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U. Diewald, S. Morigi, and M. Rumpf.
A cascadic geometric filtering approach to subdivision.
Computer Aided Geometric Design, 19:675694, 2002.
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A new approach to subdivision based on the evolution of surfaces under
curvature motion is presented. Such an evolution can be understood
as a natural geometric filter process where time corresponds to the
filter width. Thus, subdivision can be interpreted as the application
of a geometric filter on an initial surface. The concrete scheme
is a model of such a filtering based on a successively improved spatial
approximation starting with some initial coarse mesh and leading
to a smooth limit surface.In every subdivision step the underlying
grid is refined by some regular refinement rule and a linear finite
element problem is either solved exactly or, especially on fine grid
levels, one confines to a small number of smoothing steps within
the corresponding iterative linear solver. The approach closely connects
subdivision to surface fairing concerning the geometric smoothing
and to cascadic multigrid methods with respect to the actual numerical
procedure. The derived method does not distinguish between different
valences of nodes nor between different mesh refinement types. Furthermore,
the method comes along with a new approach for the theoretical treatment
of subdivision.
