Non-uniform sampling and reconstruction:Theory and algorithms

Vanderbilt University, Nashville TN

Investigators

Abstract

Aldroubi 0103104 The investigator and his colleagues develop a mathematical framework and fast computational schemes for the reconstruction of functions, signals or images from noisy, very large sampled data sets, acquired on nonuniform grids, by nonideal acquisition devices. The problem of nonuniform sampling and reconstruction is treated in the context of shift-invariant subspaces, Besov spaces, and in arbitrary dimensions. The theory is developed for the case when the samples are obtained from weighted averages. Density conditions for exact reconstruction are established. When the data are noisy, incomplete, or when the assumptions needed for exact reconstruction are not satisfied, bounds on the error between the reconstructed and original signal are derived in terms of the sampling densities, the averaging functionals, and the noise statistics. The development of the mathematical framework and the computational schemes requires a new set of techniques and ideas, and involves several areas of mathematics including wavelet theory, frame theory, functional analysis, and harmonic analysis. The project is motivated by problems arising in data transmission, geophysical exploration, astronomy, spectroscopy, and biomedical imaging. The problem of reconstructing a signal or an image from a set of nonuniform samples is encountered in many applications of signal or image processing. For example, the loss of data packets during transmission through internet or from satellites can be viewed as a nonuniform sampling and reconstruction problem. In geophysical exploration, the earth's magnetic field is measured by a combination of airborn, fast moving acquisition devices, as well as scattered stationary devices resulting in highly nonuniform sampling patterns, and a huge data set. The goal is to reconstruct the magnetic field and use it to reveal geological features. In fact, modern digital data processing of signals or images always uses a sampled version of the original analog signals or images. However, the sampling devices are never ideal, and the collected data consist of average samples. Moreover these data are often very large, incomplete, and corrupted by noise. The question then arises whether and how the original signal can be recovered from the data. Therefore the investigator aims to 1) quantify the conditions under which it is possible to reconstruct a signal exactly from different sets of nonuniform average-samples; 2) use these analytical results to develop explicit, and computationally efficient reconstruction schemes; and 3) analyze the performance of the algorithms under adverse conditions, or when the data are incomplete or corrupted by noise. The development of a theory and algorithms that perform well under stringent and realistic situations will help the analysis, processing and management of very large data sets obtained digitally by new acquisition modalities, and transmitted or received by communication networks such as the internet, cellular phones, and other distributed communication systems.