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Exponential splines and minimal-support bases for curve representation


Delgado-Gonzalo, R; Thévenaz, P; Unser, M (2012). Exponential splines and minimal-support bases for curve representation. Computer Aided Geometric Design, 29(2):109-128.

Abstract

Our interest is to characterize the spline-like integer-shift-invariant bases capable of reproducing exponential polynomial curves. We prove that any compact-support function that reproduces a subspace of the exponential polynomials can be expressed as the convolution of an exponential B-spline with a compact-support distribution. As a direct consequence of this factorization theorem, we show that the minimal-support basis functions of that subspace are linear combinations of derivatives of exponential B-splines. These minimal-support basis functions form a natural multiscale hierarchy, which we utilize to design fast multiresolution algorithms and subdivision schemes for the representation of closed geometric curves. This makes them attractive from a
computational point of view. Finally, we illustrate our scheme by constructing minimal-support bases that reproduce ellipses and higher-order harmonic curves.

Abstract

Our interest is to characterize the spline-like integer-shift-invariant bases capable of reproducing exponential polynomial curves. We prove that any compact-support function that reproduces a subspace of the exponential polynomials can be expressed as the convolution of an exponential B-spline with a compact-support distribution. As a direct consequence of this factorization theorem, we show that the minimal-support basis functions of that subspace are linear combinations of derivatives of exponential B-splines. These minimal-support basis functions form a natural multiscale hierarchy, which we utilize to design fast multiresolution algorithms and subdivision schemes for the representation of closed geometric curves. This makes them attractive from a
computational point of view. Finally, we illustrate our scheme by constructing minimal-support bases that reproduce ellipses and higher-order harmonic curves.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:Special Collections > SystemsX.ch
Special Collections > SystemsX.ch > Research, Technology and Development Projects > DynamiX
Special Collections > SystemsX.ch > Research, Technology and Development Projects
Dewey Decimal Classification:570 Life sciences; biology
Language:English
Date:2012
Deposited On:11 Jul 2013 07:55
Last Modified:05 Apr 2016 16:50
Publisher:Elsevier
ISSN:0167-8396
Publisher DOI:https://doi.org/10.1016/j.cagd.2011.10.005

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