Sampling in a Union of Frame Generated Subspaces

TL;DR

This paper extends the sampling theory based on unions of subspaces to include shift-invariant spaces described by frame generators, broadening the applicability of the structured non-linear sampling model.

Contribution

It generalizes the union of subspaces sampling framework to shift-invariant spaces using frames, ensuring stability and injectivity conditions hold.

Findings

The sampling operator remains one-to-one in the generalized setting.

Stability conditions are validated for shift-invariant spaces with frame generators.

The theory supports broader classes of signals in compressed sampling.

Abstract

A new paradigm in Sampling theory has been developed recently by Lu and Do. In this new approach the classical linear model is replaced by a non-linear, but structured model consisting of a union of subspaces. This is the natural approach for the new theory of compressed sampling, representation of sparse signals and signals with finite rate of innovation. In this article we extend the theory of Lu and Do, for the case that the subspaces in the union are shift-invariant spaces. We describe the subspaces by means of frame generators instead of orthonormal bases. We show that, the one to one and stability conditions for the sampling operator, are valid for this more general case.