Wavelet Bi-frames with Uniform Symmetry for Curve Multiresolution Processing

This paper is about the construction of wavelet bi-frames with each framelets being symmetric. When filter banks are used for surface multiresolution processing, it is required that the corresponding decomposition and reconstruction algorithms, including the algorithms for boundary vertices, have high symmetry which makes it possible to design the corresponding multiresolution algorithms for extraordinary vertices. When the multiresolution algorithms derived from univariate wavelet bi-frames are used as the boundary algorithms for surface multiresolution processing, it is required that not only the scaling functions but also all framelets are symmetric. In addition, for curve/surface multiresolution processing, it is also desirable that the algorithms should be given by templates so that the algorithms can be easily implemented. In this paper, first, by associating appropriately the lowpass and highpass outputs to the nodes of Z, we show that both biorthogonal wavelet multiresolution algorithms and bi-frame multiresolution algorithms can be represented by templates. Then, using the idea of lifting scheme, we provide frame algorithms given by several iterative steps with each step represented by a symmetric template. Finally, with the given iterative algorithms, we construct bi-frames based on their smoothness and vanishing moments. Two types of symmetric bi-frames are studied. In order to provide a clearer picture on the procedure for bi-frame construction, in this paper we also consider template-based construction of biorthogonal wavelets. The approach of the template-based bi-frame construction introduced in this paper can easily be extended to the construction of bivariate bi-frames with high symmetry for surface multiresolution processing.