Deriving Box-Spline Subdivision Schemes

Authors:
N. A. Dodgson;U. H. Augsdörfer;T. J. Cashman;M. A. Sabin
Affiliations:
The Computer Laboratory, University of Cambridge, UK;The Computer Laboratory, University of Cambridge, UK;The Computer Laboratory, University of Cambridge, UK;Numerical Geometry Ltd., UK
Venue:
Proceedings of the 13th IMA International Conference on Mathematics of Surfaces XIII
Year:
2009

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Abstract

We describe and demonstrate an arrow notation for deriving box-spline subdivision schemes. We compare it with the z -transform, matrix, and mask convolution methods of deriving the same. We show how the arrow method provides a useful graphical alternative to the three numerical methods. We demonstrate the properties that can be derived easily using the arrow method: mask, stencils, continuity in regular regions, safe extrusion directions. We derive all of the symmetric quadrilateral binary box-spline subdivision schemes with up to eight arrows and all of the symmetric triangular binary box-spline subdivision schemes with up to six arrows. We explain how the arrow notation can be extended to handle ternary schemes. We introduce two new binary dual quadrilateral box-spline schemes and one new $\sqrt2$ box-spline scheme. With appropriate extensions to handle extraordinary cases, these could each form the basis for a new subdivision scheme.