Computer Aided Geometric Design
Box splines
SMA '97 Proceedings of the fourth ACM symposium on Solid modeling and applications
C2 Surfaces Built from Zero Sets of the 7-Direction Box Spline
Proceedings of the 6th IMA Conference on the Mathematics of Surfaces
Linear and Cubic Box Splines for the Body Centered Cubic Lattice
VIS '04 Proceedings of the conference on Visualization '04
IEEE Transactions on Visualization and Computer Graphics
On visual quality of optimal 3D sampling and reconstruction
GI '07 Proceedings of Graphics Interface 2007
Quasi-interpolation projectors for box splines
Journal of Computational and Applied Mathematics
Optimal sampling lattices and trivariate box splines
Optimal sampling lattices and trivariate box splines
Box Spline Reconstruction On The Face-Centered Cubic Lattice
IEEE Transactions on Visualization and Computer Graphics
Symmetric box-splines on root lattices
Symmetric box-splines on root lattices
Symmetric box-splines on the A*n lattice
Journal of Approximation Theory
IEEE Transactions on Information Theory
Hex-splines: a novel spline family for hexagonal lattices
IEEE Transactions on Image Processing
GPU isosurface raycasting of FCC datasets
Graphical Models
Hi-index | 7.29 |
Root lattices are efficient sampling lattices for reconstructing isotropic signals in arbitrary dimensions, due to their highly symmetric structure. One root lattice, the Cartesian grid, is almost exclusively used since it matches the coordinate grid; but it is less efficient than other root lattices. Box-splines, on the other hand, generalize tensor-product B-splines by allowing non-Cartesian directions. They provide, in any number of dimensions, higher-order reconstructions of fields, often of higher efficiency than tensored B-splines. But on non-Cartesian lattices, such as the BCC (Body-Centered Cubic) or the FCC (Face-Centered Cubic) lattice, only some box-splines and then only up to dimension three have been investigated. This paper derives and completely characterizes efficient symmetric box-spline reconstruction filters on all irreducible root lattices that exist in any number of dimensionsn=2 (n=3 for D"n and D"n^* lattices). In all cases, box-splines are constructed by convolution using the lattice directions, generalizing the known constructions in two and three variables. For each box-spline, we document the basic properties for computational use: the polynomial degree, the continuity, the linear independence of shifts on the lattice and optimal quasi-interpolants for fast approximation of fields.