BIOMEDICAL IMAGING GROUP (BIG)
Laboratoire d'imagerie biomédicale (LIB)

Sampling: Beyond the Myth of a Perfect Reconstruction

M. Unser

Plenary talk, Proceedings of the Fourth International Conference on Sampling Theory and Applications (SampTA'01), Orlando FL, USA, May 13-17, 2001, pp. 15.

The key idea that we want to convey in this talk is that the principle of a perfect reconstruction in sampling is too strong a constraint; it requires mathematical assumptions on the class of input signals and measurement systems that are rarely met in practice. Instead, we should consider a more realistic form of the problem where the goal is to get the best approximation possible from the signal measurements given a chosen set of basis functions (model), or, alternatively, some reconstruction device. We will develop this idea in the special case of uniform sampling, presenting research results on the approximation of functions in "shift-invariant" spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) pre-filters that are not necessarily ideal lowpass. We will summarize and discuss the results available for the determination of the approximation error and of the minimum sampling rate when the input of the system is essentially arbitrary; e.g., non-bandlimited. Finally, we will review some variations of sampling that can be understood from the same unifying perspective. These include wavelets, multi-wavelets, Papoulis generalized sampling, finite elements, and frames.

For more details and bibliography, we refer to M. Unser, "Sampling—50 Years After Shannon," Proceedings of the IEEE, vol. 88, no. 4, pp. 569-587, April 2000.

See also the tutorials and reviews. 

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