Numerical Methods for Digital Signal Reconstruction

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The investigator develops efficient numerical methods formulti-dimensional signal reconstruction using tools from harmonicanalysis such as local trigonometric transforms and wavelets.The reconstruction problems that are considered range fromapproximation of nonstationary signals from scattered data torecovery of missing data in digital image sequences. Successfulsolution of these reconstruction problems requires a prioriinformation about the signal, otherwise these problems areill-posed. Often this a priori knowledge can be derived fromphysical properties of the process, for instance in terms of the"local frequency behavior" of the signal. Local trigonometrictransforms and wavelets have become powerful methods foranalysis, compression and denoising of digital signals. Theinvestigator derives a mathematical framework for the applicationof these concepts to signal and image reconstruction (beyond thewell-traversed area of denoising). The design of thesereconstruction methods is accompanied by the development ofmulti-level regularization techniques in order to provide aflexible framework for the incorporation of different kinds of apriori knowledge about the signal. The investigator applies thenew numerical methods to the approximation of geophysical signalsfrom scattered data and to error correction in received coded bitstreams. This project is motivated by a number of multi-dimensionalsignal reconstruction problems arising in areas as diverse asexploration geophysics, medical imaging, and telecommunication.In order to optimally compress, transmit or analyze an incompleteor distorted signal, it is necessary to first reconstruct theoriginal signal from the available data. This reconstructionprocess includes the restoration of lost data as well as therecovery of a signal from scattered noisy measurements. Themathematical framework being developed here allows the design offlexible and robust algorithms for the reconstruction ofmulti-dimensional signals. Due to huge amount of data andpossibly required real-time processing the reconstructionalgorithms must also be fast and efficient. The derivednumerical methods are important for reliable error correction indata transmission, for efficient reconstruction of geophysicalsignals, and for other applications.