REVERSING SUBDIVISION USING LOCAL LINEAR CONDITIONS: GENERATING MULTIRESOLUTIONS ON REGULAR TRIANGULAR MESHES
In a previous work  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  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 , Loop's subdivision , and a quasi-interpolation based subdivision , 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  in a nonstandard way.