MMSE Denoising of Sparse Lévy Processes via Message Passing

Azin Amini, Ulugbek Kamilov, Michaël Unser
IEEE, 2012
Article

Résumé

Many recent algorithms for sparse signal recovery can be interpreted as maximum-a-posteriori (MAP) estimators relying on some specific priors. From this Bayesian perspective, state-of-the-art methods based on discrete-gradient regularizers, such as total-variation (TV) minimization, implicitly assume the signals to be sampled instances of Lévy processes with independent Laplace-distributed increments. By extending the concept to more general Lévy processes, we propose an efficient minimum-mean-squared error (MMSE) estimation method based on message-passing algorithms on factor graphs. The resulting algorithm can be used to benchmark the performance of the existing or design new algorithms for the recovery of sparse signals.

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https://infoscience.epfl.ch/record/211479?ln=fr

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Publications associées (5)

Mmse Denoising Of Sparse Levy Processes Via Message Passing

Arash Amini, Ulugbek Kamilov, Michaël Unser

Many recent algorithms for sparse signal recovery can be interpreted as maximum-a-posteriori (MAP) estimators relying on some specific priors. From this Bayesian perspective, state-of-the-art methods based on discrete-gradient regularizers, such as total-variation (TV) minimization, implicitly assume the signals to be sampled instances of Levy processes with independent Laplace-distributed increments. By extending the concept to more general Levy processes, we propose an efficient minimum-mean-squared error (MMSE) estimation method based on message-passing algorithms on factor graphs. The resulting algorithm can be used to benchmark the performance of the existing or design new algorithms for the recovery of sparse signals.

Ieee2012

Chargement

MMSE Estimation of Sparse Levy Processes

Arash Amini, Ulugbek Kamilov, Pedram Pad, Michaël Unser

We investigate a stochastic signal-processing framework for signals with sparse derivatives, where the samples of a Levy process are corrupted by noise. The proposed signal model covers the well-known Brownian motion and piecewise-constant Poisson process; moreover, the Levy family also contains other interesting members exhibiting heavy-tail statistics that fulfill the requirements of compressibility. We characterize the maximum-a-posteriori probability (MAP) and minimum mean-square error (MMSE) estimators for such signals. Interestingly, some of the MAP estimators for the Levy model coincide with popular signal-denoising algorithms (e.g., total-variation (TV) regularization). We propose a novel non-iterative implementation of the MMSE estimator based on the belief-propagation (BP) algorithm performed in the Fourier domain. Our algorithm takes advantage of the fact that the joint statistics of general Levy processes are much easier to describe by their characteristic function, as the probability densities do not always admit closed-form expressions. We then use our new estimator as a benchmark to compare the performance of existing algorithms for the optimal recovery of gradient-sparse signals.

Ieee-Inst Electrical Electronics Engineers Inc2013

Chargement

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We consider continuous-time sparse stochastic processes from which we have only a finite number of noisy/noiseless samples. Our goal is to estimate the noiseless samples (denoising) and the signal in-between (interpolation problem). By relying on tools from the theory of splines, we derive the joint a priori distribution of the samples and show how this probability density function can be factorized. The factorization enables us to tractably implement the maximum a posteriori and minimum mean-square error (MMSE) criteria as two statistical approaches for estimating the unknowns. We compare the derived statistical methods with well-known techniques for the recovery of sparse signals, such as the l(1) norm and Log (l(1)-l(0) relaxation) regularization methods. The simulation results show that, under certain conditions, the performance of the regularization techniques can be very close to that of the MMSE estimator.

Ieee-Inst Electrical Electronics Engineers Inc2013

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