Bi-frames with 4-fold axial symmetry for quadrilateral surface multiresolution processing

When bivariate filter banks and wavelets are used for surface multiresolution processing, it is required that the decomposition and reconstruction algorithms for regular vertices derived from them have high symmetry. This symmetry requirement makes it possible to design the corresponding multiresolution algorithms for extraordinary vertices. Recently lifting-scheme based biorthogonal bivariate wavelets with high symmetry have been constructed for surface multiresolution processing. If biorthogonal wavelets have certain smoothness, then the analysis or synthesis scaling function, or both have big supports in general. In particular, when the synthesis lowpass filter is a commonly used scheme such as Loop’s scheme or Catmull-Clark’s scheme, the corresponding analysis lowpass filter has a big support and the corresponding analysis scaling function and wavelets have poor smoothness. Big supports of scaling functions, or in other words, big templates of multiresolution algorithms are undesirable for surface processing. On the other hand, frame provides a flexibility for the construction of “basis” systems. This paper concerns the construction of wavelet (or affine) bi-frames with high symmetry. In this paper we study the construction of wavelet bi-frames with 4-fold symmetry for quadrilateral surface multiresolution processing, with both the dyadic and √ 2 refinements considered. The constructed bi-frames have 4 framelets (or frame generators) for the dyadic refinement, and 2 framelets for the √ 2 refinement. Namely, with either the dyadic or √ 2 refinement, a frame system constructed in this paper has one more generator only than a wavelet system. The constructed bi-frames have better smoothness and smaller supports than biorthogonal wavelets. Furthermore, all the frame algorithms considered in this paper are given by templates so that one can easily implement them. AMS 2000 Math Subject Classification: 42C40, 65T60, 68U07, 65D17