A unified method for super-resolution recovery and real exponential-sum separation

TL;DR

This paper introduces a unified mathematical approach for super-resolution recovery and exponential-sum separation, applicable in microscopy, astronomy, MRS, and nuclear chemistry, using simple computational steps.

Contribution

It develops a unified theory and solution for both problems based on a new mathematical model inspired by light wave diffraction.

Findings

Applicable to fluorescence microscopy and astronomy

Effective in magnetic resonance spectroscopy data analysis

Provides a computationally simple method

Abstract

In this paper, motivated by diffraction of traveling light waves, a simple mathematical model is proposed, both for the multivariate super-resolution problem and the problem of blind-source separation of real-valued exponential sums. This model facilitates the development of a unified theory and a unified solution of both problems in this paper. Our consideration of the super-resolution problem is aimed at applications to fluorescence microscopy and observational astronomy, and the motivation for our consideration of the second problem is the current need of extracting multivariate exponential features in magnetic resonance spectroscopy (MRS) for the neurologist and radiologist as well as for providing a mathematical tool for isotope separation in Nuclear Chemistry. The unified method introduced in this paper can be easily realized by processing only finitely many data, sampled at locations that are not necessarily prescribed in advance, with computational scheme consisting only of matrix - vector multiplication, peak finding, and clustering.