Sparsity and `Something Else': An Approach to Encrypted Image Folding

A property of sparse representations in relation to their capacity for information storage is discussed. It is shown that this feature can be used for an application that we term Encrypted Image Folding. The proposed procedure is realizable through any suitable transformation. In particular, in this paper we illustrate the approach by recourse to the Discrete Cosine Transform and a combination of redundant Cosine and Dirac dictionaries. The main advantage of the proposed technique is that both storage and encryption can be achieved simultaneously using simple processing steps.Comment: Revised manuscript- Software for implementing the Encrypted Image Folding proposed in this paper is available on http://www.nonlinear-approx.info

Block-Sparse Recovery via Convex Optimization

Given a dictionary that consists of multiple blocks and a signal that lives in the range space of only a few blocks, we study the problem of finding a block-sparse representation of the signal, i.e., a representation that uses the minimum number of blocks. Motivated by signal/image processing and computer vision applications, such as face recognition, we consider the block-sparse recovery problem in the case where the number of atoms in each block is arbitrary, possibly much larger than the dimension of the underlying subspace. To find a block-sparse representation of a signal, we propose two classes of non-convex optimization programs, which aim to minimize the number of nonzero coefficient blocks and the number of nonzero reconstructed vectors from the blocks, respectively. Since both classes of problems are NP-hard, we propose convex relaxations and derive conditions under which each class of the convex programs is equivalent to the original non-convex formulation. Our conditions depend on the notions of mutual and cumulative subspace coherence of a dictionary, which are natural generalizations of existing notions of mutual and cumulative coherence. We evaluate the performance of the proposed convex programs through simulations as well as real experiments on face recognition. We show that treating the face recognition problem as a block-sparse recovery problem improves the state-of-the-art results by 10% with only 25% of the training data.Comment: IEEE Transactions on Signal Processin

A Hierarchical Bayesian Model for Frame Representation

In many signal processing problems, it may be fruitful to represent the signal under study in a frame. If a probabilistic approach is adopted, it becomes then necessary to estimate the hyper-parameters characterizing the probability distribution of the frame coefficients. This problem is difficult since in general the frame synthesis operator is not bijective. Consequently, the frame coefficients are not directly observable. This paper introduces a hierarchical Bayesian model for frame representation. The posterior distribution of the frame coefficients and model hyper-parameters is derived. Hybrid Markov Chain Monte Carlo algorithms are subsequently proposed to sample from this posterior distribution. The generated samples are then exploited to estimate the hyper-parameters and the frame coefficients of the target signal. Validation experiments show that the proposed algorithms provide an accurate estimation of the frame coefficients and hyper-parameters. Application to practical problems of image denoising show the impact of the resulting Bayesian estimation on the recovered signal quality

On sparsity averaging

Recent developments in Carrillo et al. (2012) and Carrillo et al. (2013) introduced a novel regularization method for compressive imaging in the context of compressed sensing with coherent redundant dictionaries. The approach relies on the observation that natural images exhibit strong average sparsity over multiple coherent frames. The associated reconstruction algorithm, based on an analysis prior and a reweighted $\ell_1$ scheme, is dubbed Sparsity Averaging Reweighted Analysis (SARA). We review these advances and extend associated simulations establishing the superiority of SARA to regularization methods based on sparsity in a single frame, for a generic spread spectrum acquisition and for a Fourier acquisition of particular interest in radio astronomy.Comment: 4 pages, 3 figures, Proceedings of 10th International Conference on Sampling Theory and Applications (SampTA), Code available at https://github.com/basp-group/sopt, Full journal letter available at http://arxiv.org/abs/arXiv:1208.233

Sparse Signal Separation in Redundant Dictionaries

We formulate a unified framework for the separation of signals that are sparse in "morphologically" different redundant dictionaries. This formulation incorporates the so-called "analysis" and "synthesis" approaches as special cases and contains novel hybrid setups. We find corresponding coherence-based recovery guarantees for an l1-norm based separation algorithm. Our results recover those reported in Studer and Baraniuk, ACHA, submitted, for the synthesis setting, provide new recovery guarantees for the analysis setting, and form a basis for comparing performance in the analysis and synthesis settings. As an aside our findings complement the D-RIP recovery results reported in Cand\`es et al., ACHA, 2011, for the "analysis" signal recovery problem: minimize_x ||{\Psi}x||_1 subject to ||y - Ax||_2 \leq {\epsilon}, by delivering corresponding coherence-based recovery results.Comment: Proc. of IEEE International Symposium on Information Theory (ISIT), Boston, MA, July 201