Deriving box-spline subdivision schemes

  • Dodgson N
  • Augsdörfer U
  • Cashman T
 et al. 
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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 √2 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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