Multiresolution analysis for surfaces of arbitrary topological type
Multiresolution analysis for surfaces of arbitrary topological type
Multiresolution analysis for surfaces of arbitrary topological type
ACM Transactions on Graphics (TOG)
Approximation properties of multivariate wavelets
Mathematics of Computation
Analysis and construction of optimal multivariate biorthogonal wavelets with compact support
SIAM Journal on Mathematical Analysis
Proceedings of the 27th annual conference on Computer graphics and interactive techniques
Subdivision Methods for Geometric Design: A Constructive Approach
Subdivision Methods for Geometric Design: A Constructive Approach
Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets
SIAM Journal on Matrix Analysis and Applications
Quincunx fundamental refinable functions and Quincunx biorthogonal wavelets
Mathematics of Computation
Triangular √3-subdivision schemes: the regular case
Journal of Computational and Applied Mathematics
Composite primal/dual √3-subdivision schemes
Computer Aided Geometric Design
√2 Subdivision for quadrilateral meshes
The Visual Computer: International Journal of Computer Graphics
Biorthogonal loop-subdivision wavelets
Computing - Geometric modelling dagstuhl 2002
Efficient wavelet construction with Catmull–Clark subdivision
The Visual Computer: International Journal of Computer Graphics
Designing composite triangular subdivision schemes
Computer Aided Geometric Design - Special issue: Geometric modelling and differential geometry
Nonseparable two- and three-dimensional wavelets
IEEE Transactions on Signal Processing
Multiwavelet bases with extra approximation properties
IEEE Transactions on Signal Processing
√3-Subdivision-Based Biorthogonal Wavelets
IEEE Transactions on Visualization and Computer Graphics
Computer Aided Geometric Design
Computer Aided Geometric Design
Bi-frames with 4-fold axial symmetry for quadrilateral surface multiresolution processing
Journal of Computational and Applied Mathematics
Wavelet bi-frames with uniform symmetry for curve multiresolution processing
Journal of Computational and Applied Mathematics
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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.