Exponential splines and minimal-support bases for curve representation

Authors:
R. Delgado-Gonzalo;P. Thévenaz;M. Unser
Affiliations:
Biomedical Imaging Group, ícole polytechnique fédérale de Lausanne (EPFL), Switzerland;Biomedical Imaging Group, ícole polytechnique fédérale de Lausanne (EPFL), Switzerland;Biomedical Imaging Group, ícole polytechnique fédérale de Lausanne (EPFL), Switzerland
Venue:
Computer Aided Geometric Design
Year:
2012

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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.