title: Hex-splines: a novel spline family for hexagonal lattices.
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.
authors.affiliation: Biomedical Imaging Group, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland. dimitri.vandeville@epfl.ch
authors.forename: Dimitri
authors.initials: D
authors.lastname: Van De Ville
authors.forename: Thierry
authors.initials: T
authors.lastname: Blu
authors.forename: Michael
authors.initials: M
authors.lastname: Unser
authors.forename: Wilfried
authors.initials: W
authors.lastname: Philips
authors.forename: Ignace
authors.initials: I
authors.lastname: Lemahieu
authors.forename: Rik
authors.initials: R
authors.lastname: Van de Walle
mesh_terms: D000465:Algorithms; D003196:Computer Graphics; D003198:Computer Simulation; D007089:Image Enhancement / Q000379:methods*; D007090:Image Interpretation, Computer-Assisted / Q000379:methods*; D016247:Information Storage and Retrieval / Q000379:methods; D009716:Numerical Analysis, Computer-Assisted; D010363:Pattern Recognition, Automated / Q000379:methods*; D015203:Reproducibility of Results; D018401:Sample Size; D012680:Sensitivity and Specificity; D012815:Signal Processing, Computer-Assisted