Reconstruction from Error-Affected Sampled Data in Shift-Invariant Spaces

Abstract

In the following chapters we provide error estimates for signals reconstructed from corrupt data. Two different types of error are considered. First, we address the problem of reconstructing a continuous function defined on <b>R</b>^d from a countable collection of samples corrupted by noise. The additive noise is assumed to be i.i.d. with mean 0 and variance σ². We sample the continuous function <i>f</i> on the uniform lattice (1/m)<b>Z</b>^d and show for large enough m that the variance of the error between the frame reconstruction <i>f</i>_ε from noisy samples of <i>f</i> and the function <i>f</i> satisfy var(<i>f</i>_ε (<i>x</i>)-<i>f</i>(<i>x</i>))≈ (σ²/m^d)<i>C</i>_x. Second, we address the problem of non-uniform sampling and reconstruction in the presence of jitter. In sampling applications, the set X={<i>x_j: j ∈ J</i>} on which a signal <i>f</i> is sampled is not precisely known. Two main questions are considered. First, if sampling a function <i>f</i> on the countable set X leads to unique and stable reconstruction of <i>f</i>, then when does sampling on the set X'={<i>x_j</i>+δ_<i>j</i>: <i>j</i> ∈ <i>J</i>} also lead to unique and stable reconstruction? Second, if we attempt to recover a sampled function <i> f</i> using the reconstruction operator corresponding to the sampling set X (because the precise sample points are unknown), is the recovered function a good approximation of the original <i>f</i>?