Jean-Christophe Pesquet

Time-invariant orthonormal wavelet representations

A simple construction of an orthonormal basis starting with a so-called mother wavelet, together with an efficient implementation gained the wavelet decomposition easy acceptance and generated a great research interest in its applications. An orthonormal basis may not, however, always be a suitable representation of a signal, particularly when time (or space) invariance is a required property. The conventional way around this problem is to use a redundant decomposition. We address the time-invariance problem for orthonormal wavelet transforms and propose an extension to wavelet packet decompositions. We show that it,is possible to achieve time invariance and preserve the orthonormality. We subsequently propose an efficient approach to obtain such a decomposition. We demonstrate the importance of our method by considering some application examples in signal reconstruction and time delay estimation.

A proximal decomposition method for solving convex variational inverse problems

A broad range of inverse problems can be abstracted into the problem of minimizing the sum of several convex functions in a Hilbert space. We propose a proximal decomposition algorithm for solving this problem with an arbitrary number of nonsmooth functions and establish its weak convergence. The algorithm fully decomposes the problem in that it involves each function individually via its own proximity operator. A significant improvement over the methods currently in use in the area of inverse problems is that it is not limited to two nonsmooth functions. Numerical applications to signal and image processing problems are demonstrated.

Bayesian approach to best basis selection

Wavelet packets and local trigonometric bases provide an efficient framework and fast algorithms to obtain a best basis or best representation of deterministic signals. Applying these deterministic techniques to stochastic processes may, however, lead to variable results. We revisit this problem and introduce a prior model on the underlying signal in noise and account for the contaminating noise model as well. We thus develop a Bayesian-based approach to the best basis problem, while preserving the classical tree search efficiency.

A nonlocal structure tensor-based approach for multicomponent image recovery problems.

Nonlocal total variation (NLTV) has emerged as a useful tool in variational methods for image recovery problems. In this paper, we extend the NLTV-based regularization to multicomponent images by taking advantage of the structure tensor (ST) resulting from the gradient of a multicomponent image. The proposed approach allows us to penalize the nonlocal variations, jointly for the different components, through various ℓ1,p-matrix-norms with p ≥ 1. To facilitate the choice of the hyperparameters, we adopt a constrained convex optimization approach in which we minimize the data fidelity term subject to a constraint involving the ST-NLTV regularization. The resulting convex optimization problem is solved with a novel epigraphical projection method. This formulation can be efficiently implemented because of the flexibility offered by recent primal-dual proximal algorithms. Experiments are carried out for color, multispectral, and hyperspectral images. The results demonstrate the interest of introducing a nonlocal ST regularization and show that the proposed approach leads to significant improvements in terms of convergence speed over current state-of-the-art methods, such as the alternating direction method of multipliers.

On the Uniform Quantization of a Class of Sparse Sources

We consider the uniform scalar quantization of a class of mixed distributed memoryless sources, namely sources having a Bernoulli Generalized Gaussian (BGG) distribution. Both for low and high resolutions, asymptotic expressions of the distortion for a pth-order moment error measure, and close approximations of the entropy are provided for these sources. Operational rate-distortion functions at high bit rate and their slope factors at low bit rate are derived. The dependence of these results on p and the distribution parameters as well as the relation to the Shannon optimal rate-distortion bound are then discussed. The application of these results to transform coding in two simple cases is finally highlighted.

Estimation of noisy signals using time-invariant wavelet packets

Time-invariance plays a key role in many applications (e.g. signal reconstruction, identification, classification). Orthonormal wavelet and wavelet packet representations are known to be time-variant. We propose an extended wavelet packet decomposition, establish its time-invariance and give an efficient implementation. Some illustrating examples are also provided.<<ETX>>

Solving inverse problems with overcomplete transforms and convex optimization techniques

Many algorithms have been proposed during the last decade in order to deal with inverse problems. Of particular interest are convex optimization approaches that consist of minimizing a criteria generally composed of two terms: a data fidelity (linked to noise) term and a prior (regularization) term. As image properties are often easier to extract in a transform domain, frame representations may be fruitful. Potential functions are then chosen as priors to fit as well as possible empirical coefficient distributions. As a consequence, the minimization problem can be considered from two viewpoints: a minimization along the coefficients or along the image pixels directly. Some recently proposed iterative optimization algorithms can be easily implemented when the frame representation reduces to an orthonormal basis. Furthermore, it can be noticed that in this particular case, it is equivalent to minimize the criterion in the transform domain or in the image domain. However, much attention should be paid when an overcomplete representation is considered. In that case, there is no longer equivalence between coefficient and image domain minimization. This point will be developed throughout this paper. Moreover, we will discuss how the choice of the transform may influence parameters and operators necessary to implement algorithms.

A block-thresholding method for multispectral image denoising

The objective of this paper is to design a new estimator for multicomponent image denoising in the wavelet transform domain. To this end, we extend the block-based thresholding method initially proposed by Cai and Silverman, which takes advantage of the spatial dependence between the wavelet coefficients. In the case of multispectral images, we develop a more general framework for block-based shrinkage, the blocks being built from various combinations of the wavelet coefficients of the different image channels at adjacent spatial positions, for a given orientation and resolution level. In the presence of possibly spectrally correlated Gaussian noise, the parameters of the resulting estimator are optimized from the data by exploiting Steins principle. Simulations show the higher performance of our estimator for denoising multispectral satellite images.

A Stochastic Majorize-Minimize Subspace Algorithm for Online Penalized Least Squares Estimation

Stochastic approximation techniques play an important role in solving many problems encountered in machine learning or adaptive signal processing. In these contexts, the statistics of the data are often unknown a priori or their direct computation is too intensive, and they have thus to be estimated online from the observed signals. For batch optimization of an objective function being the sum of a data fidelity term and a penalization (e.g., a sparsity promoting function), Majorize-Minimize (MM) methods have recently attracted much interest since they are fast, highly flexible, and effective in ensuring convergence. The goal of this paper is to show how these methods can be successfully extended to the case when the data fidelity term corresponds to a least squares criterion and the cost function is replaced by a sequence of stochastic approximations of it. In this context, we propose an online version of an MM subspace algorithm and we study its convergence by using suitable probabilistic tools. Simulation results illustrate the good practical performance of the proposed algorithm associated with a memory gradient subspace, when applied to both nonadaptive and adaptive filter identification problems.

Image reconstruction from multiple sensors using stein's principle. Application to parallel MRI

We are interested in image reconstruction when data provided by several sensors are corrupted with a linear operator and an additive white Gaussian noise. This problem is addressed by invoking Steins Unbiased Risk Estimate (SURE) techniques. The key advantage of SURE methods is that they do not require prior knowledge about the statistics of the unknown image, while yielding an expression of the Mean Square Error (MSE) only depending on the statistics of the observed data. Hence, they avoid the difficult problem of hyperparameter estimation related to some prior distribution, which traditionally needs to be addressed in variational or Bayesian approaches. Consequently, a SURE approach can be applied by directly parameterizing a wavelet-based estimator and finding the optimal parameters that minimize the MSE estimate in reconstruction problems. Simulations carried out on parallel Magnetic Resonance Imaging (pMRI) images show the improved performance of our method with respect to classical alternatives.