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An Operator Theory for Analyzing the Resolution of Multi-illumination Imaging Modalities

by P. Liu and H. Ammari

(Report number 2023-12)

Abstract
By introducing a new operator theory, we provide a unified mathematical theory for general source resolution in the multi-illumination imaging problem. Our main idea is to transform multi-illumination imaging into single-snapshot imaging with a new imaging kernel that depends on both the illumination patterns and the point spread function of the imaging system. We therefore prove that the resolution of multi-illumination imaging is approximately determined by the essential cutoff frequency of the new imaging kernel, which is roughly limited by the sum of the cutoff frequency of the point spread function and the maximum essential frequency in the illumination patterns. Our theory provides a unified way to estimate the resolution of various existing super-resolution modalities and results in the same estimates as those obtained in experiments. In addition, based on the reformulation of the multi-illumination imaging problem, we also estimate the resolution limits for resolving both complex and positive sources by sparsity-based approaches. We show that the resolution of multi-illumination imaging is approximately determined by the new imaging kernel from our operator theory and better resolution can be realized by sparsity-promoting techniques in practice but only for resolving very sparse sources. This explains experimentally observed phenomena in some sparsity-based super-resolution modalities.

Keywords: super-resolution, resolution enhancement, multi-illumination imaging, operator theory, location recovery, source number recovery

@Techreport{LA23_1049,
  author = {P. Liu and H. Ammari},
  title = {An Operator Theory for Analyzing the Resolution of Multi-illumination Imaging Modalities},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-12},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-12.pdf },
  year = {2023}
}

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