IEEE Transactions on Signal Processing
Symmetric box-splines on root lattices
Journal of Computational and Applied Mathematics
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Due to their highly symmetric structure, in arbitrary dimensions root lattices are considered as efficient sampling lattices for reconstructing isotropic signals. Among the root lattices the Cartesian lattice is widely used since it naturally matches the Cartesian coordinates. However, in low dimensions, non-Cartesian root lattices have been shown to be more efficient sampling lattices. For reconstruction we turn to a specific class of multivariate splines. Multivariate splines have played an important role in approximation theory. In particular, box-splines, a generalization of univariate uniform B-splines to multiple variables, can be used to approximate continuous fields sampled on the Cartesian lattice in arbitrary dimensions. Box-splines on non-Cartesian lattices have been used limited to at most dimension three. This dissertation investigates symmetric box-splines as reconstruction filters on root lattices (including the Cartesian lattice) in arbitrary dimensions. These box-splines are constructed by leveraging the directions inherent in each lattice. For each box-spline, its degree, continuity and the linear independence of the sequence of its shifts are established. Quasi-interpolants for quick approximation of continuous fields are derived. We show that some of the box-splines agree with known constructions in low dimensions. For fast and exact evaluation, we show that and how the splines can be efficiently evaluated via their BB(Bernstein-Bézier)-forms. This relies on a technique to compute their exact (rational) BB-coefficients. As an application, volumetric data reconstruction on the FCC (Face-Centered Cubic) lattice is implemented and compared with reconstruction on the Cartesian lattice.