$$\sqrt 5 $$ -subdivision

2009

Adv Comput Math

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.

“…Thus, I 00 (ω) defined above satisfies (11) and (12). The next proposition presents a characterization of the 4-fold axial symmetry of a filter bank in terms of its polyphase matrix.…”

Section: Biorthogonal Fir Filter Banks and Wavelets With 4-fold Axialmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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

2009

Adv Comput Math

“…The procedure outlined in this paper can be applied to other topological rules such as √ 7 and √ 5 topological rules studied in [25] and [15], respectively. Furthermore, the extension of the smoothnesspreservation results in [27,28] for the extraordinary vertices from the scalar setting to the matrix-valued subdivisions are studied in our work [4].…”

Section: Introductionmentioning

confidence: 97%

Matrix-valued symmetric templates for interpolatory surface subdivisions

Chui

Applied and Computational Harmonic Analysis

2005

The objective of this paper is to introduce a general procedure for deriving interpolatory surface subdivision schemes with "symmetric subdivision templates" (SSTs) for regular vertices. While the precise definition of "symmetry" will be clarified in the paper, the property of SSTs is instrumental to facilitate application of the standard procedure for finding symmetric weights for taking weighted averages to accommodate extraordinary (or irregular) vertices in surface subdivisions, a topic to be studied in a continuation paper. By allowing the use of matrices as weights, the SSTs introduced in this paper may be constructed to overcome the size barrier limited to scalar-valued interpolatory subdivision templates, and thus avoiding the unnecessary surface oscillation artifacts. On the other hand, while the old vertices in a (scalar) interpolatory subdivision scheme do not require a subdivision template, we will see that this is not the case for the matrix-valued setting. Here, we employ the same definition of interpolation subdivisions as in the usual scalar consideration, simply by requiring the old vertices to be stationary in the definition of matrix-valued interpolatory subdivisions. Hence, there would be another complication when the templates are extended to accommodate extraordinary vertices if the template sizes are not small. In this paper, we show that even for C 2 interpolatory subdivisions, only one "ring" is sufficient in general, for both old and new vertices. For example, for 1-to-4 split C 2 interpolatory surface subdivisions, we obtain matrix-valued symmetric interpolatory subdivision templates (SISTs) for both triangular and quadrilateral meshes with sizes that agree with those of the Loop and Catmull-Clark schemes, respectively. Matrix-valued SISTs of similar sizes are also constructed for C 2 interpolatory √ 3 and √ 2 subdivision schemes in this paper. In addition to small template sizes, an obvious feature of matrix-valued weights is the flexibility for introducing shape-control parameters. Another significance is that, in contrast to the usual scalar setting, matrix-valued SISTs can be formulated in terms of the coefficient sequence of some vector refinement equation of interpolating bivariate C 2 splines with small support. For example, by modifying the spline function vectors introduced in our previous work [C.K. Chui, Q.T. Jiang, Surface subdivision schemes generated by refinable bivariate spline function vectors, Appl. Comput. Harmon. Anal. 15 (2003) 147-162; C.K. Chui, Q.T. Jiang, Refinable bivariate quartic and quintic C 2 -splines for quadrilateral subdivisions, Preprint, 2004], C 2 symmetric interpolatory subdivision schemes associated with refinement equations of C 2 cubic and quartic splines on the 6-directional and 4-directional meshes, respectively, are also constructed in this paper.

“…The √ 5 rule was first proposed in [28] as a hierarchical sampling scheme over a regular mesh because of the unique property that the five vertices in each group that corresponds to a quadrilateral of the coarser mesh, consisting of one old vertex and four new vertices, have different x and y coordinates. A C 1 approximation √ 5-subdivision scheme was constructed in [11]. In our present paper, we will construct both interpolatory and approximation √ 5-subdivision schemes to achieve both C 1 and C 2 smoothness.…”

Section: Introductionmentioning

confidence: 99%

“…The √ 3 topological rule is designed for the triangular mesh, while the √ 2 rule for the quadrilateral mesh. In this paper, we will study, in some depth, the so-called √ 7 and √ 5 surface subdivision schemes, studied in [11,22].…”

Section: Introductionmentioning

confidence: 99%

Triangular 7 and quadrilateral 5 subdivision schemes: Regular case

Chui

Journal of Mathematical Analysis and Applications

Ndao

2008

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.