Some analytical considerations on two-scale relations

Journal of Fourier Analysis and Applications, 2009

Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space H that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function m which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space H is L 2 (R n ), the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function m satisfying a consistency condition which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function m.

2016

In the present paper, multiresolution analysis arising from Coalescence Hiddenvariable Fractal Interpolation Functions (CHFIFs) is accomplished. The availability of a larger set of free variables and constrained variables with CHFIF in multiresolution analysis based on CHFIFs provides more control in reconstruction of functions in L 2 (R) than that provided by multiresolution analysis based only on Affine Fractal Interpolation Functions (AFIFs). In our approach, the vector space of CHFIFs is introduced, its dimension is determined and Riesz bases of vector subspaces V k , k ∈ Z, consisting of certain CHFIFs in L 2 (R) C 0 (R) are constructed. As a special case, for the vector space of CHFIFs of dimension 4, orthogonal bases for the vector subspaces V k , k ∈ Z, are explicitly constructed and, using these bases, compactly supported continuous orthonormal wavelets are generated.

IEEE Transactions on Information Theory, 1992

Four-coefficient dilation equations are examined and resuits converse to a theorem of Daubechies-Lagarias are given. These results complete the characterization of those four-coefficient dilation equations having a continuous solution* Index Terms-Dilation equation, Holder continuity, joint spectral m N + 1 We now use this lemma to estimate the integral in (2.1). Note that, by (1.3) and Stirling's formula, there is a positive constant C such that radius, multiresolution analysis, scaling function, wavelet.

Applied and Computational Harmonic Analysis, 1996

The objective of this paper is to establish certain necessary and sufficient conditions for a multi-scaling function φ := (φ 1 ,. .. , φ r) T to have polynomial reproduction (p. r.) of order m in terms of the eigenvalues and their corresponding eigenvectors of two finite matrices.

Journal of Geometric Analysis, 2000

2021

In this paper, we show how to construct an orthonormal basis from Riesz basis by assuming that the fractional translates of a single function in the core subspace of the fractional multiresolution analysis form a Riesz basis instead of an orthonormal basis. In the definition of fractional multiresolution analysis, we show that the intersection triviality condition follows from the other conditions. Furthermore, we show that the union density condition also follows under the assumption that the fractional Fourier transform of the scaling function is continuous at $0$. At the culmination, we provide the complete characterization of the scaling functions associated with fractional multiresolutrion analysis.

SIAM Journal on Matrix Analysis and Applications, 1994

A dilation equation is a functional equation of the form f (t) = N k=0 c k f (2t − k), and any nonzero solution of such an equation is called a scaling function. Dilation equations play an important role in several fields, including interpolating subdivision schemes and wavelet theory. This paper obtains sharp bounds for the Hölder exponent of continuity of any continuous, compactly supported scaling function in terms of the joint spectral radius of two matrices determined by the coefficients {c 0 ,. .. , c N }. The arguments lead directly to a characterization of all dilation equations that have continuous, compactly supported solutions.

Advances in Computational Mathematics, 2008

In this paper we investigate Isotropic Multiresolution Analysis (IMRA), isotropic refinable functions, and wavelets. The main results are the characterization of IMRAs in terms of the Lax-Wiener Theorem, and the characterization of isotropic refinable functions in terms of the support of their Fourier transform. As an immediate consequence of these results, there are no compactly supported (in the space domain) isotropic refinable functions in many dimensions. Next we study the approximation properties of IMRAs. Finally, we discuss the application of IMRA wavelets to 2D and 3D-texture segmentation in natural and biomedical images.

2021

In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. This gap was filled by Gabardo and Nashed [11] by establishing a constructive algorithm based on the theory of spectral pairs for constructing non-uniform wavelet basis in L(R). In this setting, the associated translation set Λ = {0, r/N} + 2Z is no longer a discrete subgroup of R but a spectrum associated with a certain onedimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we characterize the scaling function for non-uniform multiresolution analysis on local fields of positive characteristic (LFPC). Some properties of wavelet scaling function associated with non-uniform multiresolution analysis (NUMRA) on LFPC are also established.

IEEE Transactions on Signal Processing, 1998

The discrete wavelet transform (DWT) is popular in a wide variety of applications. Its sparse sampling eliminates redundancy in the representation of signals and leads to efficient processing. However, the DWT lacks translation invariance. This makes it ill suited for many problems where the received signal is the superposition of arbitrarily shifted replicas of a transmitted signal as when multipath occurs, for example. The paper develops algorithms for the design of orthogonal and biorthogonal compact support scaling functions that are robust to translations. Our approach is to maintain the critical sampling of the DWT while designing multiresolution representations for which the coefficient energy redistributes itself mostly within each subband and not across the entire timescale plane. We obtain expedite algorithms by decoupling the optimization from the constraints on the scaling function. Examples illustrate that the designed scaling function significantly improves the robustness of the representation.

2000

A construction of multiscale decompositions relative to domains Ω ⊂ R d is given. Multiscale spaces are constructed on Ω which retain the important features of univariate multiresolution analysis including local polynomial reproduction and locally supported, stable bases.

Journal of Fourier Analysis and Applications, 2002

Two-scale homogeneous functions are functions h satisfying h(x) = λh(2x) for some constant λ. They form a class of special functions and include homogeneous polynomials. In this article we investigate two-scale homogeneous functions that are contained in a shift-invariant space S(φ) where φ is an r-vector of functions and satisfies a vector refinement equation. The structure of these functions is analyzed. In particular, we establish a one-to-one correspondence between these two-scale homogeneous functions with order λ and the left eigenvectors of a finite matrix (derived from the mask for φ) associated with eigenvalue λ. As a corollary, we show that if φ is supported in [0, N], and provides accuracy m (i. e., {1, x,. .. , x m−1 } ⊂ S(φ)), then 1, 1/2,. .. , 1/2 m−1 are eigenvalues of an rN × rN matrix. This is used to prove that among all the refinable vectors of functions with a fixed support, the B-spline vector with uniform multiple knots yields the optimal accuracy. Thus a conjecture of Plonka is confirmed. The main difficulty we overcome here is that the support of the mask may be larger than that of φ in the vector case. (The mask can even be infinitely supported). Another application is to show that subdivision operators (related to the adjoint of the Ruelle operators) do not have eigenvalues in p (Z). This improves the spectral analysis of subdivision operators. The reconstruction of φ from two-scale homogeneous functions is also considered in the scalar case. This reconstruction is possible globally only if φ is a B-spline. But if we want to reconstruct the pieces of φ on integer intervals, then we can always do so. Our study leads to further more questions concerning the relation between two-scale homogeneous functions and refinable functions.

Journal of Mathematical Imaging and Vision, 1994

It has been observed that linear, Gaussian scale-space, and nonlinear, morphological erosion and dilation scale-spaces generated by a quadratic structuring function have a lot in common. Indeed, far-reaching analogies have been reported, which seems to suggest the existence of an underlying isomorphism. However, an actual mapping appears to be missing. In the present work a one-parameter isomorphism is constructed in closed-form, which encompasses linear and both types of morphological scale-spaces as (non-uniform) limiting cases. The unfolding of the one-parameter family provides a means to transfer known results from one domain to the other. Moreover, for any fixed and non-degenerate parameter value one obtains a novel type of "pseudo-linear" multiscale representation that is, in a precise way, "in-between" the familiar ones. This is of interest in its own right, as it enables one to balance pros and cons of linear versus morphological scale-space representations in any particular situation.

1999

It has been observed that linear, Gaussian scale-space, and nonlinear, morphological erosion and dilation scale-spaces generated by a quadratic structuring function have a lot in common. Indeed, far-reaching analogies have been reported, which seems to suggest the existence of an underlying isomorphism. However, an actual mapping appears to be missing.

SIAM Journal on Mathematical Analysis, 2000