Stable Separation and Super-Resolution of Mixture Models

TL;DR

This paper introduces a convex programming algorithm for accurately separating and locating point sources in super-resolution imaging and related fields, with proven guarantees and robustness to noise.

Contribution

The paper presents a novel convex optimization method with near-optimal recovery guarantees for mixture models involving multiple band-limited point spread functions.

Findings

Exact recovery in noise-free settings

Robustness to bounded noise demonstrated

Numerical experiments confirm effectiveness

Abstract

We consider simultaneously identifying the membership and locations of point sources that are convolved with different band-limited point spread functions, from the observation of their superpositions. This problem arises in three-dimensional super-resolution single-molecule imaging, neural spike sorting, multi-user channel identification, among other applications. We propose a novel algorithm, based on convex programming, and establish its near-optimal performance guarantee for exact recovery in the noise-free setting by exploiting the spectral sparsity of the point source models as well as the incoherence between point spread functions. Furthermore, robustness of the recovery algorithm in the presence of bounded noise is also established. Numerical examples are provided to demonstrate the effectiveness of the proposed approach.