Nonuniform Average Sampling and Reconstruction in Multiply Generated Shift-Invariant Spaces

Abstract

The problem of reconstructing a function f from a set of nonuniformly distributed, weighted-average sampled values {∫R d f(x)ψXj(x) dx: j ∈ J} is studied in the context of shift-invariant subspaces of Lp(Rd) generated by p-frames. The special but important case where the weighted-average sampled values are of the form {∫R d f(x)ψ (· - Xj)dx : j ∈ J} is also studied. Fast approximation-projection iterative reconstruction algorithms are developed. The performance of the algorithms are analyzed when the data is corrupted by noise. Estimates are derived for the convergence rates of the algorithms in terms of the sampling density, the generators of the shift-invariant space, and the sampling functional (ψXj).