Computing Efficient Matrix-valued Wavelets for Meshes

In recent years, several subdivision wavelets, which are constructed directly from the subdivision templates, proposed for the triangular and quadrilateral meshes. To reduce the algorithm complexity, most deploy the local lifting and discrete inner product, which are efficient but not yet accurate. In this paper, we propose the novel approach to construct the efficient biorthogonal wavelets based on interpolatory loop subdivisions. Different from the existing subdivision wavelets, the wavelet we develop is directly constructed from the refined bivariate spline function vectors on the six-directional meshes. By applying the lifting operations, the wavelet transforms are finally local and in-place. The new wavelet transform inherits the advantage of interpolatory refinement with more levels of resolution. The numerical experiments we have tested showed that the new wavelet transform is sufficiently stable, and well performed especially in dealing with semi-regular triangular meshes.