Off-The-Grid Variational Sparse Spike Recovery: Methods and Algorithms

Abstract

Gridless sparse spike reconstruction is a rather new research field with significant results for the super-resolution problem, where we want to retrieve fine-scale details from a noisy and filtered acquisition. To tackle this problem, we are interested in optimisation under some prior, typically the sparsity i.e., the source is composed of spikes. Following the seminal work on the generalised LASSO for measures called the Beurling-Lasso (BLASSO), we will give a review on the chief theoretical and numerical breakthrough of the off-the-grid inverse problem, as we illustrate its usefulness to the super-resolution problem in Single Molecule Localisation Microscopy (SMLM) through new reconstruction metrics and tests on synthetic and real SMLM data we performed for this review.

Off-the-Grid Curve Reconstruction through Divergence Regularization: An Extreme Point Result

“…We recall some definitions and handy properties stemming from the off-the-grid literature, the interested reader may take a deeper look in the review [17] for more insights. In this case, the aim is to reconstruct spikes, i.e.…”

“…Let us also define the adjoint operator of Φ : M (X ) → H in the weak- * topology, namely the map Φ * : H → C 0 (X ), defined for all x ∈ X and p ∈ H by Φ * (p)(x) = ⟨p, φ(x)⟩ H . The choice of φ and H is dictated by the physical process of acquisition, with generic measurement kernels such as convolution, Fourier, Laplace, etc [17].…”

“…The BLASSO problem (P λ (y)) above is convex, thus one can define its dual problem [6,17] which writes down for p ∈ H:…”