Publication announcement:

Subdivision as a sequence of sampled Cp surfaces and conditions for
tuning schemes

Cedric Gerot, Loic Barthe, Neil A. Dodgson, Malcolm A. Sabin

Technical report UCAM-CL-TR-583, University of Cambridge,
Computer Laboratory, March 2004, 68 pages.

This document is now available at

http://www.cl.cam.ac.uk/TechReports/UCAM-CL-TR-583.pdf

Abstract:

We deal with practical conditions for tuning a subdivision scheme in
order to control its artifacts in the vicinity of a mark point. To do
so, we look for good behaviour of the limit vertices rather than good
mathematical properties of the limit surface. The good behaviour of the
limit vertices is characterised with the definition of C2-convergence of
a scheme. We propose necessary explicit conditions for C2-convergence of
a scheme in the vicinity of any mark point being a vertex of valency n
or the centre of an n-sided face with n greater or equal to three. These
necessary conditions concern the eigenvalues and eigenvectors of
subdivision matrices in the frequency domain. The components of these
matrices may be complex. If we could guarantee that they were real, this
would simplify numerical analysis of the eigenstructure of the matrices,
especially in the context of scheme tuning where we manipulate symbolic
terms. In this paper we show that an appropriate choice of the parameter
space combined with a substitution of vertices lets us transform these
matrices into pure real ones. The substitution consists in replacing
some vertices by linear combinations of themselves. Finally, we explain
how to derive conditions on the eigenelements of the real matrices which
are necessary for the C2-convergence of the scheme.

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University of Cambridge, Computer Laboratory,
Technical Reports (ISSN 1476-2986)
http://www.cl.cam.ac.uk/TechReports/