Papers by Jean-christophe Pesquet
The ℓ 1 /ℓ 2 ratio regularization function has shown good performance for retrieving sparse signa... more The ℓ 1 /ℓ 2 ratio regularization function has shown good performance for retrieving sparse signals in a number of recent works, in the context of blind deconvolution. Indeed, it benefits from a scale invariance property much desirable in the blind context. However, the ℓ 1 /ℓ 2 function raises some difficulties when solving the nonconvex and nonsmooth minimization problems resulting from the use of such a penalty term in current restoration methods. In this paper, we propose a new penalty based on a smooth approximation to the ℓ 1 /ℓ 2 function. In addition, we develop a proximal-based algorithm to solve variational problems involving this function and we derive theoretical convergence results. We demonstrate the effectiveness of our method through a comparison with a recent alternating optimization strategy dealing with the exact ℓ 1 /ℓ 2 term, on an application to seismic data blind deconvolution.
ABSTRACT In this paper, we consider the problem of estimating a complex-valued signal having a sp... more ABSTRACT In this paper, we consider the problem of estimating a complex-valued signal having a sparse representation in an uncountable family of vectors. The available observations are corrupted with an additive noise and the elements of the dictionary are parameterized by a scalar real variable. By a linearization technique, the original model is recast as a constrained sparse perturbed model. An optimization approach is then proposed to estimate the parameters involved in this model. The cost function includes an arbitrary Lipschitz differentiable data fidelity term accounting for the noise statistics, and an l0 penalty. A forward-backward algorithm is employed to solve the resulting non-convex and non-smooth minimization problem. This algorithm can be viewed as a generalization of an iterative hard thresholding method and its local convergence can be established. Simulation results illustrate the good practical performance of the proposed approach when applied to spectrum estimation.
ABSTRACT In this paper, we consider a class of differentiable criteria for sparse image recovery ... more ABSTRACT In this paper, we consider a class of differentiable criteria for sparse image recovery problems. The regularization is applied to a linear transform of the target image. As special cases, it includes edge preserving measures or frame analysis potentials. As shown by our asymptotic results, the considered ℓ2 — ℓ0 penalties may be employed to approximate solutions to ℓ0 penalized optimization problems. One of the advantages of the approach is that it allows us to derive an efficient Majorize-Minimize Memory Gradient algorithm. The fast convergence properties of the proposed optimization algorithm are illustrated through image restoration examples.
In this work, we consider a class of differentiable criteria for sparse image computing problems,... more In this work, we consider a class of differentiable criteria for sparse image computing problems, where a nonconvex regularization is applied to an arbitrary linear transform of the target image. As special cases, it includes edge preserving measures or frame-analysis potentials commonly used in image processing. As shown by our asymptotic results, the ℓ 2 − ℓ 0 penalties we consider may be employed to provide approximate solutions to ℓ 0penalized optimization problems. One of the advantages of the proposed approach is that it allows us to derive an efficient Majorize-Minimize subspace algorithm. The convergence of the algorithm is investigated by using recent results in nonconvex optimization. The fast convergence properties of the proposed optimization method are illustrated through image processing examples. In particular, its effectiveness is demonstrated on several data recovery problems.
2014 10th International Conference on Communications (COMM), 2014
Complex-valued data play a prominent role in a number of signal and image processing applications... more 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.
Signal Processing, 2013
Complex-valued data are encountered in many application areas of signal and image processing. In ... more 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.
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Papers by Jean-christophe Pesquet