abstract: High-order methods are known for their accuracy and computational performance when applied to solving partial differential equations and have widespread use in representing images compactly. Nonetheless, high-order methods have difficulty representing functions containing discontinuities or functions having slow spectral decay in the chosen basis. Certain sensing techniques such as MRI and SAR provide data in terms of Fourier coefficients, and thus prescribe a natural high-order basis. The field of compressed sensing has introduced a set of techniques based on $\ell^1$ regularization that promote sparsity and facilitate working with functions having discontinuities. In this dissertation, high-order methods and $\ell^1$ regularization are used to address three problems: reconstructing piecewise smooth functions from sparse and and noisy Fourier data, recovering edge locations in piecewise smooth functions from sparse and noisy Fourier data, and reducing time-stepping constraints when numerically solving certain time-dependent hyperbolic partial differential equations. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2016

Compressed Sensing (CS) is a new paradigm in signal processing which exploits the sparse or compressible nature of the signal to significantly reduce the number of measurements, without compromising on the signal reconstruction quality. Recently, many algorithms have been reported in the literature for efficient sparse signal reconstruction. Nevertheless, it is well known that the performance of any sparse reconstruction algorithm depends on many parameters like number of measurements, dimension of the sparse signal, the level of sparsity, the measurement noise power, and the underlying statistical distribution of the non-zero elements of the signal. It has been observed that a satisfactory performance of the sparse reconstruction algorithm mandates certain requirement on these parameters, which is different for different algorithms. Many applications are unlikely to fulfil this requirement. For example, imaging speed is crucial in many Magnetic Resonance Imaging (MRI) applications. This restricts the number of measurements, which in turn affects the medical diagnosis using MRI. Hence, any strategy to improve the signal reconstruction in such adverse scenario is of substantial interest in CS. Interestingly, it can be observed that the performance degradation of the sparse recovery algorithms in the aforementioned cases does not always imply a complete failure. That is, even in such adverse situations, a sparse reconstruction algorithm may provide partially correct information about the signal. In this thesis, we study this scenario and propose a novel fusion framework and an iterative framework which exploit the partial information available in the sparse signal estimate(s) to improve sparse signal reconstruction. The proposed fusion framework employs multiple sparse reconstruction algorithms, independently, for signal reconstruction. We first propose a fusion algorithm viz. FACS which fuses the estimates of multiple participating algorithms in order to improve the sparse signal reconstruction. To alleviate the inherent drawbacks of FACS and further improve the sparse signal reconstruction, we propose another fusion algorithm called CoMACS and variants of CoMACS. For low latency applications, we propose a latency friendly fusion algorithm called pFACS. We also extend the fusion framework to the MMV problem and propose the extension of FACS called MMV-FACS. We theoretically analyse the proposed fusion algorithms and derive guarantees for performance improvement. We also show that the proposed fusion algorithms are robust against both signal and measurement perturbations. Further, we demonstrate the efficacy of the proposed algorithms via numerical experiments: (i) using sparse signals with different statistical distributions in noise-free and noisy scenarios, and (ii) using real-world ECG signals. The extensive numerical experiments show that, for a judicious choice of the participating algorithms, the proposed fusion algorithms result in a sparse signal estimate which is often better than the sparse signal estimate of the best participating algorithm. The proposed fusion framework requires toemploy multiple sparse reconstruction algorithms for sparse signal reconstruction. We also propose an iterative framework and algorithm called {IFSRA to improve the performance of a given arbitrary sparse reconstruction algorithm. We theoretically analyse IFSRA and derive convergence guarantees under signal and measurement perturbations. Numerical experiments on synthetic and real-world data confirm the efficacy of IFSRA. The proposed fusion algorithms and IFSRA are general in nature and does not require any modification in the participating algorithm(s).

De nos jours, les réseaux de neurones artificiels sont largement utilisés dans de nombreusesapplications telles que le traitement d’image ou du signal. Récemment, un nouveau modèlede réseau de neurones a été proposé pour concevoir des mémoires associatives, le GBNN(Gripon-Berrou Neural Network). Ce modèle offre une capacité de stockage supérieure àcelle des réseaux de Hopfield lorsque les informations à mémoriser ont une distributionuniforme. Des méthodes améliorant leur performance pour des distributions non-uniformesainsi que des architectures matérielles mettant en œuvre les réseaux GBNN ont étéproposés. Cependant, ces solutions restent très coûteuses en ressources matérielles, et lesarchitectures proposées sont restreintes à des réseaux de tailles fixes et sont incapables depasser à l’échelle.Les objectifs de cette thèse sont les suivants : (1) concevoir des modèles inspirés du modèle GBNN et plus performants que l’état de l’art, (2) proposer des architectures moins coûteusesque les solutions existantes et (3) concevoir une architecture générique configurable mettanten œuvre les modèles proposés et capable de manipuler des réseaux de tailles variables.Les résultats des travaux de thèse sont exposés en plusieurs parties. Le concept de réseaux àclones de neurone et ses différentes instanciations sont présentés dans un premier temps. Cesréseaux offrent de meilleures performances que l’état de l’art pour un coût mémoireidentique lorsqu’une distribution non-uniforme des informations à mémoriser estconsidérée. Des optimisations de l’architecture matérielle sont ensuite introduites afin defortement réduire le coût en termes de ressources. Enfin, une architecture générique capablede passer à l’échelle et capable de manipuler des réseaux de tailles variables est proposée. / Nowadays, artificial neural networks are widely used in many applications such as image and signal processing. Recently, a new model of neural network was proposed to design associative memories, the GBNN (Gripon-Berrou Neural Network). This model offers a storage capacity exceeding those of Hopfield networks when the information to be stored has a uniform distribution. Methods improving performance for non-uniform distributions and hardware architectures implementing the GBNN networks were proposed. However, on one hand, these solutions are very expensive in terms of hardware resources and on the other hand, the proposed architectures can only implement fixed size networks and are not scalable. The objectives of this thesis are: (1) to design GBNN inspired models outperforming the state of the art, (2) to propose architectures cheaper than existing solutions and (3) to design a generic architecture implementing the proposed models and able to handle various sizes of networks. The results of these works are exposed in several parts. Initially, the concept of clone based neural networks and its variants are presented. These networks offer better performance than the state of the art for the same memory cost when a non-uniform distribution of the information to be stored is considered. The hardware architecture optimizations are then introduced to significantly reduce the cost in terms of resources. Finally, a generic scalable architecture able to handle various sizes of networks is proposed.

Cette thèse étudie et apporte des améliorations significatives sur trois techniques répandues en réduction de dimension : l'acquisition parcimonieuse (ou l'échantillonnage parcimonieux), la projection aléatoire et la représentation parcimonieuse. En acquisition parcimonieuse, la construction d’une matrice de réduction possédant à la fois de bonnes performances et une structure matérielle adéquate reste un défi de taille. Ici, nous proposons explicitement la matrice binaire optimale, avec éléments zéro-Un, en recherchant la meilleure propriété d’isométrie restreinte (RIP). Dans la pratique, un algorithme glouton efficace est successivement développé pour construire la matrice binaire optimale avec une taille arbitraire. Par ailleurs, nous étudions également un autre problème intéressant pour l'acquisition parcimonieuse, c'est celui de la performance des matrices d'acquisition parcimonieuse avec des taux de compression élevés. Pour la première fois, la limite inférieure de la performance des matrices aléatoires de Bernoulli pour des taux de compression croissants est observée et estimée. La projection aléatoire s'utilise principalement en classification mais la construction de la matrice de projection aléatoire s'avère également critique en termes de performance et de complexité. Cette thèse présente la matrice de projection aléatoire, de loin, la plus éparse. Celle-Ci est démontrée présenter la meilleure performance en sélection de caractéristiques, comparativement à d’autres matrices aléatoires plus denses. Ce résultat théorique est confirmé par de nombreuses expériences. Comme nouvelle technique pour la sélection de caractéristiques ou d’échantillons, la représentation parcimonieuse a récemment été largement appliquée dans le domaine du traitement d'image. Dans cette thèse, nous nous concentrons principalement sur ses applications de suivi d'objets dans une séquence d'images. Pour réduire la charge de calcul liée à la représentation parcimonieuse, un système simple mais efficace est proposé pour le suivi d'un objet unique. Par la suite, nous explorons le potentiel de cette représentation pour le suivi d'objets multiples. / This thesis studies three popular dimension reduction techniques: compressed sensing, random projection and sparse representation, and brings significant improvements on these techniques. In compressed sensing, the construction of sensing matrix with both good performance and hardware-Friendly structure has been a significant challenge. In this thesis, we explicitly propose the optimal zero-One binary matrix by searching the best Restricted Isometry Property. In practice, an efficient greedy algorithm is successively developed to construct the optimal binary matrix with arbitrary size. Moreover, we also study another interesting problem for compressed sensing, that is the performance of sensing matrices with high compression rates. For the first time, the performance floor of random Bernoulli matrices over increasing compression rates is observed and effectively estimated. Random projection is mainly used in the task of classification, for which the construction of random projection matrix is also critical in terms of both performance and complexity. This thesis presents so far the most sparse random projection matrix, which is proved holding better feature selection performance than other more dense random matrices. The theoretical result is confirmed with extensive experiments. As a novel technique for feature or sample selection, sparse representation has recently been widely applied in the area of image processing. In this thesis, we mainly focus our attention on its applications to visual object tracking. To reduce the computation load related to sparse representation, a simple but efficient scheme is proposed for the tracking of single object. Subsequently, the potential of sparse representation to multiobject tracking is investigated.

Magnetic Resonance Imaging (MRI) is a non-invasive medical imaging method that is used in the diagnosis of many common diseases. Compared to other medical imaging modalities, MRI has the ability to provide high-resolution 2D and 3D images in arbitrary orientations, without the use of potentially damaging ionizing radiation. Myocardial perfusion MRI is a promising non- invasive clinical way to detect cardiac disease. It can also provide quantitative analysis for blood flow within the heart. However, MRI requires longer scan times to acquire images at comparable resolutions to some other imaging modalities. Increasing image resolution, both spatially and temporally, is very important to myocardial perfusion MRI. The work presented in this dissertation focuses on the development of novel dynamic contrast-enhanced (DCE) MRI that is able to achieve both high spatial and temporal resolutions, as well as suitable spatial coverage of the heart. Three novel acquisition and reconstruction frame- works are proposed and analyzed in this dissertation. The first framework we propose uses a highly undersampled 3D Cartesian acquisition and total variation (TV) constrained reconstruction to accelerate the acquisition of myocardial perfu- sion images. This technique increases temporal resolution for contrast tracking without sacrificing spatial resolution. An analysis of the effect of different k-space trajectories using this technique is performed. The purpose of the second framework is to simplify cardiac perfusion studies. An ECG- gated saturation recovery sequence is regularly used for cardiac perfusion imaging. However, using an ungated acquisition has the potential benefit of reducing the acquisition time by eliminating the need for the ECG trigger signal. We present a novel non-Cartesian 2D multi-slice ungated acquisition, and demonstrate that it is a promising alternative to ECG-gated cardiac perfusion studies. An optimization analysis of our ungated acquisition is also presented. The third method in this dissertation combines the 2D ungated acquisition with multi-band excitation, which enables the excitation of multiple slices simultaneously. This method is able to reduce scan time not only through the ungated acquisition, but also from obtaining multiple slices at once. This allows us to achieve whole heart coverage without sacrificing temporal resolution. The contributions presented in this dissertation demonstrate the basic feasibility of car- diac perfusion MRI achieving whole-heart coverage in a clinical setting by overcoming the major existing limitations: speed of acquisition and spatial coverage.

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This research introduces a satellite characterization method based on its light curve by utilizing and adapting the methodology of compressed sensing. Compressed sensing is a mathematical theory, which is established in signal compression and which has recently been applied to an image reconstruction by single-pixel camera observation. In this thesis, compressed sensing in the use of single-pixel camera observations is compared with a satellite characterization via non-resolved light curves. The assumptions, limitations, and significant differences in utilizing compressed sensing for satellite characterization are discussed in detail. Assuming a reference observation can be used to estimate the so-called sensing matrix, compressed sensing enables to approximately reconstruct resolved satellite images revealing details about the specific satellite that has been observed based solely on non-resolved light curves. This has been shown explicitly in simulations. This result implies the great potential of compressed sensing in characterizing space objects that are so far away that traditional resolved imaging is not possible.

Compressed sensing has been a very active area of research and several elegant algorithms have been developed for the recovery of sparse signals in the past few years. However, most of these algorithms are either computationally expensive or make some assumptions that are not suitable for all real world problems. Recently, focus has shifted to Bayesian-based approaches that are able to perform sparse signal recovery at much lower complexity while invoking constraint and/or a priori information about the data. While Bayesian approaches have their advantages, these methods must have access to a priori statistics. Usually, these statistics are unknown and are often difficult or even impossible to predict. An effective workaround is to assume a distribution which is typically considered to be Gaussian, as it makes many signal processing problems mathematically tractable. Seemingly attractive, this assumption necessitates the estimation of the associated parameters; which could be hard if not impossible. In the thesis, we focus on this aspect of Bayesian recovery and present a framework to address the challenges mentioned above. The proposed framework allows Bayesian recovery of sparse signals but at the same time is agnostic to the distribution of the unknown sparse signal components. The algorithms based on this framework are agnostic to signal statistics and utilize a priori statistics of additive noise and the sparsity rate of the signal, which are shown to be easily estimated from data if not available. In the thesis, we propose several algorithms based on this framework which utilize the structure present in signals for improved recovery. In addition to the algorithm that considers just the sparsity structure of sparse signals, tools that target additional structure of the sparsity recovery problem are proposed. These include several algorithms for a) block-sparse signal estimation, b) joint reconstruction of several common support sparse signals, and c) distributed estimation of sparse signals. Extensive experiments are conducted to demonstrate the power and robustness of our proposed sparse signal estimation algorithms. Specifically, we target the problems of a) channel estimation in massive-MIMO, and b) Narrowband interference mitigation in SC-FDMA. We model these problems as sparse recovery problems and demonstrate how these could be solved naturally using the proposed algorithms.

京都大学 / 0048 / 新制・課程博士 / 博士(医学) / 甲第21001号 / 医博第4347号 / 新制||医||1027(附属図書館) / 京都大学大学院医学研究科医学専攻 / (主査)教授 溝脇 尚志, 教授 辻川 明孝, 教授 小泉 昭夫 / 学位規則第4条第1項該当 / Doctor of Medical Science / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(医学) / 甲第21662号 / 医博第4468号 / 新制||医||1035(附属図書館) / 京都大学大学院医学研究科医学専攻 / (主査)教授 溝脇 尚志, 教授 上本 伸二, 教授 増永 慎一郎 / 学位規則第4条第1項該当 / Doctor of Medical Science / Kyoto University / DFAM