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

    Leonetti, Casey Clark
    : https://etd.library.vanderbilt.edu/etd-03292007-180005
    http://hdl.handle.net/1803/11682
    : 2007-04-14

    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><sup>d</sup> from a countable collection of samples corrupted by noise. The additive noise is assumed to be i.i.d. with mean 0 and variance σ<sup>2</sup>. We sample the continuous function <i>f</i> on the uniform lattice (1/m)<b>Z</b><sup>d</sup> and show for large enough m that the variance of the error between the frame reconstruction <i>f</i><sub>ε</sub> from noisy samples of <i>f</i> and the function <i>f</i> satisfy var(<i>f</i><sub>ε</sub> (<i>x</i>)-<i>f</i>(<i>x</i>))≈ (σ<sup>2</sup>/m<sup>d</sup>)<i>C</i><sub>x</sub>. Second, we address the problem of non-uniform sampling and reconstruction in the presence of jitter. In sampling applications, the set X={<i>x<sub>j</sub>: 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<sub>j</sub></i>+δ<sub><i>j</i></sub>: <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>?
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