Beyond consistent reconstructions: optimality and sharp bounds for generalized sampling, and application to the uniform resampling problem

Ben Adcock, Anders C Hansen, Clarice Poon

Research output: Contribution to journalArticle

32 Citations (Scopus)

Abstract

Generalized sampling is a recently developed linear framework for sampling and reconstruction in separable Hilbert spaces. It allows one to recover any element in any finite-dimensional subspace given finitely many of its samples with respect to an arbitrary basis or frame. Unlike more common approaches for this problem, such as the consistent reconstruction technique of Eldar and others, it leads to numerical methods possessing both guaranteed stability and accuracy. The purpose of this paper is twofold. First, we give a complete and formal analysis of generalized sampling, the main result of which being the derivation of new, sharp bounds for the accuracy and stability of this approach. Such bounds improve upon those given previously and result in a necessary and sufficient condition, the stable sampling rate, which guarantees a priori a good reconstruction. Second, we address the topic of optimality. Under some assumptions, we show that generalized sampling is an optimal, stable method. Correspondingly, whenever these assumptions hold, the stable sampling rate is a universal quantity. In the final part of the paper we illustrate our results by applying generalized sampling to the so-called uniform resampling problem.


Original languageEnglish
Pages (from-to)3132-3167
Number of pages36
JournalSiam Journal on Mathematical Analysis
Volume45
Issue number5
DOIs
Publication statusPublished - 2013

Cite this

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abstract = "Generalized sampling is a recently developed linear framework for sampling and reconstruction in separable Hilbert spaces. It allows one to recover any element in any finite-dimensional subspace given finitely many of its samples with respect to an arbitrary basis or frame. Unlike more common approaches for this problem, such as the consistent reconstruction technique of Eldar and others, it leads to numerical methods possessing both guaranteed stability and accuracy. The purpose of this paper is twofold. First, we give a complete and formal analysis of generalized sampling, the main result of which being the derivation of new, sharp bounds for the accuracy and stability of this approach. Such bounds improve upon those given previously and result in a necessary and sufficient condition, the stable sampling rate, which guarantees a priori a good reconstruction. Second, we address the topic of optimality. Under some assumptions, we show that generalized sampling is an optimal, stable method. Correspondingly, whenever these assumptions hold, the stable sampling rate is a universal quantity. In the final part of the paper we illustrate our results by applying generalized sampling to the so-called uniform resampling problem.",
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N2 - Generalized sampling is a recently developed linear framework for sampling and reconstruction in separable Hilbert spaces. It allows one to recover any element in any finite-dimensional subspace given finitely many of its samples with respect to an arbitrary basis or frame. Unlike more common approaches for this problem, such as the consistent reconstruction technique of Eldar and others, it leads to numerical methods possessing both guaranteed stability and accuracy. The purpose of this paper is twofold. First, we give a complete and formal analysis of generalized sampling, the main result of which being the derivation of new, sharp bounds for the accuracy and stability of this approach. Such bounds improve upon those given previously and result in a necessary and sufficient condition, the stable sampling rate, which guarantees a priori a good reconstruction. Second, we address the topic of optimality. Under some assumptions, we show that generalized sampling is an optimal, stable method. Correspondingly, whenever these assumptions hold, the stable sampling rate is a universal quantity. In the final part of the paper we illustrate our results by applying generalized sampling to the so-called uniform resampling problem.

AB - Generalized sampling is a recently developed linear framework for sampling and reconstruction in separable Hilbert spaces. It allows one to recover any element in any finite-dimensional subspace given finitely many of its samples with respect to an arbitrary basis or frame. Unlike more common approaches for this problem, such as the consistent reconstruction technique of Eldar and others, it leads to numerical methods possessing both guaranteed stability and accuracy. The purpose of this paper is twofold. First, we give a complete and formal analysis of generalized sampling, the main result of which being the derivation of new, sharp bounds for the accuracy and stability of this approach. Such bounds improve upon those given previously and result in a necessary and sufficient condition, the stable sampling rate, which guarantees a priori a good reconstruction. Second, we address the topic of optimality. Under some assumptions, we show that generalized sampling is an optimal, stable method. Correspondingly, whenever these assumptions hold, the stable sampling rate is a universal quantity. In the final part of the paper we illustrate our results by applying generalized sampling to the so-called uniform resampling problem.

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