Biorthogonal wavelets with 4-fold axial symmetry for quadrilateral surface multiresolution processing

Abstract

Surface multiresolution processing is an important subject in CAGD. It also poses many challenging problems including the design of multiresolution algorithms. Unlike images which are in general sampled on a regular square or hexagonal lattice, the meshes in surfaces processing could have an arbitrary topology, namely, they consist of not only regular vertices but also extraordinary vertices, which requires the multiresolution algorithms have high symmetry. With the idea of lifting scheme, Bertram (Computing 72(1-2): 29-39, 2004) introduces a novel triangle surface multiresolution algorithm which works for both regular and extraordinary vertices. This method is also successfully used to develop multiresolution algorithms for quad surface and √ 3 triangle surface processing in Wang et al. (Vis Comput 22(9-11):874-884, 2006; IEEE Trans Vis Comput Graph 13(5):914-925, 2007) respectively.When considering the biorthogonality, these papers do not use the conventional L 2 (IR 2 ) inner product, and they do not consider the corresponding lowpass filter, highpass filters, scaling function and wavelets. Hence, some basic properties such as smoothness and approximation power of the scaling functions and wavelets for regular vertices are unclear. On the other hand, the symmetry of subdivision masks (namely, the lowpass filters of filter banks) for surface subdivision is well studied, while the symmetry of the highpass filters for surface processing is rarely considered in the literature. In this paper we introduce the notion of 4-fold symmetry for biorthogonal filter banks. We demonstrate that 4-fold symmetric filter banks result in multiresolution algorithms with Communicated by R. Q. Jia.

Multiscale representation of surfaces by tight wavelet frames with applications to denoising

“…With the idea of the lifting scheme, biorthogonal wavelets with high symmetry for surface multiresolution processing have been constructed in [9][10][11][12][13][14]. If the biorthogonal wavelets have certain smoothness, they will have big supports.…”

Symmetric canonical quincunx tight framelets with high vanishing moments and smoothness

“…Due to the importance of high dimensional problems, multivariate wavelets and framelets have been studied for many years now. For example, quincunx orthonormal wavelets have been investigated in [7,19] and quincunx biorthogonal wavelets have been studied in [7,32,37]. Using the dilation matrix M √ 2 and perturbation of the Daubechies orthonormal wavelets, a family of quincunx orthonormal wavelets with arbitrarily smoothness orders has been reported in [1].…”

“…In fact, if the dilation matrix M √ 2 is changed into N √ 2 for the family of quincunx wavelet filter banks in [1], as a known phenomenon observed in [7], their smoothness orders are no more than one and however decreases to zero. The quincunx biorthogonal wavelets constructed in some literature such as [32,37] have nice smoothness and/or full D 4 -symmetry. However the biorthogonal wavelets usually have large supports and the corresponding wavelet transforms have large condition numbers.…”

Biorthogonal Multiwavelets with Sampling Property and Application in Image Compression

“…The linear phase avails us to use simple symmetric extension methods to handle finite-length signals' boundaries. How to construct symmetric multiwavelets has been studied in many papers, such as [8][9][10][11]23,26].…”

“…Thus, parameterizations of multifilter banks are very important for studying multiwavelets. Some types of parameterizations have been given in [8][9][10][11]15,17,23,26,27]. In this section, we will discuss parameterizations of multifilter banks related to biorthogonal interpolating multiwavelets and construct some multiwavelets with certain smoothness.…”