Theoretical and Numerical Analysis of Super-Resolution Without Grid

Abstract : This thesis studies the noisy sparse spikes super-resolution problem for positive measures using the BLASSO, an infinite dimensional convex optimization problem generalizing the LASSO to measures. First, we show that the support stability of the BLASSO for N clustered spikes is governed by an object called the (2N-1)-vanishing derivatives pre-certificate. When it is non-degenerate, solving the BLASSO leads to exact support recovery of the initial measure, in a low noise regime whose size is controlled by the minimal separation distance of the spikes. In a second part, we propose the Sliding Frank-Wolfe algorithm, based on the Frank-Wolfe algorithm with an added step moving continuously the amplitudes and positions of the spikes, that solves the BLASSO. We show that, under mild assumptions, it converges in a finite number of iterations. We apply this algorithm to the 3D fluorescent microscopy problem by comparing three models based on the PALM/STORM technics.
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https://tel.archives-ouvertes.fr/tel-02002504
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Quentin Denoyelle. Theoretical and Numerical Analysis of Super-Resolution Without Grid. General Mathematics [math.GM]. PSL Research University, 2018. English. ⟨NNT : 2018PSLED030⟩. ⟨tel-02002504⟩

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