New ℓ₂ - ℓ₀ algorithm for single-molecule localization microscopy

Among the many super-resolution techniques for microscopy, single-molecule localization microscopy methods are widely used. This technique raises the difficult question of precisely localizing fluorophores from a blurred, under-resolved, and noisy acquisition. In this work, we focus on the grid-based approach in the context of a high density of fluorophores formalized by a ℓ₂ least-square term and sparsity term modeled with ℓ₀ pseudo-norm. We consider both the constrained formulation and the penalized formulation. Based on recent results, we formulate the ℓ₀ pseudo-norm as a convex minimization problem. This is done by introducing an auxiliary variable. An exact biconvex reformulation of the ℓ₂ - ℓ₀ constrained and penalized problems is proposed with a minimization algorithm. The algorithms, named CoBic (Constrained Biconvex) and PeBic (Penalized Biconvex) are applied to the problem of single-molecule localization microscopy and we compare the results with other recently proposed methods.