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

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 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.

Generalized Sampling and Infinite-Dimensional Compressed Sensing

“…Only once this problem has been solved can one tackle the issue of subsampling. Fortunately, the technique of generalized sampling (GS) was developed precisely for this problem [1,2,5,3]. We now recap this approach.…”

“…In fact, it is easy to devise pairs of bases {ϕ j } j∈N and sampling schemes {ζ j } j∈N for which the error α −α [N ] l 2 blows up as N → ∞, wheneverα [N ] is the result of the finite section method [1,5]. Another significant issue is that the finite section matrix P N U P N ∈ C N ×N may be extremely poorly conditioned, even though U and its inverse U −1 are bounded.…”

“…Another significant issue is that the finite section matrix P N U P N ∈ C N ×N may be extremely poorly conditioned, even though U and its inverse U −1 are bounded. Examples of operators U whose finite sections exhibit exponentially poor conditioning were given in [1,5]. In particular, the measurement matrix formed by sampling in the Fourier basis and reconstructing in Haar wavelets (the principal example of this paper) suffers from this phenomenon.…”

“…In particular, sampling at a rate N ≥ Θ(M ; θ) ensures that 1 − C N,M ≥ θ, and therefore stability and accuracy ofα [M ] up to the magnitude of θ. Note that Θ(M ; θ) can be easily computed numerically [5], since C N,M is just the norm of an M × M matrix (see (4.6)). Hence the conditions of Theorem 4.1 can be verified a priori via a straightforward calculation.…”

“…In [1,2,5] a new sampling theory, known as generalized sampling, was introduced for stable reconstructions of signals in arbitrary bases {ϕ j } j∈N from their samples (1.1) (see §1. 4 and §4 for further details).…”