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Mathematical foundation of sparsity-based multi-illumination super-resolution

by P. Liu and S. Yu and O. Sabet and L. Pelkmans and H. Ammari

(Report number 2022-05)

Abstract
It is well-known that the resolution of traditional optical imaging system is limited by the so-called Rayleigh resolution or diffraction limit, which is of several hundreds of nanometers. By employing fluorescence techniques, modern microscopic methods can resolve point scatterers separated by a distance much lower than the Rayleigh resolution limit. Localization-based fluorescence subwavelength imaging techniques such as PALM and STORM can achieve spatial resolution of several tens of nanometers. However, these techniques have limited temporal resolution as they require tens of thousands of exposures. Employing sparsity-based models and recovery algorithms is a natural way to reduce the number of exposures, and hence obtain high temporal resolution. Nevertheless, to date fluorescence techniques suffer from the trade-off between spatial and temporal resolutions. In a recent work by the last four authors, a newly multi-illumination imaging technique called Brownian Excitation Amplitude Modulation microscopy (BEAM) is introduced. BEAM achieves a threefold resolution improvement by applying a compressive sensing recovery algorithm over only few frames. Motivated by BEAM, our aim in this paper is to pioneer the mathematical foundation for sparsity-based multi-illumination super-resolution. More precisely, we consider several diffraction-limited images from sample exposed to different illumination patterns and recover the source by considering the sparsest solution. We estimate the minimum separation distance between point scatterers so that they could be stably recovered. By this estimation of the resolution of the sparsity recovery, we reveal the dependence of the resolution on the cut-off frequency of the imaging system, the signal-to-noise ratio, the sparsity of point scatterers, and the incoherence of illumination patterns. Our theory particularly highlights the importance of the high incoherence of illumination patterns in enhancing the resolution. It also demonstrates that super-resolution can be achieved using sparsity-based multi-illumination imaging with very few frames, whereby the spatio-temporal super-resolution becomes possible. BEAM can be viewed as the first experimental realization of our theory, which is demonstrated to hold in both the one- and two-dimensional cases.

Keywords: spatio-temporal sparsity-based super-resolution, Brownian Excitation Amplitude Modulation microscopy, multi-illumination incoherent data, compressive sensing, minimal separation distance, approximation theory in Vandermonde space

@Techreport{LYSPA22_993,
  author = {P. Liu and S. Yu and O. Sabet and L. Pelkmans and H. Ammari},
  title = {Mathematical foundation of sparsity-based multi-illumination super-resolution},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-05},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-05.pdf },
  year = {2022}
}

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