THESIS
2021
1 online resource (xiv, 205 pages) : color illustration
Abstract
The ultimate resolution limit of an imaging system is a long-standing puzzle
that originated from the 18th century. Although many heuristic resolution criteria
have been proposed after the discovery of Abbe's diffraction limit, the exact
characterization of the resolution limit is still unknown. In this thesis, we aim
to provide a rigorous mathematical theory for the resolution limits in various
setups. Specifically, we define and quantitatively characterize the computational
resolution limits for number detection and location recovery in the deconvolution
problem, the one-dimensional and multi-dimensional line spectral estimation
problems, and the imaging problem with multi-illumination. The results explicitly
elucidate the dependence of each computational resolution limit on the cutof...[
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The ultimate resolution limit of an imaging system is a long-standing puzzle
that originated from the 18th century. Although many heuristic resolution criteria
have been proposed after the discovery of Abbe's diffraction limit, the exact
characterization of the resolution limit is still unknown. In this thesis, we aim
to provide a rigorous mathematical theory for the resolution limits in various
setups. Specifically, we define and quantitatively characterize the computational
resolution limits for number detection and location recovery in the deconvolution
problem, the one-dimensional and multi-dimensional line spectral estimation
problems, and the imaging problem with multi-illumination. The results explicitly
elucidate the dependence of each computational resolution limit on the cutoff
frequency, signal-to-noise ratio, sparsity of the source, and illumination pattern.
In contrast to the classical resolution limits that are only related to the cutoff
frequency, our characterizations highlight the importance of all the aforementioned
factors to the super-resolution problem and thus are more applicable to
modern super-resolution algorithms whose performances are highly related to the
signal-to-noise ratio. Moreover, our results indicate that there are phase transitions
in these reconstruction problems. We derive number detection and location
recovery algorithms and numerically confirm the predicted phase transition phenomenons.
Finally, we propose a fast algorithm for super-resolving point sources
with a cluster structure.
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