A Stability Barrier for Reconstructions from Fourier Samples

Adcock, Ben; Hansen, Anders C.; Shadrin, Alexei

doi:10.1137/130908221

Cited by 72 publications

(43 citation statements)

References 53 publications

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“…In [4] it was proved that the matrix (2.5) is has an exponentially large condition number as N → ∞ for essentially any wavelet basis. An analogous result for polynomial bases was proved in [7].…”

Section: Remark 22supporting

confidence: 56%

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Generalized Sampling and Infinite-Dimensional Compressed Sensing

2015

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We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finite-dimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demonstrate that existing finite-dimensional techniques are ill-suited for solving a number of important problems.This work stems from recent developments in generalized sampling theorems for classical (Nyquist rate) sampling that allows for reconstructions in arbitrary bases. The main conclusion of this paper is that one can extend these ideas to allow for significant subsampling of sparse or compressible signals. The key to these developments is the introduction of two new concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize the fundamentally infinitedimensional reconstruction problem.

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Section: Remark 22supporting

confidence: 56%

“…We remark that for this choice of reconstruction basis the stable sampling rate Θ(M ; θ) is quadratic in M [2]. Moreover, a lower scaling (in particular, N = M ) results in extreme ill-conditioning [7]. We now repeat the experiment above with the function…”

Section: Example: the Effectiveness Of Generalized Samplingmentioning

confidence: 96%

Generalized Sampling and Infinite-Dimensional Compressed Sensing

2015

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“…Nonetheless, it transpires that this cannot be done in this case without compromising stability. For a more thorough analysis of stability and convergence for this reconstruction problem we refer the reader to [7].…”

Section: On Optimalitymentioning

confidence: 99%

Beyond Consistent Reconstructions: Optimality and Sharp Bounds for Generalized Sampling, and Application to the Uniform Resampling Problem

Adcock¹,

Hansen²,

Poon³

2013

SIAM J. Math. Anal.

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109

137

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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 frame. Unlike more common approaches for this problem, such as the consistent reconstruction technique of Eldar et al, it leads to completely stable 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 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 reconstruction. 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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“…In Adcock, Hansen & Shadrin (2012), it was proved that cos À q N;N Á c ÀN ; cN for some constant c > 1, and therefore the reconstruction constant CðF N ;N Þ ! c N grows exponentially fast in N. This translates into both extreme instability and divergence of the reconstruction.…”

Section: Failure Of Consistent Sampling For Problem 25mentioning

confidence: 99%