This paper studies sparse spikes deconvolution over the space of measures. We focus on the recovery properties of the support of the measure (i.e. the location of the Dirac masses) using total variation of measures (TV) regularization. This regularization is the natural extension of the 1 norm of vectors to the setting of measures. We show that support identification is governed by a specific solution of the dual problem (a so-called dual certificate) having minimum L 2 norm. Our main result shows that if this certificate is non-degenerate (see the definition below), when the signal-to-noise ratio is large enough TV regularization recovers the exact same number of Diracs. We show that both the locations and the amplitudes of these Diracs converge toward those of the input measure when the noise drops to zero. Moreover the non-degeneracy of this certificate can be checked by computing a so-called vanishing derivative pre-certificate. This proxy can be computed in closed form by solving a linear system. Lastly, we draw connections between the support of the recovered measure on a continuous domain and on a discretized grid. We show that when the signal-to-noise level is large enough, and provided the aforementioned dual certificate is non-degenerate, the solution of the discretized problem is supported on pairs of Diracs which are neighbors of the Diracs of the input measure. This gives a precise description of the convergence of the solution of the discretized problem toward the solution of the continuous grid-free problem, as the grid size tends to zero.

This paper showcases the theoretical and numerical performance of the Sliding Frank-Wolfe, which is a novel optimization algorithm to solve the BLASSO sparse spikes super-resolution problem.The BLASSO is a continuous (i.e. off-the-grid or grid-less) counterpart to the well-known 1 sparse regularisation method (also known as LASSO or Basis Pursuit). Our algorithm is a variation on the classical Frank-Wolfe (also known as conditional gradient) which follows a recent trend of interleaving convex optimization updates (corresponding to adding new spikes) with non-convex optimization steps (corresponding to moving the spikes). Our main theoretical result is that this algorithm terminates in a finite number of steps under a mild non-degeneracy hypothesis. We then target applications of this method to several instances of single molecule fluorescence imaging modalities, among which certain approaches rely heavily on the inversion of a Laplace transform. Our second theoretical contribution is the proof of the exact support recovery property of the BLASSO to invert the 1-D Laplace transform in the case of positive spikes. On the numerical side, we conclude this paper with an extensive study of the practical performance of the Sliding Frank-Wolfe on different instantiations of single molecule fluorescence imaging, including convolutive and non-convolutive (Laplace-like) operators. This shows the versatility and superiority of this method with respect to alternative sparse recovery technics.

International audienceReplacing positivity constraints by an entropy barrier is popular to approximate solutions of linear programs. In the special case of the optimal transport problem, this technique dates back to the early work of Schr\"odinger. This approach has recently been used successfully to solve optimal transport related problems in several applied fields such as imaging sciences, machine learning and social sciences. The main reason for this success is that, in contrast to linear programming solvers, the resulting algorithms are highly parallelizable and take advantage of the geometry of the computational grid (e.g. an image or a triangulated mesh). The first contribution of this article is the proof of the Γ-convergence of the entropic regularized optimal transport problem towards the Monge-Kantorovich problem for the squared Euclidean norm cost function. This implies in particular the convergence of the optimal entropic regularized transport plan towards an optimal transport plan as the entropy vanishes. Optimal transport distances are also useful to define gradient flows as a limit of implicit Euler steps according to the transportation distance. Our second contribution is a proof that implicit steps according to the entropic regularized distance converge towards the original gradient flow when both the step size and the entropic penalty vanish (in some controlled way)

We propose in this paper an extension of the NonLocal Means (NL-Means) denoising algorithm. The idea is to replace the usual square patches used to compare pixel neighborhoods with various shapes that can take advantage of the local geometry of the image. We provide a fast algorithm to compute the NL-Means with arbitrary shapes thanks to the Fast Fourier Transform. We then consider local combinations of the estimators associated with various shapes by using Stein's Unbiased Risk Estimate (SURE). Experimental results show that this algorithm improve the standard NL-Means performance and is close to state-ofthe-art methods, both in terms of visual quality and numerical results. Moreover, common visual artifacts usually observed by denoising with NL-Means are reduced or suppressed thanks to our approach.