In this paper, we study the solving of the gridless sparse optimization problem and its application to 3D image deconvolution. Based on the recent works of [14] introducing the sliding Frank–Wolfe algorithm to solve the Beurling LASSO problem, we introduce an accelerated algorithm, denoted BSFW, that preserves its convergence properties, while removing most of the costly local descents. Besides, as the solving of BLASSO still relies on a regularization parameter, we introduce a homotopy algorithm to solve the constrained BLASSO that allows to use a more practical parameter based on the image residual, e.g. its standard deviation. Both algorithms benefit from a finite termination property, i.e. they are guaranteed to find the solution in a finite number of step under mild conditions. These methods are then applied on the problem of 3D tomographic diffractive microscopy images, with the purpose of explaining the image by a small number of atoms in convolved observations. Numerical results on synthetic and real images illustrates the improvement provided by the BSFW method, the homotopy method and their combination.

The Short Time Fourier Transform (STFT) of white noise presentwell-knownproperties regarding the distribution of its zeroes. In this letter, we propose a method to locate them beyond the usual time-frequency grid, in order to gain a better understanding on the underlying point process. To do so, we use a recent method based on off-the-grid sparse analysis, that aim at finding the best set of atomic shape forming the STFT modulus. We also improve it by ensuring the resulting combination of atoms minimizes a criterion based on the theoretical behavior of the set of zeros. Numerical experiments support our approach and show, with respect to discrete methods, an improved statistical power for the detection of signals in white noise.