Non-uniform sampling and reconstruction:Theory and algorithms

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Aldroubi0103104 The investigator and his colleagues develop a mathematicalframework and fast computational schemes for the reconstructionof functions, signals or images from noisy, very large sampleddata sets, acquired on nonuniform grids, by nonideal acquisitiondevices. The problem of nonuniform sampling and reconstructionis treated in the context of shift-invariant subspaces, Besovspaces, and in arbitrary dimensions. The theory is developed forthe case when the samples are obtained from weighted averages.Density conditions for exact reconstruction are established. Whenthe data are noisy, incomplete, or when the assumptions neededfor exact reconstruction are not satisfied, bounds on the errorbetween the reconstructed and original signal are derived interms of the sampling densities, the averaging functionals, andthe noise statistics. The development of the mathematicalframework and the computational schemes requires a new set oftechniques and ideas, and involves several areas of mathematicsincluding wavelet theory, frame theory, functional analysis, andharmonic analysis. The project is motivated by problems arising in datatransmission, geophysical exploration, astronomy, spectroscopy,and biomedical imaging. The problem of reconstructing a signal oran image from a set of nonuniform samples is encountered in manyapplications of signal or image processing. For example, the lossof data packets during transmission through internet or fromsatellites can be viewed as a nonuniform sampling andreconstruction problem. In geophysical exploration, the earth'smagnetic field is measured by a combination of airborn, fastmoving acquisition devices, as well as scattered stationarydevices resulting in highly nonuniform sampling patterns, and ahuge data set. The goal is to reconstruct the magnetic field anduse it to reveal geological features. In fact, modern digitaldata processing of signals or images always uses a sampledversion of the original analog signals or images. However, thesampling devices are never ideal, and the collected data consistof average samples. Moreover these data are often very large,incomplete, and corrupted by noise. The question then ariseswhether and how the original signal can be recovered from thedata. Therefore the investigator aims to 1) quantify theconditions under which it is possible to reconstruct a signalexactly from different sets of nonuniform average-samples; 2) usethese analytical results to develop explicit, and computationallyefficient reconstruction schemes; and 3) analyze the performanceof the algorithms under adverse conditions, or when the data areincomplete or corrupted by noise. The development of a theory andalgorithms that perform well under stringent and realisticsituations will help the analysis, processing and management ofvery large data sets obtained digitally by new acquisitionmodalities, and transmitted or received by communication networkssuch as the internet, cellular phones, and other distributedcommunication systems.