On the construction of multivariate (pre)wavelets

Journal of Fourier Analysis and Applications, 1999

An abstract formulation of generalized multiresolution analyses is presented, and those GMRAs that come from multiwavelets are characterized. As an application of this abstract formulation, a constructive procedure is developed, which produces all wavelet sets in ℝnrelative to an integral expansive matrix.

Journal of Fourier Analysis and Applications, 2003

A frame multiresolution (FMRA for short) orthogonal wavelet is a single-function orthogonal wavelet such that the associated scaling space V 0 admits a normalized tight frame (under translations). In this paper, we prove that for any expansive matrix A with integer entries, there exist A-dilation FMRA orthogonal wavelets. FMRA orthogonal wavelets for some other expansive matrix with non integer entries are also discussed.

IEEE Transactions on Pattern Analysis and Machine Intelligence, 1989

Multiresolution representations are effective for analyzing the information content of images. The properties of the operator which approximates a signal at a given resolution were studied. It is shown that the difference of information between the approximation of a signal at the resolutions 2j+1 and 2j (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L2(Rn), the vector space of measurable, square-integrable n-dimensional functions. In L2(R), a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function ψ(x). This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. Wavelet representation lies between the spatial and Fourier domains. For images, the wavelet representation differentiates several spatial orientations. The application of this representation to data compression in image coding, texture discrimination and fractal analysis is discussed

Journal of Fourier Analysis and Applications, 2011

Spectral representations of the dilation and translation operators on L 2 (R) are built through appropriate bases. Orthonormal wavelets and multiresolution analysis are then described in terms of rigid operator-valued functions defined on the functional spectral spaces. The approach is useful for computational purposes.

Mathematics of Computation, 1998

Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask. We connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators. Finally, we demonstrate that a refinable function in W k−1 1 (R s ) provides approximation order k.

A sequence of increasing translation invariant subspaces can be defined by the Haar-system (or generally by wavelets). The orthogonal projection to the subspaces generates a decomposition (multiresolution) of a signal. Regarding the rate of convergence and the number of operations, this kind of decomposition is much more favorable then the conventional Fourier expansion.

Axioms, 2023

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Signal Processing, 1993

In this paper, a new multiresolution wavelet representation for two-dimensional signals is described. This wavelet representation is based on a nonrectangular decomposition of the frequency domain. The decomposition can be implemented by a digital filter bank. The application of the new representation to the coding of quincunx and rectangularly sampled images is considered and simulation examples are presented. Zusammenfassuag. In dieser Arbeit wird eine neuartige Mehrfachaufiosungs-Waveletdarstellung fur zweidimensionale Signale beschrieben. Diese Wavelet-Darstellung beruht auf einer nicht-rechteckigen Zerlegung des Frequenzbereichs. Die Zerlegung kann mittels einer digitalen Filterbank implementiert werden. Wir diskutieren die Anwendung der neuen Darstellung auf die Codierung von quincunx-and rechteckig abgetasteten Bildern and beschreiben Simulationsbeispiele. Resume. Une representation multi-resolution par ondelettes originate est decrite dans cet article. Cette representation par ondelettes est baser sur une decomposition non rectangulaire du domaine frequentiel. La decomposition pent etre implantee par on bane de filtres digitaux. Cette representation est appliquee au codage d'images echantillonnees rectangulairement et en quinconce et des exemples de simulation sont presenter. Keywords. Wavelets ; subband decomposition ; image coding. 1. Introduction Multiresolution representation of signals is used in many interesting fields including image analysis, video and image coding, and geophysics. In image coding [ 16], the original image is decomposed into several subimages at low resolutions and coding

Transactions of the American Mathematical Society, 1994

A multiresolution approximation (Km)m€Z of L2(R) is of multiplicity r > 0 if there are r functions <¡>x, ... , <j>r whose translates form a Riesz basis for V$. In the general theory we derive necessary and sufficient conditions for the translates of <j>x, ... , <j>r, y/x, ... , y/r to form a Riesz basis for V\. The resulting reconstruction and decomposition sequences lead to the construction of dual bases for V0 and its orthogonal complement W0 in Vx. The general theory is applied in the construction of spline wavelets with multiple knots. Algorithms for the construction of these wavelets for some special cases are given.

2019

In this paper we provide a complete and unifying characterization of compactly supported univariate scalar orthogonal wavelets and vector-valued or matrix-valued orthogonal multi-wavelets. This characterization is based on classical results from system theory and basic linear algebra. In particular, we show that the corresponding wavelet and multi-wavelet masks are identified with a transfer function F (z) = A+Bz(I −Dz) C, z ∈ D = {z ∈ C : |z| &lt; 1}, of a conservative linear system. The complex matrices A, B, C, D define a block circulant unitary matrix. Our results show that there are no intrinsic differences between the elegant wavelet construction by Daubechies or any other construction of vector-valued or matrix-valued multiwavelets. The structure of the unitary matrix defined by A, B, C, D allows us to parametrize in a systematic way all classes of possible wavelet and multi-wavelet masks together with the masks of the corresponding refinable functions.