Blind inverse problems with isolated spikes

TL;DR

This paper demonstrates that it is possible to stably recover an unknown integral operator and spike locations from limited measurements, with applications to optical imaging problems like super-resolution and deconvolution.

Contribution

It provides a theoretical foundation and practical approach for solving blind inverse problems involving isolated spikes, advancing optical imaging techniques.

Findings

Stable recovery of operator and spike locations under realistic assumptions

Application to challenging optical imaging problems

Practical approach grounded in theory

Abstract

Assume that an unknown integral operator living in some known subspace is observed indirectly, by evaluating its action on a few Dirac masses at unknown locations. Is this information enough to recover the operator and the impulse responses locations stably? We study this question and answer positively under realistic technical assumptions. We illustrate the well-foundedness of this theory on two challenging optical imaging problems: blind super-resolution and deconvolution. This provides a simple, practical and theoretically grounded approach to solve these long resisting problems.