Autoregressive process parameters estimation from Compressed Sensing measurements and Bayesian dictionary learning

The main contribution of this thesis is the introduction of new techniques which allow to perform signal processing operations on signals represented by means of compressed sensing. Exploiting autoregressive modeling of the original signal, we obtain a compact yet representative description of the signal which can be estimated directly in the compressed domain. This is the key concept on which the applications we introduce rely on. In fact, thanks to proposed the framework it is possible to gain information about the original signal given compressed sensing measurements. This is done by means of autoregressive modeling which can be used to describe a signal through a small number of parameters. We develop a method to estimate these parameters given the compressed measurements by using an ad-hoc sensing matrix design and two different coupled estimators that can be used in different scenarios. This enables centralized and distributed estimation of the covariance matrix of a process given the compressed sensing measurements in a efficient way at low communication cost. Next, we use the characterization of the original signal done by means of few autoregressive parameters to improve compressive imaging. In particular, we use these parameters as a proxy to estimate the complexity of a block of a given image. This allows us to introduce a novel compressive imaging system in which the number of allocated measurements is adapted for each block depending on its complexity, i.e., spatial smoothness. The result is that a careful allocation of the measurements, improves the recovery process by reaching higher recovery quality at the same compression ratio in comparison to state-of-the-art compressive image recovery techniques. Interestingly, the parameters we are able to estimate directly in the compressed domain not only can improve the recovery but can also be used as feature vectors for classification. In fact, we also propose to use these parameters as more general feature vectors which allow to perform classification in the compressed domain. Remarkably, this method reaches high classification performance which is comparable with that obtained in the original domain, but with a lower cost in terms of dataset storage. In the second part of this work, we focus on sparse representations. In fact, a better sparsifying dictionary can improve the Compressed Sensing recovery performance. At first, we focus on the original domain and hence no dimensionality reduction by means of Compressed Sensing is considered. In particular, we develop a Bayesian technique which, in a fully automated fashion, performs dictionary learning. More in detail, using the uncertainties coming from atoms selection in the sparse representation step, this technique outperforms state-of-the-art dictionary learning techniques. Then, we also address image denoising and inpainting tasks using the aforementioned technique with excellent results. Next, we move to the compressed domain where a better dictionary is expected to provide improved recovery. We show how the Bayesian dictionary learning model can be adapted to the compressive case and the necessary assumptions that must be made when considering random projections. Lastly, numerical experiments confirm the superiority of this technique when compared to other compressive dictionary learning techniques.