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Elementary factorisation of Box spline subdivision

Elementary factorisation of Box spline subdivision When a subdivision scheme is factorised into lifting steps, it admits an in–place and invertible implementation, and it can be the predictor of many multiresolution biorthogonal wavelet transforms. In the regular setting where the underlying lattice hierarchy is defined by ℤ s and a dilation matrix M, such a factorisation should deal with every vertex of each subset in ℤ s /Mℤ s in the same way. We define a subdivision scheme which admits such a factorisation as being uniformly elementary factorable. We prove a necessary and sufficient condition on the directions of the Box spline and the arity of the subdivision for the scheme to admit such a factorisation, and recall some known keys to construct it in practice. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Computational Mathematics Springer Journals

Elementary factorisation of Box spline subdivision

Advances in Computational Mathematics , Volume 45 (1) – Jun 5, 2018

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics; Mathematical and Computational Biology; Computational Science and Engineering; Visualization
ISSN
1019-7168
eISSN
1572-9044
DOI
10.1007/s10444-018-9612-x
Publisher site
See Article on Publisher Site

Abstract

When a subdivision scheme is factorised into lifting steps, it admits an in–place and invertible implementation, and it can be the predictor of many multiresolution biorthogonal wavelet transforms. In the regular setting where the underlying lattice hierarchy is defined by ℤ s and a dilation matrix M, such a factorisation should deal with every vertex of each subset in ℤ s /Mℤ s in the same way. We define a subdivision scheme which admits such a factorisation as being uniformly elementary factorable. We prove a necessary and sufficient condition on the directions of the Box spline and the arity of the subdivision for the scheme to admit such a factorisation, and recall some known keys to construct it in practice.

Journal

Advances in Computational MathematicsSpringer Journals

Published: Jun 5, 2018

References