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

Shortest Multi-Spline Bases for Generalized Sampling

A. Goujon

Online Seminars on Numerical Approximation and Applications (OSNA2'20), Passau, Federal Republic of Germany, Virtual, November 9-December 3, 2020.

Generalized sampling consists in recovering a function from the samples of its response to a collection of linear shift-invariant systems. The reconstructed function is typically picked from a finitely generated shift-invariant space that can reproduce polynomials up to a given degree M. While this property guarantees an approximation power of order (M + 1), it comes with a tradeoff on the size of the support of the basis functions. Specifically, we prove that the sum of the supports of the generators is necessarily greater or equal to (M + 1). Following this result, we introduce the notion of shortest basis of degree M, which is motivated by our desire to minimize computational cost. We then demonstrate that any shortest support basis generates a Riesz basis. Finally, we introduce a recursive algorithm for the construction of the shortest support basis for any multi-spline space. It constitutes a generalization of both polynomial and Hermite B-splines. This framework paves the way for novel applications, such as fast derivative sampling with arbitrarily high approximation power. 

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