Efficient wavelet construction with Catmull–Clark subdivision

Wang, Huawei; Qin, Kaihuai; Tang, Kai

doi:10.1007/s00371-006-0074-7

Cited by 30 publications

(29 citation statements)

References 26 publications

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“…For testing the stability of wavelet transform, we made a noise-filtering experiment, which is often used to examine the stability of approximate subdivision wavelets [10][11][12][13][14][15]. We first perturb all vertices of the mesh at highest resolution with white noise.…”

Section: Resultsmentioning

confidence: 99%

“…Li et al [11] proposed unlifted Loop subdivision wavelet by optimizing free parameters in the extended subdivisions. Wang et al [12] developed an effective wavelet construction based on general CatmullClark subdivisions and the resulted wavelets have better fitting quality than the previous Catmull-Clark like subdivision wavelets. They also constructed several new biorthogonal wavelets based on 3 subdivision over triangular meshes, and approximate and interpolatory 2 subdivision over quadrilateral meshes [13][14][15].…”

Section: Related Workmentioning

confidence: 99%

“…With this assumption, the mutual inner product of wavelets and scaling functions is defined as the sum of multiplications of corresponding coefficients (geometry coordinates of points) at finer resolution, and calculated directly from the subdivision template. It is maybe not accurate, but works well in many works [10][11][12][13][14][15]. Figure 5 shows the discrete mask of extraordinary…”

Section: Extraordinary and Boundary Points Treatmentmentioning

confidence: 99%

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Optimal Interpolatory Wavelets Transform for Multiresolution Triangular Meshes

Zhao¹,

Sun²

2011

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In recent years, several matrix-valued subdivisions have been proposed for triangular meshes. The matrix-valued subdivisions can simulate and extend the traditional scalar-valued subdivision, such as loop and 3 subdivision. In this paper, we study how to construct the matrix-valued subdivision wavelets, and propose the novel biorthogonal wavelet based on matrix-valued subdivisions on multiresolution triangular meshes. The new wavelets transform not only inherits the advantages of subdivision, but also offers more resolutions of complex models. Based on the matrix-valued wavelets proposed, we further optimize the wavelets transform for interactive modeling and visualization applications, and develop the efficient interpolatory loop subdivision wavelets transform. The transform simplifies the computation, and reduces the memory usage of matrix-valued wavelets transform. Our experiments showed that the novel wavelets transform is sufficiently stable, and performs well for noise reduction and fitting quality especially for multiresolution semi-regular triangular meshes.

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Section: Resultsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Extraordinary and Boundary Points Treatmentmentioning

confidence: 99%

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Optimal Interpolatory Wavelets Transform for Multiresolution Triangular Meshes

Zhao¹,

Sun²

2011

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“…Then an implicit interpolate can be obtained by taking a convex combination of contour representation [16], or implying the distance function [12,16]). Wang et al, [17] used Catmull-Clark subdivision for biorthogonal wavelet construction based on lifting scheme. Loop and Schafefer, [18] approximated Catmull-Clark subdivision surfaces by minimal set of bicubic patch.…”

Section: Previous Workmentioning

confidence: 99%

Reconstruction of Branching Surface and Its Smoothness by Reversible Catmull-Clark Subdivision

Jha

2009

Lecture Notes in Computer Science

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Abstract. In the current research a new algorithm has been developed to get surface from the contours having branches and a final smooth surface is obtained by reversible Catmull-Clark Subdivision. In branching, a particular layer has more than one contour, corresponds with the contour at the adjacent layer. The layer having more than one contour is converted into a 3D composite curve by inserting points between the layers. The points are inserted in such a way that the center of contours should merged to the center of the contours at the adjacent layer. This process is repeated for all layers having branching problems. In the next step, 3D composite curves are converted into different polyhedrons by the help of the contours at adjacent layers. Number of control points at different layer for contours and 3D curves may not be the same, in this case a special polyhedron construction technique has been developed. The polyhedrons are subdivided using reversible Catmull-Clark subdivision to give a smooth surface.

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“…In addition, we provide 4-fold symmetric biorthogonal FIR filter banks and construct the associated wavelets, with both the dyadic and √ 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][30][31][32][33][34][35][36][37][38][39]2004) and Wang et al :874-884, 2006; IEEE Trans Vis Comput Graph 13 (5): [914][915][916][917][918][919][920][921][922][923][924][925]2007). Our method can provide the filter banks corresponding to the multiresolution algorithms in Wang et al : [874][875][876][877][878][879][880][881][882][883][884]2006) for dyadic multiresolution quad surface processing.…”

mentioning

confidence: 99%

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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Efficient wavelet construction with Catmull–Clark subdivision

Cited by 30 publications

References 26 publications

Optimal Interpolatory Wavelets Transform for Multiresolution Triangular Meshes

Optimal Interpolatory Wavelets Transform for Multiresolution Triangular Meshes

Reconstruction of Branching Surface and Its Smoothness by Reversible Catmull-Clark Subdivision

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

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