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Sampling and Approximation Theory

M. Unser

Plenary talk, proceedings of the Summer School "New Trends and Directions in Harmonic Analysis, Approximation Theory, and Image Analysis," Inzell, Federal Republic of Germany, September 17-21, 2007, pp. 15.



This tutorial will explain the modern, Hilbert-space approach for the discretization (sampling) and reconstruction (interpolation) of images (in two or higher dimensions). The emphasis will be on quality and optimality, which are important considerations for biomedical applications.

The main point in the modern formulation is that the signal model need not be bandlimited. In fact, it makes much better sense computationally to consider spline or wavelet-like representations that involve much shorter (e.g. compactly supported) basis functions that are shifted replicates of a single prototype (e.g., B-spline). We will show how Shannon's standard sampling paradigm can be adapted for dealing with such representations. In essence, this boils down to modifying the classical "anti-aliasing" prefilter so that it is optimally matched to the representation space (in practice, this can be accomplished by suitable digital post-filtering). Another important issue will be the assessment of interpolation quality and the identification of basis functions (and interpolators) that offer the best performance for a given computational budget.

References


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