Statistical Compressed Sensing of Gaussian Mixture Models

A novel framework of compressed sensing, namely statistical compressed sensing (SCS), that aims at efficiently sampling a collection of signals that follow a statistical distribution, and achieving accurate reconstruction on average, is introduced. SCS based on Gaussian models is investigated in depth. For signals that follow a single Gaussian model, with Gaussian or Bernoulli sensing matrices of O(k) measurements, considerably smaller than the O(k log(N/k)) required by conventional CS based on sparse models, where N is the signal dimension, and with an optimal decoder implemented via linear filtering, significantly faster than the pursuit decoders applied in conventional CS, the error of SCS is shown tightly upper bounded by a constant times the best k-term approximation error, with overwhelming probability. The failure probability is also significantly smaller than that of conventional sparsity-oriented CS. Stronger yet simpler results further show that for any sensing matrix, the error of Gaussian SCS is upper bounded by a constant times the best k-term approximation with probability one, and the bound constant can be efficiently calculated. For Gaussian mixture models (GMMs), that assume multiple Gaussian distributions and that each signal follows one of them with an unknown index, a piecewise linear estimator is introduced to decode SCS. The accuracy of model selection, at the heart of the piecewise linear decoder, is analyzed in terms of the properties of the Gaussian distributions and the number of sensing measurements. A maximum a posteriori expectation-maximization algorithm that iteratively estimates the Gaussian models parameters, the signals model selection, and decodes the signals, is presented for GMM-based SCS. In real image sensing applications, GMM-based SCS is shown to lead to improved results compared to conventional CS, at a considerably lower computational cost

Statistical Compressive Sensing of Gaussian Mixture Models

A new framework of compressive sensing (CS), namely statistical compressive sensing (SCS), that aims at efficiently sampling a collection of signals that follow a statistical distribution and achieving accurate reconstruction on average, is introduced. For signals following a Gaussian distribution, with Gaussian or Bernoulli sensing matrices of O(k) measurements, considerably smaller than the O(k log(N/k)) required by conventional CS, where N is the signal dimension, and with an optimal decoder implemented with linear filtering, significantly faster than the pursuit decoders applied in conventional CS, the error of SCS is shown tightly upper bounded by a constant times the k-best term approximation error, with overwhelming probability. The failure probability is also significantly smaller than that of conventional CS. Stronger yet simpler results further show that for any sensing matrix, the error of Gaussian SCS is upper bounded by a constant times the k-best term approximation with probability one, and the bound constant can be efficiently calculated. For signals following Gaussian mixture models, SCS with a piecewise linear decoder is introduced and shown to produce for real images better results than conventional CS based on sparse models

Audio Classification from Time-Frequency Texture

Time-frequency representations of audio signals often resemble texture images. This paper derives a simple audio classification algorithm based on treating sound spectrograms as texture images. The algorithm is inspired by an earlier visual classification scheme particularly efficient at classifying textures. While solely based on time-frequency texture features, the algorithm achieves surprisingly good performance in musical instrument classification experiments

Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity

A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAP-EM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost.Comment: 30 page

Task-Driven Adaptive Statistical Compressive Sensing of Gaussian Mixture Models

A framework for adaptive and non-adaptive statistical compressive sensing is developed, where a statistical model replaces the standard sparsity model of classical compressive sensing. We propose within this framework optimal task-specific sensing protocols specifically and jointly designed for classification and reconstruction. A two-step adaptive sensing paradigm is developed, where online sensing is applied to detect the signal class in the first step, followed by a reconstruction step adapted to the detected class and the observed samples. The approach is based on information theory, here tailored for Gaussian mixture models (GMMs), where an information-theoretic objective relationship between the sensed signals and a representation of the specific task of interest is maximized. Experimental results using synthetic signals, Landsat satellite attributes, and natural images of different sizes and with different noise levels show the improvements achieved using the proposed framework when compared to more standard sensing protocols. The underlying formulation can be applied beyond GMMs, at the price of higher mathematical and computational complexity

Sparse Grouping and Invariant Representations for Estimation and Recognition

Cette thèse développe plusieurs contributions pour le traitement du signal et des images ainsi que pour la vision par ordinateur. La première partie inclut un nouvel algorithme de débruitage des sons et un algorithme de super-résolution pour l'agrandissement des images. Ces algorithmes sont basés sur de nouvelles représentations parcimonieuses par blocs. Une procédure de seuillage par bloc temps-fréquence est introduite pour le débruitage audio, qui permet de réduire le bruit sans introduire d'artefacts, avec des résultats perceptuels et numériques nettement supérieurs à l'état de l'art. Cette première partie introduit aussi un nouveau cadre mathématique et algorithmique pour des problèmes inverses, avec une régularisation non-linéaire sur des dictionnaires de blocs géométriques dans une représentation parcimonieuse. Une approche générale pour la super-résolution d'images est introduite et des résultats numériques améliorant l'état de l'art sont obtenus. La deuxième partie de la thèse introduit un algorithme (ASIFT) de mise en correspondance d'images, qui est invariant relativement à des transformations affines. Il est démontré que cet algorithme satisfait les contraintes d'invariance et qu'il peut effectuer des correspondances entre des objets observés sous des angles arbitraires. Sa complexité numérique est du même ordre que les algorithmes les plus efficaces, avec une robustesse bien supérieure grâce à son invariance affine. La troisième partie de la thèse introduit une implémentation biologiquement plausible de groupements visuels. Inspiré par les mécanismes de synchronisation neuronale en groupement perceptuel, un algorithme général basé sur des oscillateurs neuronaux est proposé pour effectuer des groupements visuels. Cet algorithme permet d'obtenir des résultats prometteurs sur plusieurs problèmes, dont le groupement de points, l'intégration de contours, et la segmentation d'images