2018/11/15 by Quentin Denoyelle, Denoyelle, Quentin, Vincent Duval +5 · 6 citations
Biochemistry, Genetics and Molecular Biology · Engineering · #Advanced Fluorescence Microscopy Techniques #FOS: Mathematics #Near-Field Optical Microscopy #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1811.06416
openalex publication_date 2018/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper showcases the theoretical and numerical performance of the Sliding\nFrank-Wolfe, which is a novel optimization algorithm to solve the BLASSO sparse\nspikes super-resolution problem. The BLASSO is a continuous (i.e. off-the-grid\nor grid-less) counterpart to the well-known 1 sparse regularisation method\n(also known as LASSO or Basis Pursuit). Our algorithm is a variation on the\nclassical Frank-Wolfe (also known as conditional gradient) which follows a\nrecent trend of interleaving convex optimization updates (corresponding to\nadding new spikes) with non-convex optimization steps (corresponding to moving\nthe spikes). Our main theoretical result is that this algorithm terminates in a\nfinite number of steps under a mild non-degeneracy hypothesis. We then target\napplications of this method to several instances of single molecule\nfluorescence imaging modalities, among which certain approaches rely heavily on\nthe inversion of a Laplace transform. Our second theoretical contribution is\nthe proof of the exact support recovery property of the BLASSO to invert the\n1-D Laplace transform in the case of positive spikes. On the numerical side, we\nconclude this paper with an extensive study of the practical performance of the\nSliding Frank-Wolfe on different instantiations of single molecule fluorescence\nimaging, including convolutive and non-convolutive (Laplace-like) operators.\nThis shows the versatility and superiority of this method with respect to\nalternative sparse recovery technics.\n