Thesis (Ph. D.)--York University, 2001. Graduate Programme in Mathematics. / Typescript. Includes bibliographical references (leaves 114-119). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNQ66358.

Η συνεισφορά της παρούσας διατριβής επικεντρώνεται στην ανάπτυξη και στη Μαθηματική θεμελίωση νέων μεθόδων συζυγών κλίσεων για βελτιστοποίηση χωρίς περιορισμούς και στη μελέτη νέων μεθόδων εκπαίδευσης νευρωνικών δικτύων και εφαρμογών τους. Αναπτύσσουμε δύο νέες μεθόδους βελτιστοποίησης, οι οποίες ανήκουν στην κλάση των μεθόδων συζυγών κλίσεων. Οι νέες μέθοδοι βασίζονται σε νέες εξισώσεις της τέμνουσας με ισχυρά θεωρητικά πλεονεκτήματα, όπως η προσέγγιση με μεγαλύτερη ακρίβεια της επιφάνεια της αντικειμενικής συνάρτησης. Επιπλέον, μία σημαντική ιδιότητα και των δύο προτεινόμενων μεθόδων είναι ότι εγγυώνται επαρκή μείωση ανεξάρτητα από την ακρίβεια της γραμμικής αναζήτησης, αποφεύγοντας τις συχνά αναποτελεσματικές επανεκκινήσεις. Επίσης, αποδείξαμε την ολική σύγκλιση των προτεινόμενων μεθόδων για μη κυρτές συναρτήσεις. Με βάση τα αριθμητικά μας αποτελέσματα καταλήγουμε στο συμπέρασμα ότι οι νέες μέθοδοι έχουν πολύ καλή υπολογιστική αποτελεσματικότητα, όπως και καλή ταχύτητα επίλυσης των προβλημάτων, υπερτερώντας σημαντικά των κλασικών μεθόδων συζυγών κλίσεων. Το δεύτερο μέρος της διατριβής είναι αφιερωμένο στην ανάπτυξη και στη μελέτη νέων μεθόδων εκπαίδευσης νευρωνικών δικτύων. Προτείνουμε νέες μεθόδους, οι οποίες διατηρούν τα πλεονεκτήματα των κλασικών μεθόδων συζυγών κλίσεων και εξασφαλίζουν τη δημιουργία κατευθύνσεων μείωσης αποφεύγοντας τις συχνά αναποτελεσματικές επανεκκινήσεις. Επιπλέον, αποδείξαμε ότι οι προτεινόμενες μέθοδοι συγκλίνουν ολικά για μη κυρτές συναρτήσεις. Τα αριθμητικά αποτελέσματα επαληθεύουν ότι οι προτεινόμενες μέθοδοι παρέχουν γρήγορη, σταθερότερη και πιο αξιόπιστη σύγκλιση, υπερτερώντας των κλασικών μεθόδων εκπαίδευσης. Η παρουσίαση του ερευνητικού μέρους της διατριβής ολοκληρώνεται με μία νέα μέθοδο εκπαίδευσης νευρωνικών δικτύων, η οποία βασίζεται σε μία καμπυλόγραμμη αναζήτηση. Η μέθοδος χρησιμοποιεί τη BFGS ενημέρωση ελάχιστης μνήμης για τον υπολογισμό των κατευθύνσεων μείωσης, η οποία αντλεί πληροφορία από την ιδιοσύνθεση του προσεγγιστικού Eσσιανού πίνακα, αποφεύγοντας οποιαδήποτε αποθήκευση ή παραγοντοποίηση πίνακα, έτσι ώστε η μέθοδος να μπορεί να εφαρμοστεί για την εκπαίδευση νευρωνικών δικτύων μεγάλης κλίμακας. Ο αλγόριθμος εφαρμόζεται σε προβλήματα από το πεδίο της τεχνητής νοημοσύνης και της βιοπληροφορικής καταγράφοντας πολύ καλά αποτελέσματα. Επίσης, με σκοπό την αύξηση της ικανότητας γενίκευσης των εκπαιδευόμενων δικτύων διερευνήσαμε πειραματικά και αξιολογήσαμε την εφαρμογή τεχνικών μείωσης της διάστασης δεδομένων στην απόδοση της γενίκευσης των τεχνητών νευρωνικών δικτύων σε μεγάλης κλίμακας δεδομένα βιοϊατρικής. / The contribution of this thesis focuses on the development and the Mathematical foundation of new conjugate gradient methods for unconstrained optimization and on the study of new neural network training methods and their applications. We propose two new conjugate gradient methods for unconstrained optimization. The proposed methods are based on new secant equations with strong theoretical advantages i.e. they approximate the surface of the objective function with higher accuracy. Moreover, they have the attractive property of ensuring sufficient descent independent of the accuracy of the line search, avoiding thereby the usual inefficient restarts. Further, we have established the global convergence of the proposed methods for general functions under mild conditions. Based on our numerical results we conclude that our proposed methods outperform classical conjugate gradient methods in both efficiency and robustness. The second part of the thesis is devoted on the study and development of new neural network training algorithms. More specifically, we propose some new training methods which preserve the advantages of classical conjugate gradient methods while simultaneously ensure sufficient descent using any line search, avoiding thereby the usual inefficient restarts. Moreover, we have established the global convergence of our proposed methods for general functions. Encouraging numerical experiments on famous benchmarks verify that the presented methods provide fast, stable and reliable convergence, outperforming classical training methods. Finally, the presentation of the research work of this dissertation is fulfilled with the presentation of a new curvilinear algorithm for training large neural networks which is based on the analysis of the eigenstructure of the memoryless BFGS matrices. The proposed method preserves the strong convergence properties provided by the quasi-Newton direction while simultaneously it exploits the nonconvexity of the error surface through the computation of the negative curvature direction without using any storage and matrix factorization. Our numerical experiments have shown that the proposed method outperforms other popular training methods on famous benchmarks. Furthermore, for improving the generalization capability of trained ANNs, we explore the incorporation of several dimensionality reduction techniques as a pre-processing step. To this end, we have experimentally evaluated the application of dimensional reduction techniques for increasing the generalization capability of neural network in large biomedical datasets.

Orientador: Alexandre Xavier Falcão / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-13T10:55:58Z (GMT). No. of bitstreams: 1 Favretto_FernandaOliveira_M.pdf: 1878530 bytes, checksum: b516fce053de83b3dbd32fa789dcb9c9 (MD5) Previous issue date: 2009 / Resumo: O registro de imagens é o processo que alinha duas ou mais imagens em um mesmo sistema de coordenadas espaciais [31]. Na área de Imagens Médicas, o problema de registro de imagens tem muitas aplicações permitindo, por exemplo, a análise da variação de fenômenos e estruturas anatômicas ao longo do tempo, pelo registro de imagens de uma mesma modalidade obtidas em diferentes instantes de tempo; ou o estudo das informações anatômicas e fisiológicas combinadas para uma dada estrutura fenômeno, pelo registro de imagens obtidas por modalidades diferentes. O objetivo deste trabalho é o desenvolvimento de uma técnica de registro para imagens tridimensionais do cérebro humano, cuja motivação é o estudo comparativo de imagens de Ressonância Magnética pré- e pós-operatórias do cérebro de pacientes de epilepsia. Um estudo recente [80] tem observado que nos casos em que houve crises recorrentes, após a remoção cirúrgica do foco da crise, os pacientes apresentaram alterações nas substâncias cinza e branca do cérebro. O registro das imagens pré- e pós- operatórias desses pacientes permite a análise dessas alterações. Foi desenvolvida uma técnica de registro rígido que realiza o alinhamento de imagens 3D de forma automática, rápida e precisa. O método baseia-se no casamento das linhas de watershed marcador de cinza extraídas da imagem móvel com uma imagem de borda realçada pelo gradiente morfológico da imagem fixa. A busca dos parâmetros de rotação e translação que compõem a função de mapeamento é feita através de uma técnica proposta neste trabalho, denominada Descendente de Gradiente em Múltiplas Escalas (MSGD) - um variante do tradicional método de Descendente de Gradiente - a qual permite passos de tamanhos escalonados dos vetores de gradiente, evitando mínimos locais indesejáveis e convergindo para o ótimo desejado mais rapidamente. O método foi avaliado em imagens 3D de ressonância magnética do cérebro humano ponderadas em T1 e obteve bons resultados. Os experimentos envolveram 2 bases de dados. A primeira base é a base de dados de controle, composta por 200 pares de imagens, onde o registro foi realizado em aproxidamente 45s e obteve erro médio de rotação de 0, 06?, 0, 08? e 0, 08? com desvio padrão de 0, 06, 0, 25 e 0, 08 nos eixos X, Y e Z, respectivamente, e erro médio de translação de 1, 67mm, 1, 55mm e 2, 27mm com desvio padrão de 1, 83, 1, 45 e 2, 27 nos eixos X, Y e Z, respectivamente. A segunda base foi uma base de dados clínicos, composta por imagens pré- e pós-operatórias de pacientes com epilepsia, que comprovou a eficácia do método em dados clínicos reais. Também foram desenvolvidas duas técnicas de visualização do registro, uma delas baseada no mosaico das imagens registradas e a outra que combina as imagens em um único volume colorido, onde as alterações de tecidos são identificadas pelas cores vermelha e verde. Portanto, as principais contribuições deste trabalho são: uma metodologia para o registro, que envolve combinação eficiente de características, métrica de similaridade e estratégia de busca; a estratégia MSGD que se mostrou promissora para outros problemas de otimização; e uma técnica de visualização das imagens registradas na forma de um volume colorido. / Abstract: Image Registration is the process that aligns two or more images in a common reference system of spacial coordinates [31]. It is an important problem with several applications in Medical Imaging, enabling, for instance, the analysis of changes in anatomy along time by the registration of images from the same modality, and the study of combined anatomic and physiologic data by the registration of images from different modalities. The objective of this work is the development of a registration method for 3D images of the human brain, and the motivation is a comparative study of pre and post-surgical images from epilepsy patients. A recent study [80] has observed that some pacients, who did not cease the seizures after surgery, presented variations in their brain tissues. The registration of pre and post-surgical images enables the analysis of these tissue's variations. We developed a rigid registration method that aligns 3D images in a fast, automatic and accurate way. The method is based on the matching between watershed lines extrated from a source image and a morphological gradient image from the target image. The search for the parameters of rotation and translation that compose the mapping function is done by a techinique proposed in this work, named Multi-Scale Gradient Descent - a variant of the tradicional method Gradient Descent - which enables gradient's vectors with scaled magnitudes, avoiding undesirable local minima and fastly converging to the desired optimum. The method was evaluated on 3D T1-weighted Magnetic Ressonance Images of the human brain. The experiments used 2 data bases: a control data base, composed by 200 pairs of images, in which the method took approximately 45s and acceptable results; and a data base of patients, composed by pre- and post-surgical images, demonstrating the effectiveness of the method for real data. We have also developed visualization techiniques for the registred images: the checkerboard image, that alternates the target and registered source in a checkerboard pattern, allowing the user to inspect the correctness, coherence and continuity of the registration; and the colorized image, that combines the target and registered source images in a single colorized volume, such that the alterations of the tissues can be identified by the red and green colors. Therefore, the main contributions of this work are: a 3D registration methodology, that involves an effective combination of feature selection, similarity measure and search strategy; a search strategy, MSGD, that seems to be promissing for other optimization problems; and a visualization techinique that uses a colorized volume to combine the registered images. / Mestrado / Processamento e Analise de Imagens / Mestre em Ciência da Computação

Orientadores: Francisco de Assis Magalhães Gomes Neto, Jose Mario Martinez / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-12T08:17:16Z (GMT). No. of bitstreams: 1 Medeiros_LuizAntoniodaSilva_D.pdf: 2528214 bytes, checksum: 42e1946a32b63c9fc5cd56b10d24d5cb (MD5) Previous issue date: 2008 / Resumo: Neste trabalho, apresentamos um algoritmo iterativo para a minimização de somas de quadrados de funções suaves, com restrições de caixa. O algoritmo é fortemente inspirado no trabalho de Birgin e Martínez [4]. A diferença principal está na escolha da direção de busca e na introdução de uma nova técnica de aceleração, usada para atualizar o passo. A cada iteração, definimos uma face ativa e resolvemos, nessa face, um subproblema quadrático irrestrito através do método evenberg-Marquardt (ver [26], [28] e [33]), obtendo uma direção de descida e uma aproximação x+ para a solução do problema. Ainda usando apenas as variáveis livres, tentamos acelerar o método definindo uma nova aproximaçaoo xa como combinação linear das últimas p - 1 aproximações da solução e do vetor x+. Os coeficientes desta combinação linear são calculados convenientemente através da resolução de um problema de Quadrados Mínimos com uma restrição de igualdade. O subproblema que determina o passo acelerado leva em conta as informações sobre a função objetivo nessas p soluções aproximadas. Como em [4], executamos uma busca linear ao longo da direção e usamos técnicas de projeção para adicionar novas restrições. Para deixar a face ativa, usamos a direção do gradiente espectral projetado [5]. Experimentos númericos são apresentados para confirmar a eficiência e robustez do novo algoritmo. / Abstract: In this work, we present an active set algorithm for minimizing the sum of squares of smooth functions, with box constraints. The algorithm is highly inspired in the work of Birgin and Mart'inez [4]. The differences are concentrated on the chosen search direction and on the use of an acceleration technique to update the step. At each iteration, we define an active face and solve an unconstrained quadratic subproblem using the Levenberg-Marquardt method (see [26], [28] and [33]), obtaining a descent direction and an approximate solution x+. Using only the free variables, we try to accelerate the method defining a new solution xa as a linear combination of the last p-1 approximate solutions together with x+. The coefficients of this linear combination are conveniently computed solving a constrained least squares problem that takes into account the objective function values of these p approximate solutions. Like in [4], we compute a line search and use projection techniques to add new constraints to the active set. The spectral projected gradient [5] is used to leave the current active face. Numerical experiments confirm that the algorithm is both efficient and robust. / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

Orientador: Denis José Schiozer / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica e Instituto de Geociências / Made available in DSpace on 2018-08-24T00:34:10Z (GMT). No. of bitstreams: 1 Pinto_MarcioAugustoSampaio_D.pdf: 5097853 bytes, checksum: bc8b7f6300987de2beb9a57c26ad806a (MD5) Previous issue date: 2013 / Resumo: Neste trabalho, um método de otimização assistido é proposto para estabelecer uma comparação refinada entre poços convencionais e inteligentes, considerando incertezas geológicas e econômicas. Para isto é apresentada uma metodologia dividida em quatro etapas: (1) representação e operação dos poços no simulador; (2) otimização das camadas/ou blocos completados nos poços convencionais e do número e posicionamento das válvulas nos poços inteligentes; (3) otimização da operação dos poços convencionais e das válvulas nos poços inteligentes, através de um método híbrido de otimização, composto pelo algoritmo genético rápido, para realizar a otimização global, e pelo método de gradiente conjugado, para realizar a otimização local; (4) uma análise de decisão considerando os resultados de todos os cenários geológicos e econômicos. Esta metodologia foi validada em modelos de reservatórios mais simples e com configuração de poços verticais do tipo five-spot, para em seguida ser aplicada em modelos de reservatórios mais complexos, com quatro poços produtores e quatro injetores, todos horizontais. Os resultados mostram uma clara diferença ao aplicar a metodologia proposta para estabelecer a comparação entre os dois tipos de poços. Apresenta também a comparação entre os resultados dos poços inteligentes com três tipos de controle, o reativo e mais duas formas de controle proativo. Os resultados mostram, para os casos utilizados nesta tese, uma ampla vantagem em se utilizar pelo menos uma das formas de controle proativo, ao aumentar a recuperação de óleo e VPL, reduzindo a produção e injeção de água na maioria dos casos / Abstract: In this work, an assisted optimization method is proposed to establish a refined comparison between conventional and intelligent wells, considering geological and economic uncertainties. For this, it is presented a methodology divided into four steps: (1) representation and operation of wells in the simulator, (2) optimization of the layers /blocks with completion in conventional wells and the number and placement of the valves in intelligent wells; (3) optimization of the operation of the conventional and valves in the intelligent, through a hybrid optimization method, comprising by fast genetic algorithm, to perform global optimization, and the conjugate gradient method, to perform local optimization; (4) decision analysis considering the results of all geological and economic scenarios. This method was validated in simple reservoir models and configuration of vertical wells with five-spot type, and then applied to a more complex reservoir model, with four producers and four injectors wells, all horizontal. The results show a clear difference in applying the proposed methodology to establish a comparison between the two types of wells. It also shows the comparison between the results of intelligent wells with three types of control, reactive and two ways of proactive control. The results show, for the cases used in this work, a large advantage to use intelligent wells with at least one form of proactive control, to enhance oil recovery and NPV, reducing water production and injection in most cases / Doutorado / Reservatórios e Gestão / Doutor em Ciências e Engenharia de Petróleo

Over the last 25 years, interior-point methods (IPMs) have emerged as a viable class of algorithms for solving various forms of conic optimization problems. Most IPMs use a modified Newton method to determine the search direction at each iteration. The system of equations corresponding to the modified Newton system can often be reduced to the so-called normal equation, a system of equations whose matrix ADA' is positive definite, yet often ill-conditioned. In this thesis, we first investigate the theoretical properties of the maximum weight basis (MWB) preconditioner, and show that when applied to a matrix of the form ADA', where D is positive definite and diagonal, the MWB preconditioner yields a preconditioned matrix whose condition number is uniformly bounded by a constant depending only on A. Next, we incorporate the results regarding the MWB preconditioner into infeasible, long-step, primal-dual, path-following algorithms for linear programming (LP) and convex quadratic programming (CQP). In both LP and CQP, we show that the number of iterative solver iterations of the algorithms can be uniformly bounded by n and a condition number of A, while the algorithmic iterations of the IPMs can be polynomially bounded by n and the logarithm of the desired accuracy. We also expand the scope of the LP and CQP algorithms to incorporate a family of preconditioners, of which MWB is a member, to determine an approximate solution to the normal equation. For the remainder of the thesis, we develop a new preconditioning strategy for solving systems of equations whose associated matrix is positive definite but ill-conditioned. Our so-called adaptive preconditioning strategy allows one to change the preconditioner during the course of the conjugate gradient (CG) algorithm by post-multiplying the current preconditioner by a simple matrix, consisting of the identity matrix plus a rank-one update. Our resulting algorithm, the Adaptive Preconditioned CG (APCG) algorithm, is shown to have polynomial convergence properties. Numerical tests are conducted to compare a variant of the APCG algorithm with the CG algorithm on various matrices.

The purpose of this dissertation is to design identification methods for identifying a permanent magnet synchronous motor. The whole identification and motor control is carried out in d-q coordinates, and the program used for processing and control was the matlab simulink, together with the real time platform DSpace. The work focuses on two main areas of identification, off-line identification and on-line identification. For offline identification the frequency analysis was used with the lock rotor test to get three main parameters. They are the quadrature and direct inductances and stator resistance. In the online mode, the identified parameters were extended to magnet flux _f identified by MRAS method. The remaining parameters were again identified by frequency analysis, which was adapted into online mode, and simultaneously applied to the identification of several part in one time. The next method is Newton method, which is used for estimating stator resistance of the motor, without the need to apply any signal.

Avec l'avènement de machine parallèles multi-coeurs, de nombreux algorithmes doivent être modifiés ou conçus pour s'adapter à ces architectures. Ces algorithmes consistent pour la plupart à diviser le problème original en plusieurs petits sous-problèmes et à les distribuer sur les différentes unités de calcul disponibles. La résolution de ces petits sous-problèmes peut être exécutée en parallèle, des communications entre les unités de calcul étant indispensables pour assurer la convergence de ces méthodes.Ma thèse propose de nouveaux algorithmes parallèles pour résoudre de grands systèmes linéaires.Les algorithmes proposés sont ici basés sur la méthode du gradient. Deux points fondamentaux de la méthode du gradient sont la direction de descente de la solution approchée et la valeur du pas de descente, qui détermine la modification à effectuer à chaque itération. Nous proposons dans cette thèse de calculer la direction et le pas indépendamment et localement sur chaque unité de calcul, ce qui nécessite moins de synchronisation entre les processeurs, et par suite rend chaque itération simple et plus rapide, et rend son extension dans un contexte asynchrone possible.Avec les paramètres d'échelle appropriés pour le pas des longueurs, la convergence peut être démontrée pour les deux versions synchrone et asynchrone des algorithmes. De nombreux tests numériques illustrent l’efficacité de ces méthodes.L'autre partie de ma thèse propose d'utiliser une méthode d'extrapolation pour accélérer les méthodes itératives classiques avec retard. Bien que les séquences de vecteur générées par des méthodes itératives asynchrones générales classiques ne peut être accélérée, nous sommes en mesure de démontrer que, une fois le modèle de calcul et de communication fixés au cours de l’exécution, la séquence de vecteurs générés peut être accéléré. De nombreux tests numériques illustrent l’efficacité de ces accélérations dans le cas des méthodes avec retard. / With the increase of architectures composed of multi-cores, many algorithms need to revisited and be modified to exploit the power of these new architectures. These algorithms divide the original problem into “small pieces” and distribute these pieces to different processors at disposal, thus communications among them are indispensible to assure the convergence. My thesis mainly focus on solving large sparse systems of linear equations in parallel with new methods. These methods are based on the gradient methods. Two key parameters of the gradient methods are descent direction and step-length of descent for each iteration. Our methods compute the directions locally, which requires less synchronization and computation, leading to faster iterations and make easy asynchronization possible. Convergence can be proved in both synchronized or asynchronized cases. Numerical tests demonstrate the efficiency of these methods. The other part of my thesis deal with the acceleration of the vector sequences generated by classical iterative algorithms. Though general chaotic sequences may not be accelerated, it is possible to prove that with any fixed retard pattern, then the generated sequence can be accelerated. Different numerical tests demonstrate its efficiency.

Este trabalho apresenta um método de reconstrução de imagens de ultrassom por problemas inversos que tem como penalidade para o erro entre solução e dados a norma L2, ou euclidiana, e como penalidade de regularização a norma L1. A motivação para o uso da regularização L1 é que se trata de um tipo de regularização promotora de esparsidade na solução. A esparsidade da regularização L1 contorna o problema de excesso do artefatos, observado em outras implementações de reconstrução por problemas inversos em ultrassom. Este problema é consequência principalmente da limitação da representação discreta do objeto contínuo no modelo de aquisição. Por conta desta limitação, objetos refletores na área imageada quase sempre localizam-se em posições que não correspondem precisamente a uma das posições do modelo discreto, gerando dados que não correspondem aos dados modelados. As formulações do problema com regularização L2 e com regularização L1 são apresentadas e comparadas dos pontos de vista geométrico e Bayesiano. O algoritmo de otimização proposto é uma implementação do algoritmo Iteratively Reweighted Least Squares (IRLS) e utiliza o método do Gradiente Conjugado (CG - Conjugate Gradient) a cada iteração, sendo chamado de IRLS-CG. São realizadas simulações com phantoms computacionais que mostram que o método permite reconstruir imagens a partir da aquisição de dados com refletores em posições não modeladas sem a observação de artefatos. As simulações também mostram melhor resolução espacial do método proposto com relação ao algoritmo delay-and-sum (DAS). Também se observou melhor desempenho computacional do CG com relação à matriz inversa nas iterações do IRLS. / This work presents an inverse problem based method for ultrasound image reconstruction which uses the L2-norm (or euclidean norm) as a penalty for the error between the data and the solution, and the L1-norm as a regularization penalty. The motivation for the use of of L1 regularization is the sparsity promoting property of this type of regularization. The sparsity of L1 regularization circumvents the problem of excess of artifatcts that is observed in other approaches of inverse problem based reconstrucion in ultrasound. Such problem is mainly a consequence of the limitation in the discrete representation of a continuous object in the acquisition model. Due to this limitation, reflecting objects in the imaged area are often localized in positions that do not correspond precisely to one of the positions in the discrete model, therefore generating data that do not correspond to the model data. The formulations of the problem with L2 regularization and with L1 regularization are presented and compared in geometric and Bayesian terms. The optimization algorithm proposed is an implementation of Iteratively Reweighted Least Squares (IRLS) and uses the Conjugate Gradient (CG) method inside each iteration, thus being called IRLS-CG. Simulations with computer phantoms are realized showing that the proposed method allows for the reconstruction of images, without observable artifacts, from data with reflectors located in non-modeled positions. Simulations also show a better spatial resolution in the proposed method when compared to the delay-and-sum (DAS) algorithm. It was also observed better computational performance of CG when compared to the matrix inversion in the iterations of IRLS.

Este trabalho apresenta um método de reconstrução de imagens de ultrassom por problemas inversos que tem como penalidade para o erro entre solução e dados a norma L2, ou euclidiana, e como penalidade de regularização a norma L1. A motivação para o uso da regularização L1 é que se trata de um tipo de regularização promotora de esparsidade na solução. A esparsidade da regularização L1 contorna o problema de excesso do artefatos, observado em outras implementações de reconstrução por problemas inversos em ultrassom. Este problema é consequência principalmente da limitação da representação discreta do objeto contínuo no modelo de aquisição. Por conta desta limitação, objetos refletores na área imageada quase sempre localizam-se em posições que não correspondem precisamente a uma das posições do modelo discreto, gerando dados que não correspondem aos dados modelados. As formulações do problema com regularização L2 e com regularização L1 são apresentadas e comparadas dos pontos de vista geométrico e Bayesiano. O algoritmo de otimização proposto é uma implementação do algoritmo Iteratively Reweighted Least Squares (IRLS) e utiliza o método do Gradiente Conjugado (CG - Conjugate Gradient) a cada iteração, sendo chamado de IRLS-CG. São realizadas simulações com phantoms computacionais que mostram que o método permite reconstruir imagens a partir da aquisição de dados com refletores em posições não modeladas sem a observação de artefatos. As simulações também mostram melhor resolução espacial do método proposto com relação ao algoritmo delay-and-sum (DAS). Também se observou melhor desempenho computacional do CG com relação à matriz inversa nas iterações do IRLS. / This work presents an inverse problem based method for ultrasound image reconstruction which uses the L2-norm (or euclidean norm) as a penalty for the error between the data and the solution, and the L1-norm as a regularization penalty. The motivation for the use of of L1 regularization is the sparsity promoting property of this type of regularization. The sparsity of L1 regularization circumvents the problem of excess of artifatcts that is observed in other approaches of inverse problem based reconstrucion in ultrasound. Such problem is mainly a consequence of the limitation in the discrete representation of a continuous object in the acquisition model. Due to this limitation, reflecting objects in the imaged area are often localized in positions that do not correspond precisely to one of the positions in the discrete model, therefore generating data that do not correspond to the model data. The formulations of the problem with L2 regularization and with L1 regularization are presented and compared in geometric and Bayesian terms. The optimization algorithm proposed is an implementation of Iteratively Reweighted Least Squares (IRLS) and uses the Conjugate Gradient (CG) method inside each iteration, thus being called IRLS-CG. Simulations with computer phantoms are realized showing that the proposed method allows for the reconstruction of images, without observable artifacts, from data with reflectors located in non-modeled positions. Simulations also show a better spatial resolution in the proposed method when compared to the delay-and-sum (DAS) algorithm. It was also observed better computational performance of CG when compared to the matrix inversion in the iterations of IRLS.