On the construction of multivariate (pre)wavelets (original) (raw)

On multiresolution analysis (MRA) wavelets in ℝ n

The Journal of Fourier Analysis and Applications, 2000

We prove that for any expansive n x n integral matrix A with l det A [ = 2, there exist A-dilation minimally supported frequency (MSF) wavelets that are associated with a multiresolution analysis (MRA). The condition I det A I = 2 was known to be necessary, and we prove that it is sufficient. A wavelet set is the support set of the Fourier transform of an MSF wavelet. We give some concrete examples of MRA wavelet sets in the plane. The same technique of proof is also applied to yield an existence result for A-dilation MRA subspace wavelets. An orthonormal wavelet for a dilation factor a > 0 in R is a single function ~p E L2(R) with the property that {a~r n, lEZ} is an orthonormal basis for L2(R). The proof of the existence of wavelets for any dilation factor a > 1 can be found in [6]. Similarly, one can consider wavelets in R n. If A is a real expansive matrix (equivalently, all the eigenvalues of A are required to have absolute value greater than 1), an A-dilation wavelet is a single function ~p E L2(R n) (product Lebesgue measure) with the property that {[detAl~r~(Amt-k) : m EZ, k EZ n} is an orthonormal basis for L2(Rn). In the article [7], Dai, Larson and Speegle proved the existence of wavelets for any expansive dilation matrix A. This was surprising since prior to this, several researchers had suspected that single function wavelets did not exist for A = 21 in the case n > 1. The method used in [7] was the construction of special wavelets of the form 1 ff~-I (~XE) (*) Math Subject Classifications. 42C15, 46E15.

Irregular multiresolution analysis and associated wavelet

Arabian Journal of Mathematics, 2014

We introduce two generalizations, the first of which generalizes the concept of multiresolution analysis. We define the irregular generalized multiresolution analysis (IGMRA). This structure is defined taking translations on sets that are not necessarily regular lattices, for which certain density requirements are required, and without using dilations, also allows each subspace of IGMRA to be generated by outer frames of translations of different functions. The second generalization concerns the concept of association of wavelets to these new structures. We take frames of translations of a countable set of functions, which we called generalized wavelets, and define the concept of association of these generalized wavelets to those previously defined IGMRA. In the next stage, we prove two existence theorems. In the first theorem, we prove existence of IGMRA, and in the second existence of generalized wavelets associated with it. In the latter, we show that we are able to associate frames of translations with optimal localization properties, to IGMRA. In the last section of this paper, concrete examples of these structures are presented for L 2 (R) and for L 2 (R 2). Mathematics Subject Classification 42C40 • 42C30 1 Introduction From the classic concept of multiresolution analysis (MRA), introduced and further developed by Meyer [27,28], and Mallat [24,25], which provides a systematic way to construct orthonormal wavelet bases of L 2 (R), research in this area has been extended in various ways. These concepts are generalized to L 2 (R d) [14],

BLaC-wavelets: a multiresolution analysis with non-nested spaces

Proceedings of Seventh Annual IEEE Visualization '96, 1996

In the last ve y ears, there has been numerous applications of wavelets and m ultiresolution analysis in many elds of computer graphics as di erent as geometric modelling, volume visualization or illumination modelling. Classical multiresolution analysis is based on the k n o wledge of a nested set of functional spaces in which t he s u ccessive a p proximations of a given function converge to t hat f u nction, and can be e ciently computed. This paper rst proposes a theoretical framework which e n ables multiresolution analysis even if the f u nctional spaces are not nested, as long a s t hey still have t he property that t he s u ccessive a p proximations converge to t he g i v en function. Based on this concept we n ally introduce a new multiresolution analysis with exact reconstruction for large data sets de ned on uniform grids. We construct a one-parameter family of multiresolution analyses which is a blending o f H a a r and linear multiresolution.

Wavelets with Frame Multiresolution Analysis

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.

A Theory for Multiresolution Signal Decomposition: The Wavelet Representation

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

Spectral Models for Orthonormal Wavelets and Multiresolution Analysis of L 2(ℝ)

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.

Approximation properties of multivariate wavelets

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.