing inverse methods. The computation time and noise sensitivity are studied for each of these methods. Numerical and experimental applications will thus be presented to illustrate the results.