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.
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Dodgson, N.A., Augsdörfer, U.H., Cashman, T.J., Sabin, M.A. (2009). Deriving Box-Spline Subdivision Schemes. In: Hancock, E.R., Martin, R.R., Sabin, M.A. (eds) Mathematics of Surfaces XIII. Mathematics of Surfaces 2009. Lecture Notes in Computer Science, vol 5654. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03596-8_7
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DOI: https://doi.org/10.1007/978-3-642-03596-8_7
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