BIOMEDICAL IMAGING GROUP (BIG)
Laboratoire d'imagerie biomédicale (LIB)

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. 

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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",
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