Abstract
We propose a new solver for the sparse spikes deconvolution problem over the space of Radon measures. A common approach to off-the-grid deconvolution considers semidefinite (SDP) relaxations of the total variation (the total mass of the measure) minimization problem. The direct resolution of this SDP is however intractable for large scale settings, since the problem size grows as fc2d where fc is the cutoff frequency of the filter. Our first contribution introduces a penalized formulation of this semidefinite lifting, which has low-rank solutions. Our second contribution is a conditional gradient optimization scheme with non-convex updates. This algorithm leverages both the low-rank and the convolutive structure of the problem, resulting in an O(fcd log fc) complexity per iteration. Numerical simulations are promising and show that the algorithm converges in exactly k steps, k being the number of Diracs composing the solution.
Highlights
1.1 Sparse spikes deconvolutionSparse super-resolution problems consist in recovering pointwise sources from low-resolution and possibly noisy measurements
Such issues arise naturally in fields like astronomical imaging [28], fluorescency microscopy [19, 30] or seismic imaging [21], where it may be crucial to correct the physical blur introduced by sensing devices, due to diffraction or photonic noise for instance
We aim at retrieving a d-dimensional discrete Radon measure μ0 =
Summary
Sparse super-resolution problems consist in recovering pointwise sources from low-resolution and possibly noisy measurements. Such issues arise naturally in fields like astronomical imaging [28], fluorescency microscopy [19, 30] or seismic imaging [21], where it may be crucial to correct the physical blur introduced by sensing devices, due to diffraction or photonic noise for instance. Let δx be the Dirac measure at position x ∈ Td, where T = R/Z is the torus. We aim at retrieving a d-dimensional discrete Radon measure μ0 =. Td), given the measurements y = Φμ0 + w = y0 + w where Φ is some known convolution operator, and w some unknown noise
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