Abstract
In a previous work [1] we investigated how to reverse subdivision
rules using local linear conditions based upon least squares approximation. We
outlined a general approach for reversing subdivisions and showed how to use
the approach to construct multiresolutions with finite decomposition and
reconstruction filters. These multiresolutions correspond to biorthogonal
wavelet systems that use inner products implicitly defined by the construction.
We gave evidence through a number of example subdivision rules that the
approach works for curves and tensor-product surfaces. In [14] some of this
material was put to work on non-tensor-product surface meshes of arbitrary
connectivity. The price to be paid for such connectivity is a limitation on
the flexibility one has in formulating the linear conditions for reversal and
the complexity in assessing the face topology of the mesh. The full sweep of
the general approach is lost in the irregularity of the application. In this
work we take regular, triangular meshes and use one interpolating and two
noninterpolating subdivisions: the Butterfly subdivision [6], Loop's
subdivision [12], and a quasi-interpolation based subdivision [11], as
examples. We visit the general approach for curves once again and, using these
example subdivisions, show that the approach can be applied with success to
produce finite filter multiresolutions in the triangular mesh case as well. In
the process, we introduce graphical insights that provide a mask-based
development in place of our previous matrix-based development, suggesting that
our construction is not limited to triangle mesh geometry. To overcome a
limitation we encountered in symbolic algebra systems, we invoke the lifting
process [19] in a nonstandard way.
Notes
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