Non-uniform Sampling in Shift-invariant Spaces

This article discusses modern techniques for non-uniform sampling and reconstruction of functions in shift-invariant spaces. The reconstruction of a function or signal or image f from its non-uniform samples f(xj) is a common task in many applications in data or signal or image processing. The non-uniformity of the sampling set is often a fact of life and prevents the use of the standard methods from Fourier analysis. The non-uniform sampling problem is usually treated in the context of band-limited functions and with tools from complex analysis. However, many applied problems impose different a priori constraints on the type of function. These constraints are taken into consideration by investigating the problem in general shift-invariant spaces rather than for band-limited functions only. This generalization requires a new set of techniques and ideas. While some of the tools are implicitly used in certain questions in approximation theory, wavelet theory and frame theory, they have received little attention in the context of sampling theory, and thus some of the current literature seems unnecessarily complicated. This article is a survey as well as a research paper and is intended to provide the main building blocks for a general theory of non-uniform sampling in shift-invariant spaces. Emphasis is on the following features: (a) Within the setting of shift-invariant spaces the sampling problem is well-defined; (b) The general theory works in arbitrary dimension and for a broad class of generators; (c) For any sufficiently dense non-uniform sampling set useful iterative reconstruction algorithms are derived; (d) In order to model the decay conditions encountered in natural signals and images, the sampling theory is developed in weighted L-spaces. This is a useful innovation over the standard L techniques. Date: January 23, 2000. 1991 Mathematics Subject Classification. Primary 41A15,42C15, 46A35, 46E15, 46N99, 47B37.