In this paper, we introduce the biorthogonal vector-valued wavelets. We prove that, like in the scalar wavelet case, the existence of a pair of biorthogonal compactly supported vector-valued scaling functions guarantees the existence of a pair of biorthogonal compactly supported vector-valued wavelet functions. An algorithm for constructing a pair of biorthogonal compactly supported vector-valued wavelet functions is presented by means of vector-valued multiresolution analysis and matrix theory. The notion of biorthogonal vector-valued wavelet packets is introduced, and their properties are investigated by virtue of time–frequency analysis and algebra theory. Three biorthogonality formulas concerning the wavelet packets are established. Relation to some physical theories such as E -infinity Cantorian space–time theory is also discussed.

The research of a class of biorthogonal compactly supported vector-valued wavelets

In this paper, we introduce vector-valued non-separable higher-dimensional wavelet packets with an arbitrary integer dilation factor. An approach for constructing vector-valued higher-dimensional wavelet packet bases is proposed. Their characteristics are investigated by means of harmonic analysis method, matrix theory and operator theory, and three orthogonality formulas concerning the wavelet packets are presented. Orthogonal decomposition relation formulas of the space L 2 ( R n ) p are derived by designing a series of subspaces of the vector-valued wavelet packets. Moreover, several orthonormal wavelet packet bases of L 2 ( R n ) p are constructed from the wavelet packets. Relation to some physical theories such as E-infinity theory and multifractal theory is also discussed.

Characteristics of a class of vector-valued non-separable higher-dimensional wavelet packet bases

In this paper, we introduce matrix-valued multiresolution analysis and orthogonal matrix-valued wavelets with arbitrary integer dilation factor m. A necessary and sufficient condition on the existence of orthogonal matrix-valued wavelets is derived by virtue of paraunitary vector filter bank theory. An algorithm for constructing compactly supported m-scale orthogonal matrix-valued wavelets is presented. The notion of orthogonal matrix-valued wavelet packets is proposed. Their properties are investigated by means of time–frequency method, operator theory and matrix theory. In particular, it is shown how to construct various orthonormal bases of space L2(R, Cr×r) from these wavelet packets, and the orthogonal decomposition relation is also given.

Construction and properties of orthogonal matrix-valued wavelets and wavelet packets☆

In this paper, the notion of orthogonal vector-valued wavelet packets of the space L 2 ( R s ,  C d ) is introduced. A method for constructing the orthogonal vector-valued wavelet packets is presented. Their properties are investigated by virtue of time–frequency analysis method, matrix theory and finite group theory, and three orthogonality formulas with respect to the wavelet packets are established. Orthogonal decomposition relation formulas of L 2 ( R s ,  C d ) are obtained by constructing a series of subspaces of vector-valued wavelet packets. In particular, it is shown how to construct various orthonormal bases of L 2 ( R s ,  C d ) from these wavelet packets.

Construction and characterizations of orthogonal vector-valued multivariate wavelet packets

Wavelet transforms using matrix-valued wavelets (MVWs) can process the components of vector-valued signals jointly. We construct some novel families of non-trivial orthogonal n×n MVWs for n=2 and 4 having several vanishing moments. Some useful uniqueness and non-existence results for filters with certain lengths and numbers of vanishing moments are proved. The matrix-based method for n=4 is used for the construction of a non-trivial symmetric quaternion wavelet with compact support. This is an important addition to the literature where existing quaternion wavelet designs suffer from some critical problems.

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Matrix-Valued and Quaternion Wavelets

In this paper, we introduce biorthogonal multiple vector-valued wavelets which are wavelets for vector fields. We proved that, like in the scalar and multiwavelet case, the existence of a pair of biorthogonal multiple vector-valued scaling functions guarantees the existence of a pair of biorthogonal multiple vector-valued wavelet functions. Finally, we investigate the construction of a class of compactly supported biorthogonal multiple vector-valued wavelets.

Existence and construction of compactly supported biorthogonal multiple vector-valued wavelets

In this paper, we study minimum-energy frame Ψ = {ψ1, ψ2, … , ψN} with arbitrary integer dilation factor d for L2(R), Ψ correspond to some refinable functions with compact support. A precise existence criterion of Ψ is given in terms of an inequality condition on the Laurent polynomial symbols of the refinable functions. We give a constructive proof that when Ψ does exist, d functions with compact support are sufficient to constitute Ψ, and present a explicit formula of constructing Ψ. Finally, we present the minimum-energy frames decomposition and reconstruction formulas which are similar to those of orthogonal wavelets. Numerical examples are given.

Minimum-energy frames associated with refinable function of arbitrary integer dilation factor

In this paper, first we introduce vector-valued multiresolution analysis with dilation factor α  ⩾ 2 and orthogonal vector-valued wavelet. A necessary and sufficient condition on the existence of orthogonal vector-valued wavelet is derived. Then, for a given L -length compactly supported orthogonal vector-valued wavelet system, by virtue of an s  ×  s orthogonal real matrix M and an s  ×  s symmetry idempotent real matrix H where M ( I s  −  H  +  H  e −i η ) is a unitary matrix for each η  ∈  R , we construct ( L  + 1)-length compactly supported orthogonal vector-valued wavelet system. Our method is of flexibility and easy to carry out. Finally, as an application we give an example.

Construction of a class of compactly supported orthogonal vector-valued wavelets

In this paper, the notion of an m -band generalized multiresolution structure (GMS) of L 2 ( R ) is introduced. We give the definition and the characterization of affine pseudoframes for subspaces. The construction of a GMS of Paley–Wiener subspaces of L 2 ( R ) is investigated. The pyramid decomposition scheme is derived based on such a GMS. As a major new contribution the construction of affine frames for L 2 ( R ) based on a GMS is presented.

Affine pseudoframes for subspaces of L2(R) associated with a generalized multiresolution structure

In this paper, first we introduce trivariate multiresolution analysis and trivariate biorthogonal wavelets. A sufficient condition on the existence of a pair of trivariate biorthogonal scaling functions is derived. Then, the pair of nonseparable or separable trivariate biorthogonal wavelets can be achieved from the pair of trivariate biorthogonal scaling functions.

Zhengxing Cheng

Papers

Existence and construction of compactly supported biorthogonal multiple vector-valued wavelets

Minimum-energy frames associated with refinable function of arbitrary integer dilation factor

Construction of a class of compactly supported orthogonal vector-valued wavelets

Affine pseudoframes for subspaces of L2(R) associated with a generalized multiresolution structure

Construction of trivariate biorthogonal compactly supported wavelets