An algorithm for non-convex off-the-grid sparse spike estimation with a minimum separation constraint

TL;DR

This paper introduces a practical algorithm for estimating sparse off-the-grid spikes from Fourier measurements, leveraging a non-convex optimization approach with a minimum separation constraint, supported by theoretical insights and experiments.

Contribution

It presents a novel projected gradient descent algorithm for non-convex sparse spike estimation under separation constraints, bridging theory and practical imaging applications.

Findings

Algorithm successfully estimates spikes from Fourier data

Theoretical analysis supports convergence and stability

Experimental results demonstrate effectiveness in imaging tasks

Abstract

Theoretical results show that sparse off-the-grid spikes can be estimated from (possibly compressive) Fourier measurements under a minimum separation assumption. We propose a practical algorithm to minimize the corresponding non-convex functional based on a projected gradient descent coupled with an initialization procedure. We give qualitative insights on the theoretical foundations of the algorithm and provide experiments showing its potential for imaging problems.