Sparse representation on graphs by tight wavelet frames and applications

http://www.cs.umsl.edu/~jiang/webpaper/surface_represen_denoising.pdf

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

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 $\sqrt 3$ 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 $L^2({{\rm I}\kern-.2em{\rm R}}^2)$ 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 the required symmetry for quad surface processing. In addition, we provide 4-fold symmetric biorthogonal FIR filter banks and construct the associated wavelets, with both the dyadic and $\sqrt 2$ refinements. Furthermore, we show that some filter banks constructed in this paper result in very simple multiresolution decomposition and reconstruction algorithms as those in Bertram (Computing 72(1---2):29---39, 2004) and Wang et al. (Vis Comput 22(9---11):874---884, 2006; IEEE Trans Vis Comput Graph 13(5):914---925, 2007). Our method can provide the filter banks corresponding to the multiresolution algorithms in Wang et al. (Vis Comput 22(9---11):874---884, 2006) for dyadic multiresolution quad surface processing. Therefore, the properties of the scaling functions and wavelets corresponding to those algorithms can be obtained by analyzing the corresponding filter banks.

http://www.cs.umsl.edu/~jiang/webpaper/bio_wavelet_4sym_quad.pdf

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

http://www.cs.umsl.edu/~jiang/webpaper/frame_triang_dyadic_6sym.pdf

Highly symmetric bi-frames for triangle surface multiresolution processing

On construction of multivariate symmetric MRA-based wavelets☆

This paper discusses the construction of highly symmetric compactly supported wavelets for hexagonal data/image and triangle surface multiresolution processing. Recently, hexagonal image processing has attracted attention. Compared with the conventional square lattice, the hexagonal lattice has several advantages, including that it has higher symmetry. It is desirable that the filter banks for hexagonal data also have high symmetry which is pertinent to the symmetric structure of the hexagonal lattice. The high symmetry of filter banks and wavelets not only leads to simpler algorithms and efficient computations, it also has the potential application for the texture segmentation of hexagonal data. While in the field of computer-aided geometric design (CAGD), when the filter banks are used for surface multiresolution processing, it is required that the corresponding decomposition and reconstruction algorithms for regular vertices have high symmetry, which make it possible to design the corresponding multiresolution algorithms for extraordinary vertices. In this paper we study the construction of six-fold axial symmetric biorthogonal filter banks and the associated wavelets, with both the dyadic and -refinements. The constructed filter banks have the desirable symmetry for hexagonal data processing. By associating the outputs (after one-level multiresolution decomposition) appropriately with the nodes of the regular triangular mesh with which the input data is associated (sampled), we represent multiresolution analysis and synthesis algorithms as templates. The six-fold axial symmetric filter banks constructed in this paper result in algorithm templates with desirable symmetry for triangle surface processing.

http://www.cs.umsl.edu/~jiang/webpaper/Bio_wavelet_6sym_dyadic_sqrt3.pdf

Qingtang Jiang

Papers

Biorthogonal wavelets with six-fold axial symmetry for hexagonal data and triangle surface multiresolution processing