Charles Dossal

Robust Sparse Analysis Regularization

This paper investigates the theoretical guarantees of l1-analysis regularization when solving linear inverse problems. Most of previous works in the literature have mainly focused on the sparse synthesis prior where the sparsity is measured as the l1 norm of the coefficients that synthesize the signal from a given dictionary. In contrast, the more general analysis regularization minimizes the l1 norm of the correlations between the signal and the atoms in the dictionary, where these correlations define the analysis support. The corresponding variational problem encompasses several well-known regularizations such as the discrete total variation and the fused Lasso. Our main contributions consist in deriving sufficient conditions that guarantee exact or partial analysis support recovery of the true signal in presence of noise. More precisely, we give a sufficient condition to ensure that a signal is the unique solution of the l1 -analysis regularization in the noiseless case. The same condition also guarantees exact analysis support recovery and l2-robustness of the l1-analysis minimizer vis-à-vis an enough small noise in the measurements. This condition turns to be sharp for the robustness of the sign pattern. To show partial support recovery and l2 -robustness to an arbitrary bounded noise, we introduce a stronger sufficient condition. When specialized to the l1-synthesis regularization, our results recover some corresponding recovery and robustness guarantees previously known in the literature. From this perspective, our work is a generalization of these results. We finally illustrate these theoretical findings on several examples to study the robustness of the 1-D total variation, shift-invariant Haar dictionary, and fused Lasso regularizations.

Local Behavior of Sparse Analysis Regularization: Applications to Risk Estimation

In this paper, we aim at recovering an unknown signal x0 from noisy L1measurements y=Phi*x0+w, where Phi is an ill-conditioned or singular linear operator and w accounts for some noise. To regularize such an ill-posed inverse problem, we impose an analysis sparsity prior. More precisely, the recovery is cast as a convex optimization program where the objective is the sum of a quadratic data fidelity term and a regularization term formed of the L1-norm of the correlations between the sought after signal and atoms in a given (generally overcomplete) dictionary. The L1-sparsity analysis prior is weighted by a regularization parameter lambda>0. In this paper, we prove that any minimizers of this problem is a piecewise-affine function of the observations y and the regularization parameter lambda. As a byproduct, we exploit these properties to get an objectively guided choice of lambda. In particular, we develop an extension of the Generalized Stein Unbiased Risk Estimator (GSURE) and show that it is an unbiased and reliable estimator of an appropriately defined risk. The latter encompasses special cases such as the prediction risk, the projection risk and the estimation risk. We apply these risk estimators to the special case of L1-sparsity analysis regularization. We also discuss implementation issues and propose fast algorithms to solve the L1 analysis minimization problem and to compute the associated GSURE. We finally illustrate the applicability of our framework to parameter(s) selection on several imaging problems.

Geometrical image estimation with orthogonal bandlet bases

This article presents the first adaptive quasi minimax estimator for geometrically regular images in the white noise model. This estimator is computed using a thresholding in an adapted orthogonal bandlet basis optimized for the noisy observed image. In order to analyze the quadratic risk of this best basis denoising, the thresholding in an orthogonal bandlets basis is recasted as a model selection process. The resulting estimator is computed with a fast algorithm whose theoretical performance can be derived. This efficiency is confirmed through numerical experiments on natural images.

Proximal Splitting Derivatives for Risk Estimation

This paper develops a novel framework to compute a projected Generalized Stein Unbiased Risk Estimator (GSURE) for a wide class of sparsely regularized solutions of inverse problems. This class includes arbitrary convex data fidelities with both analysis and synthesis mixed L1-L2 norms. The GSURE necessitates to compute the (weak) derivative of a solution w.r.t.~the observations. However, as the solution is not available in analytical form but rather through iterative schemes such as proximal splitting, we propose to iteratively compute the GSURE by differentiating the sequence of iterates. This provides us with a sequence of differential mappings, which, hopefully, converge to the desired derivative and allows to compute the GSURE. We illustrate this approach on total variation regularization with Gaussian noise and to sparse regularization with poisson noise, to automatically select the regularization parameter.

Bandlet Image Estimation with Model Selection

To estimate geometrically regular images in the white noise model and obtain an adaptive near asymptotic minimaxity result, we consider a model selection based bandlet estimator. This bandlet estimator combines the best basis selection behaviour of the model selection and the approximation properties of the bandlet dictionary. We derive its near asymptotic minimaxity for geometrically regular images as an example of model selection with general dictionary of orthogonal bases. This paper is thus also a self contained tutorial on model selection with orthogonal bases dictionary.

Optimal Dual Certificates for Noise Robustness Bounds in Compressive Sensing

The paper deals with optimizing Lipschitz bounds relating locally the reconstruction error to the measurement error in the RIPless compressive sensing framework. Most recent theoretical papers in the field parametrize such bounds relative to certain families of vectors called dual certificates, which are fundamental to several reconstruction criteria. We show in the paper that such a family of bounds admits a unique minimizer that has a deep geometric meaning and can be explicitly constructed via a convex projection algorithm that we describe. We also give a faster greedy algorithm that provides approximate solutions. The algorithms are numerically illustrated and analyzed for different types of sensing matrices, such as random matrices or deterministic matrices issued from tomography and super-resolution.

Lipschitz Bounds for Noise Robustness in Compressive Sensing: Two Algorithms

The paper deals with numerical estimations of Lipschitz bounds relating locally the reconstruction error to the measurement error in the compressive sensing framework. Most recent theoretical papers in the field parametrize such bounds relatively to certain families of vectors called dual certificates, which are fundamental to several reconstruction criteria. The paper provides two algorithms for computing dual certificates that optimize their related reconstruction error bounds. We give a greedy algorithm that provides a fast approximate solution, and a convex-projection algorithm that computes the exact optimum.

Identifiability for Gauge Regularizations and Algorithms for Block-Sparse Synthesis in Compressive Sensing

In the paper we give a characterization of identifiability for regularizations with gauges of compact convexes. This extends the classic identifiability results from the standard l1-regularization framework in compressive sensing. We show that the standard dual certificate techniques can no longer work by themselves ouside the polytope case. We then apply the general characterization to the caseof block-sparse regularizations and obtain an identification algorithm based on a combination of the standard duality and a convex-projection technique.

Sparse-Based Morphometry: Principle and Application to Alzheimer's Disease

The detection of brain alterations is crucial for understanding pathophysiological processes. The Voxel-Based Morphometry (VBM) is one of the most popular methods to achieve this task. Despite its numerous advantages, VBM is based on a highly reduced representation of the local brain anatomy since complex anatomical patterns are reduced to local averages of tissue probabilities. In this paper, we propose a new framework called Sparse-Based Morphometry (SBM) to better represent local brain anatomies. The presented patch-based approach uses dictionary learning to detect anatomical pattern modifications based on their shape and geometry. In our experiences, we compare SBM and VBM along Alzheimer’s Disease (AD) progression.

An Evaluation of the Sparsity Degree for Sparse Recovery with Deterministic Measurement Matrices

The paper deals with the estimation of the maximal sparsity degree for which a given measurement matrix allows sparse reconstruction through ℓ1-minimization. This problem is a key issue in different applications featuring particular types of measurement matrices, as for instance in the framework of tomography with low number of views. In this framework, while the exact bound is NP hard to compute, most classical criteria guarantee lower bounds that are numerically too pessimistic. In order to achieve an accurate estimation, we propose an efficient greedy algorithm that provides an upper bound for this maximal sparsity. Based on polytope theory, the algorithm consists in finding sparse vectors that cannot be recovered by ℓ1-minimization. Moreover, in order to deal with noisy measurements, theoretical conditions leading to a more restrictive but reasonable bounds are investigated. Numerical results are presented for discrete versions of tomography measurement matrices, which are stacked Radon transforms corresponding to different tomograph views.