EPFL | Biomedical Imaging Group

Hex-Splines: A Novel Spline Family for Hexagonal Lattices

D. Van De Ville, T. Blu, M. Unser, W. Philips, I. Lemahieu, R. Van de Walle

IEEE Transactions on Image Processing, vol. 13, no. 6, pp. 758-772, June 2004.

This paper proposes a new family of bivariate, non-separable splines, called hex-splines, especially designed for hexagonal lattices. The starting point of the construction is the indicator function of the Voronoi cell, which is used to define in a natural way the first-order hex-spline. Higher order hex-splines are obtained by successive convolutions. A mathematical analysis of this new bivariate spline family is presented. In particular, we derive a closed form for a hex-spline of arbitrary order. We also discuss important properties, such as their Fourier transform and the fact they form a Riesz basis. We also highlight the approximation order. For conventional rectangular lattices, hex-splines revert to classical separable tensor-product B-splines. Finally, some prototypical applications and experimental results demonstrate the usefulness of hex-splines for handling hexagonally sampled data.

@ARTICLE(http://bigwww.epfl.ch/publications/vandeville0402.html,
AUTHOR="Van De Ville, D. and Blu, T. and Unser, M. and Philips, W. and
	Lemahieu, I. and Van de Walle, R.",
TITLE="Hex-Splines: {A} Novel Spline Family for Hexagonal Lattices",
JOURNAL="{IEEE} Transactions on Image Processing",
YEAR="2004",
volume="13",
number="6",
pages="758--772",
month="June",
note="")

© 2004 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.