Triangular <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msqrt><mml:mn>7</mml:mn></mml:msqrt></mml:math> and quadrilateral <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"><mml:msqrt><mml:mn>5</mml:mn></mml:msqrt></mml:math> subdivision schemes: Regular case

Chui, Charles K.; Jiang, Qingtang; Ndao, Rokhaya N.

doi:10.1016/j.jmaa.2007.05.070

Journal of Mathematical Analysis and Applications

2008

DOI: 10.1016/j.jmaa.2007.05.070

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Triangular 7 and quadrilateral 5 subdivision schemes: Regular case

Charles K. Chui

Qingtang Jiang

Rokhaya N. Ndao

Abstract: This paper is devoted to the study of triangular √ 7 and quadrilateral √ 5 surface subdivisions. Both approximation and interpolatory subdivision schemes are considered, with illustrative examples of both scalar-valued and matrix-valued √ 7 and √ 5 subdivision masks that satisfy the sum rule of sufficiently high orders. In particular, "optimal" Sobolev smoothness is determined and Hölder smoothness estimates are presented.

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Cited by 7 publications

References 29 publications

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

Jiang

2009

Adv Comput Math

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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 √ 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.

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

Jiang

2009

Adv Comput Math

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Hierarchical grid conversion

Mahdavi-Amiri

Harrison

Samavati

2016

Computer-Aided Design

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Quasi-interpolating bivariate dual 2-subdivision using 1D stencils

Fang

Han²,

Shen³

2022

Computer Aided Geometric Design

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Triangular 7 and quadrilateral 5 subdivision schemes: Regular case

Cited by 7 publications

References 29 publications

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

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

Hierarchical grid conversion

Quasi-interpolating bivariate dual 2-subdivision using 1D stencils

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