Hex-splines: a novel spline family for hexagonal lattices

Authors:
D. Van De Ville;T. Blu;M. Unser;W. Philips;I. Lemahieu;R. Van de Walle
Affiliations:
Biomed. Imaging Group, Swiss Fed. Inst. of Technol. Lausanne, Switzerland;-;-;-;-;-
Venue:
IEEE Transactions on Image Processing
Year:
2004

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Abstract

This paper proposes a new family of bivariate, nonseparable 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.