Complex-valued data are encountered in many application areas of signal and image processing. In the context of optimization of functions of real variables, subspace algorithms have recently attracted much interest, owing to their efficiency for solving large-size problems while simultaneously offering theoretical convergence guarantees. The goal of this paper is to show how some of these methods can be successfully extended to the complex case. More precisely, we investigate the properties of the proposed complex-valued Majorize-Minimize Memory Gradient (3MG) algorithm. Important practical applications of these results arise in inverse problems. Here, we focus on image reconstruction in Parallel Magnetic Resonance Imaging (PMRI). The linear operator involved in the observation model then includes a subsampling operator over the k-space (spatial Fourier domain) the choice of which is analyzed through our numerical results. In addition, sensitivity matrices associated with the multiple coil channels come into play. Comparisons with existing optimization methods confirm the good performance of the proposed algorithm. * (1) Part of A. Florescu's work was supported by PhD Fellowship "Investitii in cercetare-inovare-dezvoltare pentru viitor (DocInvest)", EC project POS-DRU/107/1.5/S/76813.(2) Corresponding author. (
The paper addresses a classical problem of spectral analysis, that of high accuracy estimation of line-spectra, based on zero-padded Discrete Fourier Transform (DFT), transposed in the modern approach of sparse vectors estimation. Based on the analysis of the specific dictionary, some conclusions are drawn concerning the possibility to obtain high accuracy and resolution, by using a limited temporal analysis window. The issue of uniqueness is discussed. Two algorithms are presented, based on Matching Pursuit (MP) and Orthogonal Matching Pursuit (OMP).
Abstract-Complex-valued data play a prominent role in a number of signal and image processing applications. The aim of this paper is to establish some theoretical results concerning the Cramer-Rao bound for estimating a sparse complex-valued vector. Instead of considering a countable dictionary of vectors, we address the more challenging case of an uncountable set of vectors parameterized by a real variable. We also present a proximal forward-backward algorithm to minimize an ℓ0 penalized cost, which allows us to approach the derived bounds. These results are illustrated on a spectrum analysis problem in the case of irregularly sampled observations.
International audienceThe main focus of this work is the estimation of a complex valued signal assumed to have a sparse representation in an uncountable dictionary of signals. The dictionary elements are parameterized by a real-valued vector and the available observations are corrupted with an additive noise. By applying a linearization technique, the original model is recast as a constrained sparse perturbed model. The problem of the computation of the involved multiple parameters is addressed from a nonconvex optimization viewpoint. A cost function is defined including an arbitrary Lipschitz differentiable data fidelity term accounting for the noise statistics, and an l0-like penalty. A proximal algorithm is then employed to solve the resulting nonconvex and nonsmooth minimization problem. Experimental results illustrate the good practical performance of the proposed approach when applied to 2D spectrum analysis
The paper analyses the possibility to improve the resolution of spectral lines identification based on zero-padding Discrete Fourier Transform (DFT), by using a sparse vector estimation technique. A variant of the Orthogonal Matching Pursuit (OMP) algorithm called 'Multiple Choice OMP (MCOMP)' is proposed. This algorithm considers the difficulty arisen by the errors generated by the maximum coherence criterion in atoms selection, because of the high degree of dictionary coherence. It can be applied for a limited number of frequencies, with total guarantees or probabilistic guarantees.
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