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Hex-splines: a novel spline family for hexagonal lattices

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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.
Keywords
bivariate splines, approximation theory, hexagonal lattices, sampling theory, FAST FOURIER-TRANSFORM, 2-DIMENSIONAL SIGNALS, B-SPLINES, RECONSTRUCTION, INTERPOLATION

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Chicago
Van De Ville, Dimitri, Thierry Blu, Michael Unser, Wilfried Philips, Ignace Lemahieu, and Rik Van de Walle. 2004. “Hex-splines: a Novel Spline Family for Hexagonal Lattices.” Ieee Transactions on Image Processing 13 (6): 758–772.
APA
Van De Ville, Dimitri, Blu, T., Unser, M., Philips, W., Lemahieu, I., & Van de Walle, R. (2004). Hex-splines: a novel spline family for hexagonal lattices. IEEE TRANSACTIONS ON IMAGE PROCESSING, 13(6), 758–772.
Vancouver
1.
Van De Ville D, Blu T, Unser M, Philips W, Lemahieu I, Van de Walle R. Hex-splines: a novel spline family for hexagonal lattices. IEEE TRANSACTIONS ON IMAGE PROCESSING. 2004;13(6):758–72.
MLA
Van De Ville, Dimitri, Thierry Blu, Michael Unser, et al. “Hex-splines: a Novel Spline Family for Hexagonal Lattices.” IEEE TRANSACTIONS ON IMAGE PROCESSING 13.6 (2004): 758–772. Print.
@article{306153,
  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.},
  author       = {Van De Ville, Dimitri and Blu, Thierry and Unser, Michael and Philips, Wilfried and Lemahieu, Ignace and Van de Walle, Rik},
  issn         = {1057-7149},
  journal      = {IEEE TRANSACTIONS ON IMAGE PROCESSING},
  keyword      = {bivariate splines,approximation theory,hexagonal lattices,sampling theory,FAST FOURIER-TRANSFORM,2-DIMENSIONAL SIGNALS,B-SPLINES,RECONSTRUCTION,INTERPOLATION},
  language     = {eng},
  number       = {6},
  pages        = {758--772},
  title        = {Hex-splines: a novel spline family for hexagonal lattices},
  url          = {http://dx.doi.org/10.1109/TIP.2004.827231},
  volume       = {13},
  year         = {2004},
}

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