Recent Advances in Sampling Theory and Applications

A Multidimensional Irregular Sampling Algorithm and Applications

Authors:

John J Benedetto,
Hui-Chuan Wu,

Page (NA) Paper number 3005

Abstract:

For a given spiral, a bandwidth B can be chosen and a sequence S can be constructed on the spiral with the property that all finite energy signals having bandwidth B can be reconstructed from sampled values on S. The bandwidth can be expanded as desired, and reconstruction is attained by constructing sampling sets on interleaving spirals. This solves a problem in MRI; and the algorithm can be modified to deal with irregular sampling problems in SAR. The algorithm is a consequence of our theoretical results, which in turn were inspired by seminal work on balayage in the 1960s by Beurling and Landau. Our results depend on d-dimensional Fourier frames and tiling properties of spectral synthesis sets.

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The Condition Number Of Certain Matrices And Applications

Authors:

Paulo J.S.G. Ferreira,

Page (NA) Paper number 3002

Abstract:

This paper addresses the problem of estimating the eigenvalues and condition numbers of matrices of the form R=r(t_i-t_j). We begin by mentioning some of the problems in which such matrices occur, and to which the results obtained in this paper may be applied. Examples of such problems include (i) approximation by sums of irregular translates (ii) the missing data problem and incomplete sampling series. Then we describe the method for estimating the eigenvalues and the condition number. Some open issues will also be discussed.

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On The Estimation Of The Bandwidth Of Nonuniformly Sampled Signals

Authors:

Thomas Strohmer,

Page (NA) Paper number 3023

Abstract:

In many applications signals can only be sampled at nonuniformly spaced points. An analyis of the properties of the underlying process often requires knowledge of the (essential) bandwidth of the signal. Therefore robust and efficient methods are needed that allow to estimate the bandwidth of a signal from nonuniform spaced, noisy samples. We present two procedures for bandwidth estimation. The first method is based on the discrete Bernstein inequality and Newton's divided differences and is computationally very efficient. The second method requires somewhat more computational effort, since it simultaneously estimates the bandwidth and provides a reconstruction of the signal. It is based on a multi-scale conjugate gradient algorithm for the solution of a nested sequence of Toeplitz systems and is particularly useful in case of noisy data. Examples from various applications demonstrate the performance of the proposed methods.

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Nonperiodic Sampling And Reconstruction From Averages

Authors:

David F Walnut,

Page (NA) Paper number 3016

Abstract:

In this paper, we discuss an application of sampling theory to the problem of reconstructing a function from its local averages on cubes of different sizes. This problem can be interpreted as a type of Pompeiu problem or from a signal or image processing perspective as a deconvolution problem. In both interpretations, the basic idea is to construct sets of deconvolvers which either exactly or approximately invert the convolution process. In this way, the deconvolution process involves simple linear operations on the convolution data. It is hoped that similar techniques can be used to do reconstruction from averages over other types of regions.

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The Restoration of Missing Data Using Bayesian Numerical Methods

Authors:

William J Fitzgerald, The Department of Engineering, Cambridge University, Cambridge, UK (U.K.)

Page (NA) Paper number 3038

Abstract:

This paper will outline a method for restoring missing samples in digital signals. The missing samples are imputed using a Markov Chain Monte Carlo approach and an introduction to these numerical techniques will be given. One application area will be presented from the area of digital audio restoration where clicks are a familiar problem, and can take the form of sudden unexpected bursts of impulsive noise with random but finite duration. These bursts of noise have numerous causes such as dirt, electrical interference or mechanical damage to the storage medium. The original signal is often effectively lost. Several methods of detecting clicks have been devised, with the best approaches being model based. Once a click has been detected the ''suspect'' samples are removed and replaced by interpolation. Results obtained on both synthetic and real data will be given.

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Non-Uniform Sampling In Wavelet Subspaces

Authors:

Gilbert G Walter, U-Wisconsin-Milwaukee (U.K.)

Page (NA) Paper number 3006

Abstract:

It is well known that the Shannon sampling theorem can be put into a wavelet context. But is has also been shown that for most wavelets, a sampling theorem for the associated subspaces exists. There is even a non-uniform sampling theorem as in the Shannon case. No simple Kadec 1/4 theorem holds except in special cases (such as the Franklin case where the bound is 1/2). For a particular case, the Meyer wavelets, which are bandlimited but with a smooth spectrum, a similar bound is sometimes obtainable. Unfortunately, it is much smaller than 1/4.

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