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Erschienen in:
2009 | OriginalPaper | Buchkapitel
Deriving Box-Spline Subdivision Schemes
verfasst von : N. A. Dodgson, U. H. Augsdörfer, T. J. Cashman, M. A. Sabin
Erschienen in: Mathematics of Surfaces XIII
Verlag: Springer Berlin Heidelberg
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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.