Sampling: 60 Years After Shannon
M. Unser
Plenary talk, The Sixteenth International Conference on Digital Signal Processing (DSP'09), Santorini, Greece, July 5-7, 2009, in press.
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The purpose of this talk, which is centered around the classical sampling theorem, is to present a modern, unifying perspective of sampling, while demonstrating that the research in this area is still alive and well. We concentrate on the traditional setup where the samples are taken on a uniform grid, but we explicitly take into account the non-ideal nature of the acquisition device and the fact that the measurements may be corrupted by noise. We present a powerful projection-based formulation where the goal is to reconstruct a good approximation of the original signal within a given "shift-invariant" function space (not necessarily bandlimited!). We make the link with splines and approximation theory, while providing efficient computational solutions. We consider several mathematical formulations for specifying the "optimal" reconstruction space—regularization theory, minimum mean square error estimation, invariance to basic coordinate transformations—and show that they essentially lead to the same type of solutions. This suggests a unifying implementation of the optimal reconstruction process in terms of generalized B-spline basis functions. We also makes the connection with kernel methods (reproducing kernel Hilbert spaces, radial basis functions) and indicate some directions for future research; in particular, methods that incorporate sparsity and/or non-quadratic constraints such as TV.
Biography
Michael Unser is professor and Director of EPFL's Biomedical Imaging Group, Lausanne, Switzerland. His main research area is biomedical image processing. He has a strong interest in sampling theories, multiresolution algorithms, wavelets, and the use of splines for image processing. He has published over 150 journal papers on those topics, and is one of ISI's Highly Cited authors in Engineering (http://isihighlycited.com/).
From 1985 to 1997, he was with the Biomedical Engineering and Instrumentation Program, National Institutes of Health, Bethesda USA, conducting research on bioimaging.
Dr. Unser is a fellow of the IEEE, a member of the Swiss Academy of Engineering Sciences, and the recipient of three Best Paper Awards from the IEEE Signal Processing Society.