Triangular 3-subdivision schemes: the regular case

Applied Numerical Mathematics

2012

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Nonlinear diffusion filtering and wavelet/frame shrinkage are two popular methods for signal and image denoising. The relationship between these two methods has been studied recently. In this paper we investigate the correspondence between frame shrinkage and nonlinear diffusion.We show that the frame shrinkage of Ron-Shen's continuous-linear-spline-based tight frame is associated with a fourth-order nonlinear diffusion equation. We derive high-order nonlinear diffusion equations associated with general tight frame shrinkages. These high-order nonlinear diffusion equations are different from the high-order diffusion equations studied in the literature. We also construct two sets of tight frame filter banks which result in the sixth-and eighth-order nonlinear diffusion equations.The correspondence between frame shrinkage and diffusion filtering is useful to design diffusion-inspired shrinkage functions with competitive performance. On the other hand, the study of such a correspondence leads to a new type of diffusion equations and helps to design frame-inspired diffusivity functions. The denoising results with diffusion-inspired shrinkages provided in this paper are promising.

Section: Sixth-order Nonlinear Diffusion Equationmentioning

confidence: 99%

Correspondence between frame shrinkage and high-order nonlinear diffusion

Applied Numerical Mathematics

2012

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“…See [49], [50] for the details and refer to [24] for algorithms and Matlab routines to nd the Sobolev smoothness order. …”

Section: Which Implies That (25) Holds If and Only If V (ω)mentioning

confidence: 99%

“…Example 2: Let {p, q (1) , · · · , q (6) } be the orthogonal lter bank with 6-fold rotational symmetry given by (24) for n = 1: (28) …”

Section: Which Implies That (25) Holds If and Only If V (ω)mentioning

confidence: 99%

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Orthogonal and Biorthogonal FIR Hexagonal Filter Banks With Sixfold Symmetry

IEEE Trans. Signal Process.

2008

Recently hexagonal image processing has attracted attention. The hexagonal lattice has several advantages in comparison with the rectangular lattice, the conventionally used lattice for image sampling and processing. For example, a hexagonal lattice needs fewer sampling points; it has better consistent connectivity; it has higher symmetry; its structure is plausible to human vision systems. The multiresolution analysis method has been used for hexagonal image processing. Since the hexagonal lattice has high degree of symmetry, it is desirable that the hexagonal lter banks designed for multiresolution hexagonal image processing also have high order of symmetry which is pertinent to the symmetry structure of the hexagonal lattice. The orthogonal or prefect reconstruction (PR) hexagonal lter banks which are available in the literature have only 3-fold symmetry.In this paper we investigate the construction of orthogonal and PR FIR hexagonal lter banks with 6-fold symmetry. We obtain block structures of 7-size re nement (7-channel 2-D) orthogonal and PR FIR hexagonal lter banks with 6-fold rotational symmetry.√ 7-re nement orthogonal and biorthogonal wavelets based on these block structures are constructed. In this paper, we also consider FIR hexagonal lter banks with axial (line) symmetry, and we present a block structure of FIR hexagonal lter banks with pseudo 6-fold axial symmetry.Index TermsHexagonal lattice, hexagonal image, lter bank with 6-fold symmetry, orthogonal hexagonal lter bank, biorthogonal hexagonal lter bank, √ 7-re nement wavelet, 7-size re nement multiresolution decomposition/reconstruction.

“…For the 1-to-4 split rule, the dilation matrix to be selected is simply 2I 2 , both for triangular and quadrilateral meshes. Other topological rules of interest include the √ 3 [22,23,20,28,21,6] and the √ 2 split [40,41,12,14,24] rules, with dilation matrices given, for example, by…”

Section: Figure 2: Subdivision Templates Of the Catmull-clark Scheme mentioning

confidence: 99%

Matrix-valued subdivision schemes for generating surfaces with extraordinary vertices

Chui

Computer Aided Geometric Design

2006

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Subdivision templates of numerical values are replaced by templates of matrices in this paper to allow the introduction of shape control parameters for the feasibility of achieving desirable geometric shapes at those points on the subdivision surfaces that correspond to extraordinary control vertices. Formulation of the matrix-valued subdivision surface is derived. Based on refinable bivariate spline function vectors for matrix-valued subdivisions, the notion of characteristic map introduced by Reif is extended from (scalar) surface subdivisions to matrix-valued subdivisions. The C 1 -and C k -continuity of Reif and Prautzsch for matrix-valued subdivisions are discussed. To illustrate the general theory, the smoothness of matrix-valued triangular subdivision schemes for extraordinary vertices with valences 3 and 4 is analyzed. The issue of effective choices of the shape control parameters will also be discussed in this paper.