Cette thèse est consacrée à l'élaboration de nouvelles méthodes d'acquisition et de traitement de données en IRM de diffusion (IRMd) afin de caractériser la diffusion des molécules d'eau dans les fibres de matière blanche à l'échelle d'un voxel. Plus particulièrement, nous travaillons sur un moyen de reconstruction précis de l'Ensemble Average Propagator (EAP), qui représente la fonction de probabilité de diffusion des molécules d'eau. Plusieurs modèles de diffusion tels que le tenseur de diffusion ou la fonction de distribution d'orientation sont très utilisés dans la communauté de l'IRMd afin de quantifier la diffusion des molécules d'eau dans le cerveau. Ces modèles sont des représentations partielles de l'EAP et ont été développés en raison du petit nombre de mesures nécessaires à leurs estimations. Cependant, il est important de pouvoir reconstruire précisément l'EAP afin d'acquérir une meilleure compréhension des mécanismes du cerveau et d'améliorer le diagnostique des troubles neurologiques. Une estimation correcte de l'EAP nécessite l'acquisition de nombreuses images de diffusion sensibilisées à des orientations différentes dans le q-space. Ceci rend son estimation trop longue pour être utilisée dans la plupart des scanners cliniques. Dans cette thèse, nous utilisons des techniques de reconstruction parcimonieuses et en particulier la technique connue sous le nom de Compressive Sensing (CS) afin d’accélérer le calcul de l'EAP. Les multiples aspects de la théorie du CS et de son application à l'IRMd sont présentés dans cette thèse. / This thesis is dedicated to the development of new acquisition and processing methods in diffusion MRI (dMRI) to characterize the diffusion of water molecules in white matter fiber bundles at the scale of a voxel. In particular, we focus our attention on the accurate recovery of the Ensemble Average Propagator (EAP), which represents the full 3D displacement of water molecule diffusion. Diffusion models such that the Diffusion Tensor or the Orientation Distribution Function (ODF) are largely used in the dMRI community in order to quantify water molecule diffusion. These models are partial EAP representations and have been developed due to the small number of measurement required for their estimations. It is thus of utmost importance to be able to accurately compute the EAP and order to acquire a better understanding of the brain mechanisms and to improve the diagnosis of neurological disorders. Estimating the full 3D EAP requires the acquisition of many diffusion images sensitized todifferent orientations in the q-space, which render the estimation of the EAP impossible in most of the clinical dMRI scanner. A surge of interest has been seen in order to decrease this time for acquisition. Some works focus on the development of new and efficient acquisition sequences. In this thesis, we use sparse coding techniques, and in particular Compressive Sensing (CS) to accelerate the computation of the EAP. Multiple aspects of the CS theory and its application to dMRI are presented in this thesis.

In many scenarios, events such as singularities and transients that carry important information about a signal undergo spreading during acquisition or transmission and it is important to localize the events. For example, edges in an image, point sources in a microscopy or astronomical image are blurred by the point-spread function (PSF) of the acquisition system, while in a speech signal, the epochs corresponding to glottal closure instants are shaped by the vocal tract response. Such events can be extracted with the help of techniques that promote sparsity, which enables separation of the smooth components from the transient ones. In this thesis, we consider development of such sparsity promoting techniques. The contributions of the thesis are three-fold: (i) an auditory-motivated continuous wavelet design and representation, which helps identify singularities; (ii) a sparsity-driven deconvolution technique; and (iii) a sparsity-driven deconvolution technique for reconstruction of nite-rate-of-innovation (FRI) signals. We use the speech signal for illustrating the performance of the techniques in the first two parts and super-resolution microscopy (2-D) for the third part. In the rst part, we develop a continuous wavelet transform (CWT) starting from an auditory motivation. Wavelet analysis provides good time and frequency localization, which has made it a popular tool for time-frequency analysis of signals. The CWT is a multiresolution analysis tool that involves decomposition of a signal using a constant-Q wavelet filterbank, akin to the time-frequency analysis performed by basilar membrane in the peripheral human auditory system. This connection motivated us to develop wavelets that possess auditory localization capabilities. Gammatone functions are extensively used in the modeling of the basilar membrane, but the non-zero average of the functions poses a hurdle. We construct bona de wavelets from the Gammatone function called Gammatone wavelets and analyze their properties such as admissibility, time-bandwidth product, vanishing moments, etc.. Of particular interest is the vanishing moments property, which enables the wavelet to suppress smooth regions in a signal leading to sparsi cation. We show how this property of the Gammatone wavelets coupled with multiresolution analysis could be employed for singularity and transient detection. Using these wavelets, we also construct equivalent lterbank models and obtain cepstral feature vectors out of such a representation. We show that the Gammatone wavelet cepstral coefficients (GWCC) are effective for robust speech recognition compared with mel-frequency cepstral coefficients (MFCC). In the second part, we consider the problem of sparse blind deconvolution (SBD) starting from a signal obtained as the convolution of an unknown PSF and a sparse excitation. The BD problem is ill-posed and the goal is to employ sparsity to come up with an accurate solution. We formulate the SBD problem within a Bayesian framework. The estimation of lter and excitation involves optimization of a cost function that consists of an `2 data- fidelity term and an `p-norm (p 2 [0; 1]) regularizer, as the sparsity promoting prior. Since the `p-norm is not differentiable at the origin, we consider a smoothed version of the `p-norm as a proxy in the optimization. Apart from the regularizer being non-convex, the data term is also non-convex in the filter and excitation as they are both unknown. We optimize the non-convex cost using an alternating minimization strategy, and develop an alternating `p `2 projections algorithm (ALPA). We demonstrate convergence of the iterative algorithm and analyze in detail the role of the pseudo-inverse solution as an initialization for the ALPA and provide probabilistic bounds on its accuracy considering the presence of noise and the condition number of the linear system of equations. We also consider the case of bounded noise and derive tight tail bounds using the Hoe ding inequality. As an application, we consider the problem of blind deconvolution of speech signals. In the linear model for speech production, voiced speech is assumed to be the result of a quasi-periodic impulse train exciting a vocal-tract lter. The locations of the impulses or epochs indicate the glottal closure instants and the spacing between them the pitch. Hence, the excitation in the case of voiced speech is sparse and its deconvolution from the vocal-tract filter is posed as a SBD problem. We employ ALPA for SBD and show that excitation obtained is sparser than the excitations obtained using sparse linear prediction, smoothed `1=`2 sparse blind deconvolution algorithm, and majorization-minimization-based sparse deconvolution techniques. We also consider the problem of epoch estimation and show that epochs estimated by ALPA in both clean and noisy conditions are closer to the instants indicated by the electroglottograph when with to the estimates provided by the zero-frequency ltering technique, which is the state-of-the-art epoch estimation technique. In the third part, we consider the problem of deconvolution of a specific class of continuous-time signals called nite-rate-of-innovation (FRI) signals, which are not bandlimited, but specified by a nite number of parameters over an observation interval. The signal is assumed to be a linear combination of delayed versions of a prototypical pulse. The reconstruction problem is posed as a 2-D SBD problem. The kernel is assumed to have a known form but with unknown parameters. Given the sampled version of the FRI signal, the delays quantized to the nearest point on the sampling grid are rst estimated using proximal-operator-based alternating `p `2 algorithm (ALPAprox), and then super-resolved to obtain o -grid (O. G.) estimates using gradient-descent optimization. The overall technique is termed OG-ALPAprox. We show application of OG-ALPAprox to a particular modality of super-resolution microscopy (SRM), called stochastic optical reconstruction microscopy (STORM). The resolution of the traditional optical microscope is limited by di raction and is termed as Abbe's limit. The goal of SRM is to engineer the optical imaging system to resolve structures in specimens, such as proteins, whose dimensions are smaller than the di raction limit. The specimen to be imaged is tagged or labeled with light-emitting or uorescent chemical compounds called uorophores. These compounds speci cally bind to proteins and exhibit uorescence upon excitation. The uorophores are assumed to be point sources and the light emitted by them undergo spreading due to di raction. STORM employs a sequential approach, wherein each step only a few uorophores are randomly excited and the image is captured by a sensor array. The obtained image is di raction-limited, however, the separation between the uorophores allows for localizing the point sources with high precision. The localization is performed using Gaussian peak- tting. This process of random excitation coupled with localization is performed sequentially and subsequently consolidated to obtain a high-resolution image. We pose the localization as a SBD problem and employ OG-ALPAprox to estimate the locations. We also report comparisons with the de facto standard Gaussian peak- tting algorithm and show that the statistical performance is superior. Experimental results on real data show that the reconstruction quality is on par with the Gaussian peak- tting.

On étudie l’application des algorithmes de décomposition matricielles tel que la Factorisation Matricielle Non-négative (FMN), aux représentations fréquentielles de signaux audio musicaux. Ces algorithmes, dirigés par une fonction d’erreur de reconstruction, apprennent un ensemble de fonctions de base et un ensemble de coef- ficients correspondants qui approximent le signal d’entrée. On compare l’utilisation de trois fonctions d’erreur de reconstruction quand la FMN est appliquée à des gammes monophoniques et harmonisées: moindre carré, divergence Kullback-Leibler, et une mesure de divergence dépendente de la phase, introduite récemment. Des nouvelles méthodes pour interpréter les décompositions résultantes sont présentées et sont comparées aux méthodes utilisées précédemment qui nécessitent des connaissances du domaine acoustique. Finalement, on analyse la capacité de généralisation des fonctions de bases apprises par rapport à trois paramètres musicaux: l’amplitude, la durée et le type d’instrument. Pour ce faire, on introduit deux algorithmes d’étiquetage des fonctions de bases qui performent mieux que l’approche précédente dans la majorité de nos tests, la tâche d’instrument avec audio monophonique étant la seule exception importante. / We study the application of unsupervised matrix decomposition algorithms such as Non-negative Matrix Factorization (NMF) to frequency domain representations of music audio signals. These algorithms, driven by a given reconstruction error function, learn a set of basis functions and a set of corresponding coefficients that approximate the input signal. We compare the use of three reconstruction error functions when NMF is applied to monophonic and harmonized musical scales: least squares, Kullback-Leibler divergence, and a recently introduced “phase-aware” divergence measure. Novel supervised methods for interpreting the resulting decompositions are presented and compared to previously used methods that rely on domain knowledge. Finally, the ability of the learned basis functions to generalize across musical parameter values including note amplitude, note duration and instrument type, are analyzed. To do so, we introduce two basis function labeling algorithms that outperform the previous labeling approach in the majority of our tests, instrument type with monophonic audio being the only notable exception.

On étudie l’application des algorithmes de décomposition matricielles tel que la Factorisation Matricielle Non-négative (FMN), aux représentations fréquentielles de signaux audio musicaux. Ces algorithmes, dirigés par une fonction d’erreur de reconstruction, apprennent un ensemble de fonctions de base et un ensemble de coef- ficients correspondants qui approximent le signal d’entrée. On compare l’utilisation de trois fonctions d’erreur de reconstruction quand la FMN est appliquée à des gammes monophoniques et harmonisées: moindre carré, divergence Kullback-Leibler, et une mesure de divergence dépendente de la phase, introduite récemment. Des nouvelles méthodes pour interpréter les décompositions résultantes sont présentées et sont comparées aux méthodes utilisées précédemment qui nécessitent des connaissances du domaine acoustique. Finalement, on analyse la capacité de généralisation des fonctions de bases apprises par rapport à trois paramètres musicaux: l’amplitude, la durée et le type d’instrument. Pour ce faire, on introduit deux algorithmes d’étiquetage des fonctions de bases qui performent mieux que l’approche précédente dans la majorité de nos tests, la tâche d’instrument avec audio monophonique étant la seule exception importante. / We study the application of unsupervised matrix decomposition algorithms such as Non-negative Matrix Factorization (NMF) to frequency domain representations of music audio signals. These algorithms, driven by a given reconstruction error function, learn a set of basis functions and a set of corresponding coefficients that approximate the input signal. We compare the use of three reconstruction error functions when NMF is applied to monophonic and harmonized musical scales: least squares, Kullback-Leibler divergence, and a recently introduced “phase-aware” divergence measure. Novel supervised methods for interpreting the resulting decompositions are presented and compared to previously used methods that rely on domain knowledge. Finally, the ability of the learned basis functions to generalize across musical parameter values including note amplitude, note duration and instrument type, are analyzed. To do so, we introduce two basis function labeling algorithms that outperform the previous labeling approach in the majority of our tests, instrument type with monophonic audio being the only notable exception.

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