A Theory of Computational Resolution Limit for Line Spectral Estimation

Liu, Ping; Hai, Zhang

doi:10.1109/tit.2021.3075149

Cited by 22 publications

(43 citation statements)

References 47 publications

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“…So the line spectral problem is equivalent to the super-resolution problem considered here. The results in [24] show that, for the single measurement case, when the point scatterers are separated by…”

Section: Comparison With the Single Illumination Casementioning

confidence: 96%

“…In [24], the authors estimate the so-called computational resolution limit for the line spectral estimation problem of the single measurement case. The line spectral estimation problem is to estimate the locations of some line spectra from the Fourier data (in a bounded domain) of one of their linear combination.…”

Section: Comparison With the Single Illumination Casementioning

confidence: 99%

“…In this section, we present the main technique that is used in the proofs of the main results of the paper, namely the approximation theory in Vandermonde space. This theory was first introduced in [24,25]. Instead of considering the non-linear approximation problem there, we consider a different approximation problem, which is relevant to the stability analysis of (2.4).…”

Section: Non-linear Approximation Theory In Vandermonde Spacementioning

confidence: 99%

“…More recently, to analyze the resolution for recovering multiple point scatterers, in [23][24][25] the authors defined "computational resolution limits" which characterize the minimum required distance between point scatterers so that their number and locations can be stably resolved under certain noise level. By developing a non-linear approximation theory in a socalled Vandermonde space, they derived bounds for computational resolution limits for a deconvolution problem [25] and a line spectral problem [24] (equivalent to the super-resolution problem considered here). In particular, they showed in [24] that the computational resolution limit for number and location recovery should be respectively 2n−1 ), stably recovering the scatterer locations is impossible in the worst case.…”

Section: Introductionmentioning

confidence: 99%

“…By developing a non-linear approximation theory in a socalled Vandermonde space, they derived bounds for computational resolution limits for a deconvolution problem [25] and a line spectral problem [24] (equivalent to the super-resolution problem considered here). In particular, they showed in [24] that the computational resolution limit for number and location recovery should be respectively 2n−1 ), stably recovering the scatterer locations is impossible in the worst case. This resolution limit indicates that superresolution is possible for the single measurement case but requires very high signal-to-noise ratio (according to the exponent 1 2n−1 ).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Liu¹,

Yu²,

Sabet³

et al. 2022

Preprint

Self Cite

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1 of the resolution on the cut-off frequency of the imaging system, the signal-tonoise 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.

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Section: Comparison With the Single Illumination Casementioning

confidence: 96%

Section: Comparison With the Single Illumination Casementioning

confidence: 99%

Section: Non-linear Approximation Theory In Vandermonde Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Liu¹,

Yu²,

Sabet³

et al. 2022

Preprint

Self Cite

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show abstract

Improved Resolution Estimate for the Two-Dimensional Super-Resolution and a New Algorithm for Direction of Arrival Estimation with Uniform Rectangular Array

Liu,

Ammari

2023

Found Comput Math

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In this paper, we develop a new technique to obtain improved estimates for the computational resolution limits in two-dimensional super-resolution problems and present a new idea for developing two-dimensional super-resolution algorithms. To be more specific, our main contributions are fourfold: (1) Our work improves the resolution estimates for number detection and location recovery in two-dimensional super-resolution problems; (2) As a consequence, we derive a stability result for a sparsity-promoting algorithm in two-dimensional super-resolution problems [or direction of arrival Problems (DOA)]. The stability result exhibits the optimal performance of sparsity promoting in solving such problems; (3) Inspired by the new techniques, we propose a new coordinate-combination-based model order detection algorithm for two-dimensional DOA estimation and theoretically demonstrate its optimal performance, and (4) we also propose a new coordinate-combination-based MUSIC algorithm for super-resolving sources in two-dimensional DOA estimation. It has excellent performance and enjoys some advantages compared to the conventional DOA algorithms.

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A mathematical theory of the computational resolution limit in one dimension

Liu

Zhang

2022

Applied and Computational Harmonic Analysis

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A Theory of Computational Resolution Limit for Line Spectral Estimation

Cited by 22 publications

References 47 publications

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

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

Improved Resolution Estimate for the Two-Dimensional Super-Resolution and a New Algorithm for Direction of Arrival Estimation with Uniform Rectangular Array

A mathematical theory of the computational resolution limit in one dimension

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