This thesis presents compressive sensing algorithms that utilize system dynamics in the sparse signal recovery process. These dynamics may arise due to a time-varying signal, streaming measurements, or an adaptive signal transform. Compressive sensing theory has shown that under certain conditions, a sparse signal can be recovered from a small number of linear, incoherent measurements. The recovery algorithms, however, for the most part are static: they focus on finding the solution for a fixed set of measurements, assuming a fixed (sparse) structure of the signal. In this thesis, we present a suite of sparse recovery algorithms that cater to various dynamical settings. The main contributions of this research can be classified into the following two categories: 1) Efficient algorithms for fast updating of L1-norm minimization problems in dynamical settings. 2) Efficient modeling of the signal dynamics to improve the reconstruction quality; in particular, we use inter-frame motion in videos to improve their reconstruction from compressed measurements. Dynamic L1 updating: We present homotopy-based algorithms for quickly updating the solution for various L1 problems whenever the system changes slightly. Our objective is to avoid solving an L1-norm minimization program from scratch; instead, we use information from an already solved L1 problem to quickly update the solution for a modified system. Our proposed updating schemes can incorporate time-varying signals, streaming measurements, iterative reweighting, and data-adaptive transforms. Classical signal processing methods, such as recursive least squares and the Kalman filters provide solutions for similar problems in the least squares framework, where each solution update requires a simple low-rank update. We use homotopy continuation for updating L1 problems, which requires a series of rank-one updates along the so-called homotopy path. Dynamic models in video: We present a compressive-sensing based framework for the recovery of a video sequence from incomplete, non-adaptive measurements. We use a linear dynamical system to describe the measurements and the temporal variations of the video sequence, where adjacent images are related to each other via inter-frame motion. Our goal is to recover a quality video sequence from the available set of compressed measurements, for which we exploit the spatial structure using sparse representations of individual images in a spatial transform and the temporal structure, exhibited by dependencies among neighboring images, using inter-frame motion. We discuss two problems in this work: low-complexity video compression and accelerated dynamic MRI. Even though the processes for recording compressed measurements are quite different in these two problems, the procedure for reconstructing the videos is very similar.

La paralysie cérébrale (PC) est la première cause de l’handicap moteur de l’enfant en France (2 naissances pour 1000). Il s’agit d’une pathologie causée par des atteintes non progressives survenues lors du développement du cerveau chez le foetus ou le nourrisson. L’équin de la cheville est la déformation musculo-squelettique la plus fréquente chez les enfants atteints par la PC. Malgré des thérapies médico-chirurgicales multiples, le taux de récidive post-opératoire demeure très élevé(48%). Une des principales raisons des échecs des thérapies est le manque de connaissance de la biomécanique articulaire et musculaire. Les techniques d’imagerie en IRM dynamique permettent aujourd’hui d’explorer l’appareil musculo-squelettique au cours du mouvement dans les 3 dimensions de l’espace avec une grande précision (<1mm). Cependant, ces techniques viennent avec leur propre liste de problèmes tels que la résolution réduite, l’anisotropie et les artefacts de mouvement. Dans cette thèse, nous abordons ces problèmes en combinant l’information spatiale de l’IRM conventionnel avec l’information temporelle fournie par les séquences IRM dynamique. Nous avons réussi à atteindre l’objectif principal de ces travaux de recherche en développant des algorithmes robustes combinant des aspects informatiques et mathématiques (dont le recalage d’images basé sur l’intensité était le facteur clé) qui nous ont permis de reconstruire les mouvements articulaires et donc d’établir une analyse biomécanique de la cheville en plus de la reconstruction spatio-temporelle de la séquence dynamique en utilisant une approche logeuclidienne. Les algorithmes proposés ont été appliqués sur la base de données actuellement disponible (contenant 6 sujets normaux) et devraient être également appliqués sur une base plus large contenant des sujets pathologiques de la même tranche d’âges afin de comparer les deux populations et de caractériser la pathologie. / Cerebral Palsy (CP) is a common birth pathology in children leading to ankle joint deformity, also known as the Spastic Equinus (SE) deformity, which causes abnormal function of the joint. While the management of ankle disorders focuses on restoring the joint functions, the underlying pathomechanics is not clearly understood yet. To better understand the biomechanics of the pediatric ankle joint, it is crucial to establish in vivo normative joint biomechanics before focusing on pathomechanics studies. Dynamic MRI has made it possible to non-invasively capture the ankle joint during a complete motion cycle. However, dynamic MRI comes with its own set of unique challenges such as low resolution, anisotropy, and motion artifacts. This motivates our choice for combining spatial information of conventional static MRI with temporal information of dynamic MRI sequences. The global aim of this research work is to build computational frameworks and to develop robust intensity-based approaches for estimating the joint motion and deformations from 3D+t MRI data, and thus for deriving the joint kinematics and the joint contact mechanics during a single cycle of dorsiplantarflexion. Due to a lack of sufficient Imaging data in the pediatric cohort, the proposed algorithms are applied on dynamic MRI data (portraying both passive and active ankle motions) from 6 healthy children.

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Die hier verwendete Methode der dynamischen Magnetresonanztomographie (dMRT) erlaubte bei Hirntumorpatienten erstmals, gleichzeitig neben dem regionalen zerebralen Blutvolumen (rCBV) und dem regionalen zerebralen Blutfluß weitere Parameter, wie Permeabilitäten, die interstitiellen Volumina und das Zellvolumen, zu bestimmen. Anhand dieser Parameter sollte erstens geprüft werden, inwieweit diese zu einer besseren Malignitätseinstufung von Hirntumoren beitragen. Zweitens sollte geklärt werden, inwiefern sich die untersuchten Tumorgruppen voneinander unterscheiden lassen. Drittens war es Ziel, ein in-vivo-Grading für die Gliome zu entwickeln. Es wurden 60 Patienten mit verschiedenen Tumoren, wie Gliome, Metastasen, Meningeome und Lymphome, untersucht. Die aus der dMRT-Untersuchung erhaltenen Daten wurden mit einem pharmakokinetischen Modell ausgewertet. Für jeden Patienten wurden die oben genannten Parameter in Form von Bildern dargestellt und quantitativ berechnet. Für die Tumordifferenzierung eignete sich das mittlere rCBV am besten: Innerhalb der Gliome konnte signifikant zwischen den Grad-II- und Grad-III-Gliomen und den Grad-II- und Grad-IV-Gliomen unterschieden werden. Weiterhin konnten die Meningeome signifikant von den anderen untersuchten Tumorentitäten abgegerenzt werden. Das in-vivo-Grading der Gliome erlaubte in 71 % der Fälle eine korrekte Zuordnung zum WHO-Grad. Die Parameterbilder lieferten neben Informationen für die Tumordifferenzierung auch beispielsweise Hinweise auf den heterogenen Tumoraufbau. Des weiteren ermöglichten sie, Narbengewebe gegenüber Tumorgewebe abzugrenzen und Folgen einer Strahlentherapie zu beobachten. Schließlich waren Aussagen über die Gefäßarchitektur und das Wachstum unterschiedlicher Tumorgruppen möglich. Die mit der hier verwendeten Methode der dMRT erhaltenen Parameter boten mehrere Vorteile: Eine Differenzierung einzelner Tumorgruppen war möglich. Für die Gliome konnte mittels des in-vivo-Gradings eine quantitative Malignitätseinschätzung erfolgen. Die gewonnenen Informationen über den heterogenen Tumoraufbau erlauben bessere Biopsieergebnisse. Zusätzlich wurden Hinweise auf die Tumorpathophysiologie erhalten und es erschien möglich Veränderungen nach Therapie zu beobachten. / A method of dynamic magnet resonance imaging (dMRI) was used, which allowed for the first time to determine simultaneously several parameters in patients with brain tumors. These parameters were the regional cerebral blood volume (rCBV), the regional cerebral blood flow, and in addition, permeabilities, interstitial volumes, and the cell volume. First, it should be determined to what extent these parameters allow a better classification of the malignancy of brain tumors. Second, it should be evaluated how far it is possible to differentiate the examined tumor groups from each other. Third, a method for an in-vivo-grading specifically for gliomas should be developed. Altogether 60 patients with different tumors such as gliomas, metastasis, meningiomas, and lymphomas were examined. The data of the dMRI examination were evaluated using a pharmacokinetic model. For every patient, the parameters mentioned above were shown in maps and calculated quantitatively. The mean rCBV resulted in the best tumor differentiation: within the group of gliomas it was possible to differentiate significantly between grade-II- and grade-III-gliomas and grade-II- and grade-IV-gliomas. Furthermore, meningiomas were differentiated significantly from the other tumors. In respect to the group of gliomas, the tumor grades determined by the developed in-vivo-grading corresponded with the WHO grade of each glioma in 71 % of the cases. The parameter maps were not only usefull for tumor differentiation, but also yielded information concerning the heterogenous tumor structure. Additionally, these maps allowed to differentiate scar tissue from tumor tissue and effects of a radiotherapy could be observed. Finally, information about the vessel architecture and the growth of different tumor groups could be obtained. The parameters determined by the dMRI method used here offered several advantages: it was possible to differentiate between single tumor groups. For the gliomas, a quantitative malignancy classification resulted from the in-vivo-grading. The information concerning the structure of the heterogeneity of the tumor allows for better biopsy results. Additionally, information was also obtained concerning the pathophysiology of the tumors and it seemed possible to observe changes after a therapy.