Sparsity and cosparsity for audio declipping: a flexible non-convex approach

This work investigates the empirical performance of the sparse synthesis versus sparse analysis regularization for the ill-posed inverse problem of audio declipping. We develop a versatile non-convex heuristics which can be readily used with both data models. Based on this algorithm, we report that, in most cases, the two models perform almost similarly in terms of signal enhancement. However, the analysis version is shown to be amenable for real time audio processing, when certain analysis operators are considered. Both versions outperform state-of-the-art methods in the field, especially for the severely saturated signals

Certifying the restricted isometry property is hard

This paper is concerned with an important matrix condition in compressed sensing known as the restricted isometry property (RIP). We demonstrate that testing whether a matrix satisfies RIP is NP-hard. As a consequence of our result, it is impossible to efficiently test for RIP provided P \neq NP

A robust parallel algorithm for combinatorial compressed sensing

In previous work two of the authors have shown that a vector $x \in \mathbb{R}^n$ with at most $k < n$ nonzeros can be recovered from an expander sketch $Ax$ in $\mathcal{O}(\mathrm{nnz}(A)\log k)$ operations via the Parallel-$\ell_0$ decoding algorithm, where $\mathrm{nnz}(A)$ denotes the number of nonzero entries in $A \in \mathbb{R}^{m \times n}$. In this paper we present the Robust-$\ell_0$ decoding algorithm, which robustifies Parallel-$\ell_0$ when the sketch $Ax$ is corrupted by additive noise. This robustness is achieved by approximating the asymptotic posterior distribution of values in the sketch given its corrupted measurements. We provide analytic expressions that approximate these posteriors under the assumptions that the nonzero entries in the signal and the noise are drawn from continuous distributions. Numerical experiments presented show that Robust-$\ell_0$ is superior to existing greedy and combinatorial compressed sensing algorithms in the presence of small to moderate signal-to-noise ratios in the setting of Gaussian signals and Gaussian additive noise

A Compressed Sensing Algorithm for Sparse-View Pinhole Single Photon Emission Computed Tomography

Single Photon Emission Computed Tomography (SPECT) systems are being developed with multiple cameras and without gantry rotation to provide rapid dynamic acquisitions. However, the resulting data is angularly undersampled, due to the limited number of views. We propose a novel reconstruction algorithm for sparse-view SPECT based on Compressed Sensing (CS) theory. The algorithm models Poisson noise by modifying the Iterative Hard Thresholding algorithm to minimize the Kullback-Leibler (KL) distance by gradient descent. Because the underlying objects of SPECT images are expected to be smooth, a discrete wavelet transform (DWT) using an orthogonal spline wavelet kernel is used as the sparsifying transform. Preliminary feasibility of the algorithm was tested on simulated data of a phantom consisting of two Gaussian distributions. Single-pinhole projection data with Poisson noise were simulated at 128, 60, 15, 10, and 5 views over 360 degrees. Image quality was assessed using the coefficient of variation and the relative contrast between the two objects in the phantom. Overall, the results demonstrate preliminary feasibility of the proposed CS algorithm for sparse-view SPECT imaging

Mixed Operators in Compressed Sensing

Applications of compressed sensing motivate the possibility of using different operators to encode and decode a signal of interest. Since it is clear that the operators cannot be too different, we can view the discrepancy between the two matrices as a perturbation. The stability of L1-minimization and greedy algorithms to recover the signal in the presence of additive noise is by now well-known. Recently however, work has been done to analyze these methods with noise in the measurement matrix, which generates a multiplicative noise term. This new framework of generalized perturbations (i.e., both additive and multiplicative noise) extends the prior work on stable signal recovery from incomplete and inaccurate measurements of Candes, Romberg and Tao using Basis Pursuit (BP), and of Needell and Tropp using Compressive Sampling Matching Pursuit (CoSaMP). We show, under reasonable assumptions, that the stability of the reconstructed signal by both BP and CoSaMP is limited by the noise level in the observation. Our analysis extends easily to arbitrary greedy methods.Comment: CISS 2010 (44th Annual Conference on Information Sciences and Systems