Optimal Spline Generators for Derivative Sampling

Aziznejad, Shayan; Naderi, Alireza; Unser, Michaël

doi:10.1109/sampta45681.2019.9030990

Cited by 1 publication

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“…For a single generator φ such that S(φ) reproduces polynomials of degree up to M, Schoenberg stated that |supp(φ)| ≥ M + 1 [22]. The result was proved in [40] for N = 2. We now extend the proof to any N ∈ N \ {0}.…”

Section: Shortest Basesmentioning

confidence: 98%

Shortest-support Multi-Spline Bases for Generalized Sampling

Goujon¹,

Aziznejad²,

Naderi³

et al. 2020

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Generalized sampling consists in the recovery of a function f , from the samples of the responses of a collection of linear shift-invariant systems to the input f . The reconstructed function is typically a member of a finitely generated integer-shift-invariant space that can reproduce polynomials up to a given degree M . While this property allows for an approximation power of order (M + 1), it comes with a tradeoff on the length of the support of the basis functions. Specifically, we prove that the sum of the length of the support of the generators is at least (M +1). Following this result, we introduce the notion of shortest basis of degree M , which is motivated by our desire to minimize the computational costs. We then demonstrate that any basis of shortest support generates a Riesz basis. Finally, we introduce a recursive algorithm to construct the shortest-support basis for any multi-spline space. It provides 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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Section: Shortest Basesmentioning

confidence: 98%