We propose a global optimization approach to solve ℓ0-norm penalized least-squares problems, using a dedicated branch-and-bound methodology. A specific tree search strategy is built, with branching rules inspired from greedy exploration techniques. We show that the subproblem involved at each node can be evaluated via ℓ1-norm-based optimization problems with box constraints, for which an active-set algorithm is built. Our method is able to solve exactly moderate-size, yet difficult, sparse approximation problems, without resorting to mixed-integer programming (MIP) optimization. In particular, it outperforms the generic MIP solver CPLEX.

We address sparse approximation in the particular case where the dictionary is built upon the discretization of a continuous parameter. The resulting dictionary being highly correlated, equivalence between ℓ0and suboptimal solutions (e.g. greedy algorithms and convex relaxation) is not guaranteed. To tackle this issue, continuous parameter estimation has been proposed using a dictionary based on polar interpolation [1], [2]. Alternately, the exact ℓ0-norm optimization problem can be addressed on moderate size problems through Mixed Integer Programming (MIP) [3]. We propose to merge these two approaches in a new MIP formulation adapted to polar interpolation. Improvements on polar interpolation and refinements on its use in the ℓ1-norm framework are also proposed. Methods are evaluated on simulated spike train deconvolution problems, where the proposed ℓ0-norm approach with continuous dictionary achieves the best results, although with higher computing time.

The blind structured illumination microscopy strategy proposed by Mudry et al. is fully re-founded in this paper, unveiling the central role of the sparsity of the illumination patterns in the mechanism that drives super-resolution in the method. A numerical analysis shows that the resolving power of the method can be further enhanced with optimized one-photon or two-photon speckle illuminations. A much improved numerical implementation is provided for the reconstruction problem under the image positivity constraint. This algorithm rests on a new preconditioned proximal iteration faster than existing solutions, paving the way to 3D and real-time 2D reconstruction.

In this communication, a fast reconstruction algorithm is proposed for fluorescence blind structured illumination microscopy (SIM) under the sample positivity constraint. This new algorithm is by far simpler and faster than existing solutions, paving the way to 3D and real-time 2D reconstruction.

This paper deals with the estimation of the arrival times of overlapping ultrasonic echoes. We focus on approaches based on discrete sparse deconvolution. Such methods are limited by the time resolution imposed by the model discretization, which is usually considered at the data sampling rate. In order to get closer to the continuous-time model, we propose to increase the time precision by introducing an up-sampling factor in the discrete model. The problem is then recast as a Multiple Input Single Output (MISO) deconvolution problem. Then, we propose to revisit standard sparse deconvolution algorithms for MISO systems. Specific and efficient algorithmic implementation is derived in such setting. Algorithms are evaluated on synthetic data, showing improvements in robustness toward discretization errors and competitive computational time compared to the standard approaches.

Line spectra estimation is addressed for irregularly sampled astrophysical data. A formulation with a large number of discretized frequencies is used and sparseness is encouraged via a Bernoulli-Gaussian (BG) model on the corresponding amplitudes. Contrary to classic BG models, here the frequency parameters are not constrained on a fixed grid, theoretically enabling unlimited frequency precision. We propose a posterior mean unsupervised estimation scheme combined with an hybrid MCMC algorithm, that allows us to derive crucial information in an astrophysical context, such as confidence levels and variances for each detected spectral line. Simulations confirm the validity of this model with satisfactory estimation results, in addition to a more solid behaviour than parametric methods towards classic astrophysical perturbations

High resolution spectral analysis has recently been addressed as an inverse problem, and solutions are currently proposed through the regularization framework. In this paper, we focus on regularized spectral analysis of unevenly sampled data for line spectra estimation. First, we study the structural differences of the model between regular sampling, missing data (where the sampling is regular, but with missing data) and irregular sampling cases. Then, consequences for the computation of the solution are emphasized. We propose an approximation of the irregular sampling model to compute the nonquadratic regularization solution at a cost comparable to the other sampling cases. Finally, algorithmic implementation is discussed and applications to simulated data are presented.