Wavelet bi-frames with uniform symmetry for curve multiresolution processing

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