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Three-Directional Box-Splines: Characterization and Efficient Evaluation

L. Condat, D. Van De Ville

IEEE Signal Processing Letters, vol. 13, no. 7, pp. 417-420, July 2006.

We propose a new characterization of three-directional box-splines, which are well adapted for interpolation and approximation on hexagonal lattices. Inspired by a construction already applied with success for exponential splines [1] and hex-splines [2], we characterize a box-spline as a convolution of a generating function, which is a Green function of the spline's associated differential operator, and a discrete filter that plays the role of a localization operator. This process leads to an elegant analytical expression of three-directional box-splines. It also brings along a particularly efficient implementation.


  1. M. Unser, T. Blu, "Cardinal Exponential Splines: Part I—Theory and Filtering Algorithms," IEEE Transactions on Signal Processing, vol. 53, no. 4, pp. 1425-1438, April 2005.

  2. 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.

AUTHOR="Condat, L. and Van De Ville, D.",
TITLE="Three-Directional Box-Splines: {C}haracterization and Efficient
JOURNAL="{IEEE} Signal Processing Letters",

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