Blind Compressed Sensing Over a Structured Union of Subspaces

This paper addresses the problem of simultaneous signal recovery and dictionary learning based on compressive measurements. Multiple signals are analyzed jointly, with multiple sensing matrices, under the assumption that the unknown signals come from a union of a small number of disjoint subspaces. This problem is important, for instance, in image inpainting applications, in which the multiple signals are constituted by (incomplete) image patches taken from the overall image. This work extends standard dictionary learning and block-sparse dictionary optimization, by considering compressive measurements, e.g., incomplete data). Previous work on blind compressed sensing is also generalized by using multiple sensing matrices and relaxing some of the restrictions on the learned dictionary. Drawing on results developed in the context of matrix completion, it is proven that both the dictionary and signals can be recovered with high probability from compressed measurements. The solution is unique up to block permutations and invertible linear transformations of the dictionary atoms. The recovery is contingent on the number of measurements per signal and the number of signals being sufficiently large; bounds are derived for these quantities. In addition, this paper presents a computationally practical algorithm that performs dictionary learning and signal recovery, and establishes conditions for its convergence to a local optimum. Experimental results for image inpainting demonstrate the capabilities of the method

A fast patch-dictionary method for whole image recovery

Various algorithms have been proposed for dictionary learning. Among those for image processing, many use image patches to form dictionaries. This paper focuses on whole-image recovery from corrupted linear measurements. We address the open issue of representing an image by overlapping patches: the overlapping leads to an excessive number of dictionary coefficients to determine. With very few exceptions, this issue has limited the applications of image-patch methods to the local kind of tasks such as denoising, inpainting, cartoon-texture decomposition, super-resolution, and image deblurring, for which one can process a few patches at a time. Our focus is global imaging tasks such as compressive sensing and medical image recovery, where the whole image is encoded together, making it either impossible or very ineffective to update a few patches at a time. Our strategy is to divide the sparse recovery into multiple subproblems, each of which handles a subset of non-overlapping patches, and then the results of the subproblems are averaged to yield the final recovery. This simple strategy is surprisingly effective in terms of both quality and speed. In addition, we accelerate computation of the learned dictionary by applying a recent block proximal-gradient method, which not only has a lower per-iteration complexity but also takes fewer iterations to converge, compared to the current state-of-the-art. We also establish that our algorithm globally converges to a stationary point. Numerical results on synthetic data demonstrate that our algorithm can recover a more faithful dictionary than two state-of-the-art methods. Combining our whole-image recovery and dictionary-learning methods, we numerically simulate image inpainting, compressive sensing recovery, and deblurring. Our recovery is more faithful than those of a total variation method and a method based on overlapping patches

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