TV-min and Greedy Pursuit for Constrained Joint Sparsity and Application to Inverse Scattering

This paper proposes a general framework for compressed sensing of constrained joint sparsity (CJS) which includes total variation minimization (TV-min) as an example. TV- and 2-norm error bounds, independent of the ambient dimension, are derived for the CJS version of Basis Pursuit and Orthogonal Matching Pursuit. As an application the results extend Cand`es, Romberg and Tao's proof of exact recovery of piecewise constant objects with noiseless incomplete Fourier data to the case of noisy data.Comment: Mathematics and Mechanics of Complex Systems (2013

Compressive Sensing Theory for Optical Systems Described by a Continuous Model

A brief survey of the author and collaborators' work in compressive sensing applications to continuous imaging models.Comment: Chapter 3 of "Optical Compressive Imaging" edited by Adrian Stern published by Taylor & Francis 201

Conditioning of Random Block Subdictionaries with Applications to Block-Sparse Recovery and Regression

The linear model, in which a set of observations is assumed to be given by a linear combination of columns of a matrix, has long been the mainstay of the statistics and signal processing literature. One particular challenge for inference under linear models is understanding the conditions on the dictionary under which reliable inference is possible. This challenge has attracted renewed attention in recent years since many modern inference problems deal with the "underdetermined" setting, in which the number of observations is much smaller than the number of columns in the dictionary. This paper makes several contributions for this setting when the set of observations is given by a linear combination of a small number of groups of columns of the dictionary, termed the "block-sparse" case. First, it specifies conditions on the dictionary under which most block subdictionaries are well conditioned. This result is fundamentally different from prior work on block-sparse inference because (i) it provides conditions that can be explicitly computed in polynomial time, (ii) the given conditions translate into near-optimal scaling of the number of columns of the block subdictionaries as a function of the number of observations for a large class of dictionaries, and (iii) it suggests that the spectral norm and the quadratic-mean block coherence of the dictionary (rather than the worst-case coherences) fundamentally limit the scaling of dimensions of the well-conditioned block subdictionaries. Second, this paper investigates the problems of block-sparse recovery and block-sparse regression in underdetermined settings. Near-optimal block-sparse recovery and regression are possible for certain dictionaries as long as the dictionary satisfies easily computable conditions and the coefficients describing the linear combination of groups of columns can be modeled through a mild statistical prior.Comment: 39 pages, 3 figures. A revised and expanded version of the paper published in IEEE Transactions on Information Theory (DOI: 10.1109/TIT.2015.2429632); this revision includes corrections in the proofs of some of the result

Compressive Inverse Scattering I. High Frequency SIMO Measurements

Inverse scattering from discrete targets with the single-input-multiple-output (SIMO), multiple-input-single-output (MISO) or multiple-input-multiple-output (MIMO) measurements is analyzed by compressed sensing theory with and without the Born approximation. High frequency analysis of (probabilistic) recoverability by the $L^1$-based minimization/regularization principles is presented. In the absence of noise, it is shown that the $L^1$-based solution can recover exactly the target of sparsity up to the dimension of the data either with the MIMO measurement for the Born scattering or with the SIMO/MISO measurement for the exact scattering. The stability with respect to noisy data is proved for weak or widely separated scatterers. Reciprocity between the SIMO and MISO measurements is analyzed. Finally a coherence bound (and the resulting recoverability) is proved for diffraction tomography with high-frequency, few-view and limited-angle SIMO/MISO measurements.Comment: A new section on diffraction tomography added; typos fixed; new figures adde