Sampling: 60 Years After Shannon
M. Unser
Plenary talk, Sixteenth International Conference on Digital Signal Processing (DSP'09), Σαντορίνη (Santorini), Ελληνική Δημοκρατία (Hellenic Republic), July 5-7, 2009.
The purpose of this talk 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 argue in favor of a variational formulation where the optimal signal reconstruction is specified via a functional optimization problem. The cost to minimize is the sum of a discrete data term and a regularization functional that penalizes non-desirable solutions. We show that, when the regularization is quadratic, the optimal signal reconstruction (among all possible functions) is a generalized spline whose type is tied to the regularization operator. This leads to an optimal discretization and an efficient signal reconstruction in terms of generalized B-spline basis functions. A possible variation is to penalize the L1-norm of the derivative of the function (total variation), which can also be achieved within the spline framework via a suitable knot deletion process.
The theory of compressed sensing provides an alternative approach to sampling that is qualitatively similar to total-variation regularization. Here the idea to favor solutions that have a sparse representation in a wavelet basis. Practically, this is achieved by imposing a regularization constraint on the ℓ1-norm of the wavelet coefficients. We show that the corresponding inverse problem can be solved efficiently via a multi-scale variant of the ISTA algorithm (iterative skrinkage-thresholding). We illustrate the method with two concrete imaging examples: the deconvolution of 3-D fluorescence micrographs, and the reconstruction of magnetic resonance images from arbitrary (non-uniform) k-space trajectories.
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.
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