Rank <i>M</i> Wavelets with <i>N</i> Vanishing Moments

Heller, P.N.

doi:10.1137/s0895479893245486

Cited by 82 publications

(58 citation statements)

References 11 publications

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“…It suggests the possibility that from a given scaling filter, one may construct a full wavelet matrix with an asymptotically smaller DWT complexity. Several approaches [15,20,30] have been presented to construct the m − 1 wavelet filters from the scaling filter. In this paper we will give an optimal solution for the problem of designing wavelet filters, where the optimality is measured in the sense of DWT computational complexity.…”

Section: A Fast Wavelet Transform Based On Factorizationmentioning

confidence: 99%

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An Algebraic Structure of Orthogonal Wavelet Space

Jiang

Wells

2000

Applied and Computational Harmonic Analysis

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In this paper we study the algebraic structure of the space of compactly supported orthonormal wavelets over real numbers. Based on the parameterization of wavelet space, one can define a parameter mapping from the wavelet space of rank 2 (or 2-band, scale factor of 2) and genus g to the (g − 1) dimensional real torus (the products of unit circles). By the uniqueness and exactness of factorization, this mapping is well defined and one-to-one. Thus we can equip the rank 2 orthogonal wavelet space with an algebraic structure of the torus. Because of the degenerate phenomenon of the paraunitary matrix, the parameterization map is not onto. However, there exists an onto mapping from the torus to the closure of the wavelet space. And with such mapping, a more complete parameterization is obtained. By utilizing the factorization theory, we present a fast implementation of discrete wavelet transform (DWT). In general, the computational complexity of a rank m orthogonal DWT is O(m 2 g). In this paper we start with a given scaling filter and construct additional (m − 1) wavelet filters so that the DWT can be implemented in O(mg). With a fixed scaling filter, the approximation order, the orthogonality, and the smoothness remain unchanged; thus our fast DWT implementation is quite general.

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Section: A Fast Wavelet Transform Based On Factorizationmentioning

confidence: 99%

“…A more general construction method is described in [20]. The next theorem is Theorem 4.1 of [15] (see also [27]). …”

Section: A Fast Wavelet Transform Based On Factorizationmentioning

confidence: 99%

An Algebraic Structure of Orthogonal Wavelet Space

Jiang

Wells

2000

Applied and Computational Harmonic Analysis

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“…where N is as in (6.4) [11][12][13]16]. A most important fact is that the spectral radius of T P is the same as that of the restriction of the operator to certain invariant finite-dimensional subspaces.…”

Section: Critical and Sobolev Exponents Of M-band Waveletsmentioning

confidence: 99%

“…The interest in MRA structures with dilation factor greater than 2 [4,14,15,23,30] is motivated by the theory of M-band channel subband coding schemes [3,14,15,27] and by the attempt to obtain sharper time-frequency localization and greater flexibility in the construction of wavelets. The design of the filters is quite different from the classical case (M = 2): it is, in general, more difficult, and the wavelets are no longer determined, in an essentially unique manner, by a pair of biorthogonal MRAs.…”

Section: Introductionmentioning

confidence: 99%

M-Band Burt–Adelson Biorthogonal Wavelets

Maggioni¹

2000

Applied and Computational Harmonic Analysis

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For every integer M > 2 we introduce a new family of biorthogonal MRAs with dilation factor M, generated by symmetric scaling functions with small support. This construction generalizes Burt-Adelson biorthogonal 2-band wavelets. For M ∈ {3, 4} we are able to find simple explicit expressions for two different families of wavelets associated with these MRAs: one with better localization and the other with interesting symmetry-antisymmetry properties. We study the regularity of our scaling functions by determining their Sobolev exponent, for every value of the parameter and every M. We also study the critical exponent when M = 3.

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“…The regularity of φ has been studied widely; see for instance [CL], [BDS], [D], [HW1], [HW2], [So], [S] and [WL]. In general, to study the regularity of φ we need to consider the symbol (2) first.…”

Section: Introductionmentioning

confidence: 99%