Approximation properties of multivariate wavelets

Jia, Rong-Qing

doi:10.1090/s0025-5718-98-00925-9

Cited by 131 publications

(96 citation statements)

References 18 publications

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“…Under some condition, sum rule order is equivalent to the approximation order of φ, see [48]. Let σ 1 , σ 2 be the eigenvalues of the …”

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

confidence: 99%

Orthogonal and Biorthogonal FIR Hexagonal Filter Banks With Sixfold Symmetry

Jiang

2008

IEEE Trans. Signal Process.

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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.

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“…Under some condition, sum rule order is equivalent to the approximation order of φ, see [48]. Let σ 1 , σ 2 be the eigenvalues of the …”

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

confidence: 99%

Orthogonal and Biorthogonal FIR Hexagonal Filter Banks With Sixfold Symmetry

Jiang

2008

IEEE Trans. Signal Process.

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“…The partial derivative of a differentiable function f with respect to the jth coordinate is denoted , and Π r be the space of all polynomials of total degree at most r. We say that a mask a satisfies the sum rules of order r + 1 ( [18]) with respect to the lattice 2Z…”

Section: Wavelets Based On the Loop Schemementioning

confidence: 99%

Wavelets from the Loop Scheme

Han

Shen

2005

J Fourier Anal Appl

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Abstract. A new wavelet-based geometric mesh compression algorithm was developed recently in the area of computer graphics by Khodakovsky, Schröder, and Sweldens in their interesting paper [23]. The new wavelets used in [23] were designed from the Loop scheme by using ideas and methods of [26,27], where orthogonal wavelets with exponential decay and pre-wavelets with compact support were constructed. The wavelets have the same smoothness order as that of the basis function of the Loop scheme around the regular vertices which has a continuous second derivative; the wavelets also have smaller supports than those wavelets obtained by constructions in [26,27] or any other compactly supported biorthogonal wavelets derived from the Loop scheme (e.g. [11,12]). Hence, the wavelets used in [23] have a good time frequency localization. This leads to a very efficient geometric mesh compression algorithm as proposed in [23]. As a result, the algorithm in [23] outperforms several available geometric mesh compression schemes used in the area of computer graphics. However, it remains open whether the shifts and dilations of the wavelets form a Riesz basis of L 2 (R 2 ). Riesz property plays an important role in any waveletbased compression algorithm and is critical for the stability of any wavelet-based numerical algorithms. We confirm here that the shifts and dilations of the wavelets used in [23] for the regular mesh, as expected, do indeed form a Riesz basis of L 2 (R 2 ) by applying the more general theory established in this paper.

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“…Now suppose φ is the normalized solution of the refinement equation (1.2) with a mask a and a dilation matrix M . It was proved by Jia in [17] that if the shifts of φ are stable, then S(φ) provides approximation order k if and only if the mask a satisfies the sum rules of order k. Note that a fundamental function has stable shifts. Thus, in particular, if φ is a fundamental refinable function with a mask a and a dilation matrix M , then S(φ) provides approximation order k if and only if a satisfies the sum rules of order k with the dilation matrix M .…”

Section: Approximation Order and Smoothness Ordermentioning

confidence: 99%

“…where the finite dimensional space W is the minimal invariant subspace of T a,M generated by ∆ k j δ, j = 1, · · · , s. For discussion on subdivision operators and transition operators, the reader is referred to [11,17,20].…”

Section: Theorem 22 Let a Be A Quincunx Interpolatory Mask Supportementioning

confidence: 99%

“…The concept of vanishing moments of a quincunx biorthogonal wavelet plays an important role in applications. See [2,4,6,17] and references therein for discussions on vanishing moments and their relation to sum rules. Given a primal mask, it is desirable to construct a dual mask with high order of sum rules and relatively small support.…”

Section: Quincunx Biorthogonal Waveletsmentioning

confidence: 99%

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Quincunx fundamental refinable functions and quincunx biorthogonal wavelets

Han

Jia

2000

Math. Comp.

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Abstract. We analyze the approximation and smoothness properties of quincunx fundamental refinable functions. In particular, we provide a general way for the construction of quincunx interpolatory refinement masks associated with the quincunx lattice in R 2 . Their corresponding quincunx fundamental refinable functions attain the optimal approximation order and smoothness order. In addition, these examples are minimally supported with symmetry. For two special families of such quincunx interpolatory masks, we prove that their symbols are nonnegative. Finally, a general way of constructing quincunx biorthogonal wavelets is presented. Several examples of quincunx interpolatory masks and quincunx biorthogonal wavelets are explicitly computed.

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Approximation properties of multivariate wavelets

Cited by 131 publications

References 18 publications

Orthogonal and Biorthogonal FIR Hexagonal Filter Banks With Sixfold Symmetry

Orthogonal and Biorthogonal FIR Hexagonal Filter Banks With Sixfold Symmetry

Wavelets from the Loop Scheme

Quincunx fundamental refinable functions and quincunx biorthogonal wavelets

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