A mathematical theory of the computational resolution limit in one dimension

Liu, Ping; Zhang, Hai

doi:10.1016/j.acha.2021.09.002

Cited by 18 publications

(13 citation statements)

References 43 publications

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“…On the other hand, the resolution analysis in this paper also follows the line of the authors' previous researches on exploring the super-resolution capability for different imaging configurations [33][34][35][36]. Specifically, to analyze the resolution for recovering multiple point sources from a single measurement, in [34][35][36] the authors defined "computational resolution limits" which characterize the minimum required distance between point sources so that their number and locations can be stably resolved under certain noise level. Based on a new approximation theory in a so-called Vandermonde space, they derived bounds for the resolution limits of one-and multi-dimensional super-resolution problems [34][35][36].…”

Section: Related Workmentioning

confidence: 88%

“…Specifically, to analyze the resolution for recovering multiple point sources from a single measurement, in [34][35][36] the authors defined "computational resolution limits" which characterize the minimum required distance between point sources so that their number and locations can be stably resolved under certain noise level. Based on a new approximation theory in a so-called Vandermonde space, they derived bounds for the resolution limits of one-and multi-dimensional super-resolution problems [34][35][36]. In particular, they showed that the computational resolution limit for number and location recovery should be…”

Section: Related Workmentioning

confidence: 99%

“…The results are slightly different from the ones in [35] and therefore, we present their detailed proofs here for the sake of completeness. We first introduce some lemmas and theorems from [36]. We denote for integer k ≥ 1,…”

Section: Theorems On Source Number and Location Recovery For The One-...mentioning

confidence: 99%

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Dynamic super-resolution in particle tracking problems

Liu¹,

Ammari²

2022

Preprint

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Particle tracking in a live cell environment is concerned with reconstructing the trajectories, locations, or velocities of the targeting particles, which holds the promise of revealing important new biological insights. The standard approach of particle tracking consists of two steps: first reconstructing statically the source locations in each time step, and second applying tracking techniques to obtain the trajectories and velocities. In contrast to the standard approach, the dynamic reconstruction seeks to simultaneously recover the source locations and velocities from all frames, which enjoys certain advantages. In this paper, we provide a rigorous mathematical analysis for the resolution limit of reconstructing source number, locations, and velocities by general dynamical reconstruction in particle tracking problems, by which we demonstrate the possibility of achieving super-resolution for the dynamic reconstruction. We show that when the location-velocity pairs of the particles are separated beyond certain distances (the resolution limits), the number of particles and the location-velocity pair can be stably recovered. The resolution limits are related to the cut-off frequency of the imaging system, signal-to-noise ratio, and the sparsity of the source. By these estimates we also derive a stability result for a sparsity-promoting dynamic reconstruction. In addition, we further show that the reconstruction of velocities has a better resolution limit which improves constantly as the particles moving. This result is derived by a crucial observation that the inherent cut-off frequency for the velocity recovery can be viewed as the total observation time multiplies the cut-off frequency of the imaging system, which may lead to a better resolution limit as compared to the one for each diffraction-limited frame. It is anticipated that this crucial observation can inspire many new reconstruction algorithms that significantly improve the resolution of particle tracking in practice.In addition, we propose super-resolution algorithms for recovering the number and values of the velocities in the tracking problem and demonstrate theoretically or numerically their super-resolution capability.

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Section: Related Workmentioning

confidence: 88%

Section: Related Workmentioning

confidence: 99%

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Dynamic super-resolution in particle tracking problems

Liu¹,

Ammari²

2022

Preprint

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“…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%

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

Liu¹,

Yu²,

Sabet³

et al. 2022

Preprint

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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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Super-Resolution of Positive Sources on an Arbitrarily Fine Grid

Morgenshtern

2021

J Fourier Anal Appl

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A mathematical theory of the computational resolution limit in one dimension

Cited by 18 publications

References 43 publications

Dynamic super-resolution in particle tracking problems

Dynamic super-resolution in particle tracking problems

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

Super-Resolution of Positive Sources on an Arbitrarily Fine Grid

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