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A mathematical theory of super-resolution and diffraction limit

by P. Liu and H. Ammari

(Report number 2023-01)

Abstract
This paper is devoted to elucidating the essence of super-resolution and deals mainly with the stability of super-resolution and the diffraction limit. The first discovery is two location-amplitude identities characterizing the relations between source locations and amplitudes in the super-resolution problem. These identities allow us to directly derive the super-resolution capability for number, location, and amplitude recovery in the super-resolution problem and improve state-of-the-art estimations to an unprecedented level to have practical significance. The nonlinear inverse problems studied in this paper are known to be very challenging and have only been partially solved in recent years. However, thanks to this paper, we now have a clear and simple picture of all of these problems, which allows us to solve them in a unified way in just a few pages. The second crucial result of this paper is the theoretical proof of a two-point diffraction limit in spaces of general dimensionality under only an assumption on the noise level. This solves the long-standing puzzle and debate about the diffraction limit for imaging (and line spectral estimation) in very general circumstances. Our results also show that, for the resolution of any two point sources, when the signal-to-noise ratio is larger than two, one can definitely exceed the Rayleigh limit, which is far beyond common sense. We also find the optimal algorithm that achieves the optimal resolution when distinguishing two sources. By this work, we hope to inspire a start of a new period where examining the resolution based on the signal-to-noise ratio becomes a feasible method in the field of imaging.

Keywords: super-resolution, resolution limit, diffraction limit, line spectral estimation, Vandermonde matrix, phase transition

@Techreport{LA23_1038,
  author = {P. Liu and H. Ammari},
  title = {A mathematical theory of super-resolution and diffraction limit},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-01},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-01.pdf },
  year = {2023}
}

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