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Hex-Splines
 Investigator: Dimitri Van De Ville
 Summary: We characterize the new family of hex-splines which are specifically designed for hexagonal lattices. The hex-spline basis functions are compactly supported; they are constructed from the N-fold convolution of a hexagonal tile.
 Introduction
 Hex-splines are a new type of bivariate splines that are especially designed for hexagonal lattices. Inspired by the indicator function of the Voronoi cell, they are able to preserve the isotropy of the hexagonal lattice (as opposed to their B-spline counterparts). They can be constructed for any order and are piecewise-polynomial (on a triangular mesh). Analytical formulæ have been worked out in both spatial and Fourier domains. For orthogonal lattices, the hex-splines revert to the classical tensor-product B-splines. While the standard approach to represent two-dimensional data uses orthogonal lattices, hexagonal lattices provide several advantages, including a higher degree of symmetry and a better packing density.
 Main Contributions
 We present a thorough mathematical analysis of this new bivariate spline family. In particular, we derive a closed form for a hex-spline of arbitrary order. We also discuss important properties, such as their Fourier transform, the fact that they form a Riesz basis, approximation order, among others. Additionally, we discuss how to advantageously apply them for image processing. We show examples of interpolation and least-squares resampling. This page contains some examples and a Maple 7 worksheet to generate the analytical form of any hex-spline.
 Collaborations: Prof. Michael Unser, Dr. Thierry Blu
 Period: 2002-2005
 Major Publications
 [1] D. Van De Ville, T. Blu, M. Unser, W. Philips, I. Lemahieu, R. Van de Walle, "Hex-Splines: A Novel Spline Family for Hexagonal Lattices," IEEE Transactions on Image Processing, vol. 13, no. 6, pp. 758-772, June 2004.
 [2] D. Van De Ville, T. Blu, M. Unser, "Recursive Filtering for Splines on Hexagonal Lattices," Proceedings of the Twenty-Eighth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'03), Hong Kong SAR, People's Republic of China, April 6-10, 2003, vol. III, pp. 301-304.
 [3] D. Van De Ville, W. Philips, I. Lemahieu, R. Van de Walle, "Suppression of Sampling Moire in Color Printing by Spline-Based Least-Squares Prefiltering," Pattern Recognition Letters, vol. 24, no. 11, pp. 1787-1794, July 2003.
 [4] D. Van De Ville, T. Blu, M. Unser, "On the Approximation Power of Splines: Orthogonal Versus Hexagonal Lattices," Proceedings of the Fifth International Workshop on Sampling Theory and Applications (SampTA'03), Strobl, Austria, May 26-30, 2003, pp. 109-111.
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