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

Bastien Laville,Gilles Aubert,Laure Blanc-Féraud

doi:10.3390/jimaging7120266

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.

Highlights

We propose to conduct a comprehensive review on the so-called off-the-grid variational methods to solve the sparse spike recovery problem
+ λṼ (μ) where Φμ = Ω φ( a, x )dμ( a, x ) for φ( a, x ) = aφ( x ) and Ṽ is the total variation (TV)-norm on the spatial component of the measure μ
It is worth noticing that other imaging systems exist, for which the inverse problems to solve are a bit different from deconvolution: e.g., Nuclear Magnetic Resonance spectroscopy with Fourier measurements [40], MA-TIRF with Laplace [13]

Summary

Introduction

We will exhibit the main theoretical and numerical results in the literature, underlining the interest of these methods for various domains dealing with inverse problems. As part of this review and our former work on gridless methods, we developed an implementation of the more consistent numerical methods with a focus on efficiency and computation time. With this implementation, we were able to apply off-the-grid method to fluorescence microscopy super-resolution problem. The codes and the computed result are an addition to the offthe-grid literature, and constitute further evidence supporting the relevance of this domain in the inverse problem field

Methods

Results

Conclusion