Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces

Abstract

Abstract. This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift-invariant spaces. It is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shiftinvariant spaces by bringing together wavelet theory, frame theory, reproducing kernel Hilbert spaces, approximation theory, amalgam spaces, and sampling. Inspired by applications taken from communication, astronomy, and medicine, the following aspects will be emphasized: (a) The sampling problem is well defined within the setting of shift-invariant spaces. (b) The general theory works in arbitrary dimension and for a broad class of generators. (c) The reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms. These algorithms converge geometrically and are robust in the presence of noise. (d) To model the natural decay conditions of real signals and images, the sampling theory is developed in weighted L p -spaces.

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“…Under certain conditions, functions belonging to the more general shift-invariant spaces (SIS) can be perfectly reconstructed from uniform (Aldroubi et al, 1994) as well as nonuniform samples Feichtinger, 1998, Aldroubi andGröchenig, 2001). Furthermore, Gontier and Vetterli (2014) have provided sufficient conditions for reconstructing the input of an IF neuron belonging to a SIS from the associated output sequence.…”

“…PW Ω , which is equivalent to the fact that and Sell, 1982), also known as the harmonic Fourier basis (Aldroubi and Gröchenig, 2001), where…”

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“…Thus, it converges to an element c ′ [n] ∈ ℓ 2 (Z). The inequality in (24) then implies that n∈Z c[n]φ n,α (t) converges to n∈Z c ′ [n]φ n,α (t) ∈ V α (φ) as n tends to infinity. Finally, if n∈Z c[n]φ n,α (t) = 0, then (25) implies that c[n] = 0 for all n ∈ Z.…”

“…Clearly, if A ≤ 2πG 2 φ,α (u) ≤ B, then it follows from (26) and (13) that {φ n,α (t)} n∈Z suffices (8), and (24) and (25) …”

“…Consequently, {φ n,α (t)} n∈Z is a Riesz basis (or a frame) for V α (φ) if and only if {φ(t − n)} n∈Z is a Riesz basis (or a frame) for V(φ), for which there are many known results, e.g., nonuniform sampling and reconstruction, wavelets and multiresolution analysis, oversampling, compressed sensing, and frames of translates [14,[24][25][26][27][28][29].…”