Pairs of Dual Wavelet Frames from Any Two Refinable Functions

Starting from any two compactly supported refinable functions in L₂(R) with dilation factor d,we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L₂(R). Moreover, the number of vanishing moments of each of these wavelet frames is equal to the approximation order of the dual MRA; this is the highest possible. In particular, when we consider symmetric refinable functions, the constructed dual wavelets are also symmetric or antisymmetric. As a consequence, for any compactly supported refinable function in L₂(R), it is possible to construct, explicitly and easily, wavelets that are finite linear combinations of translates (d · – k), and that generate a wavelet frame with an arbitrarily preassigned number of vanishing moments.We illustrate the general theory by examples of such pairs of dual wavelet frames derived from B-spline functions.     

Symmetric multivariate orthogonal refinable functions

In this paper, we shall investigate the symmetry property of a multivariate orthogonal M-refinable function with a general dilation matrix M. For an orthogonal M-refinable function such that is symmetric about a point (centro-symmetric) and provides the approximation order k, we show that must be an orthogonal M-refinable function that generates a generalized coiflet of order k. Next, we show that there does not exist a real-valued compactly supported orthogonal 2I_s-refinable function in any dimension such that is symmetric about a point and generates a classical coiflet. Finally, we prove that if a real-valued compactly supported orthogonal dyadic refinable function L₂(R^s) has the axis symmetry, then cannot be a continuous function and can provide the approximation order at most one. The results in this paper may provide a better picture about symmetric multivariate orthogonal refinable functions. In particular, one of the results in this paper settles a conjecture in [D. Stanhill, Y.Y. Zeevi, IEEE Trans. Signal Process. 46 (1998), 183–190] about symmetric orthogonal dyadic refinable functions.     

Approximation by frame-like wavelet systems

A wide class of MRA-based wavelet systems which are not frames in L₂(Rd), generally speaking, is studied. Frame-type expansions over a pair of dual wavelet systems (with the series converging in different senses) and their approximation order are investigated.     

A class of multiwavelets and projected frames from two-direction wavelets

This article aims at studying two-direction refinable functions and two-direction wavelets in the setting ?^s, s > 1. We give a sufficient condition for a two-direction refinable function belonging to L²(?^s). Then, two theorems are given for constructing biorthogonal (orthogonal) two-direction refinable functions in L²(?^s) and their biorthogonal (orthogonal) two-direction wavelets, respectively. From the constructed biorthogonal (orthogonal) two-direction wavelets, symmetric biorthogonal (orthogonal) multiwaveles in L²(?^s can be obtained easily. Applying the projection method to biorthogonal (orthogonal) two-direction wavelets in L²(?^s, we can get dual (tight) two-direction wavelet frames in L²(?^m, where. m ≥ s From the projected dual (tight) two-direction wavelet frames in L²(?^m, symmetric dual (tight) frames in L²(?^m can be obtained easily. In the end, an example is given to illustrate theoretical results.     

Stability of Gabor frames with arbitrary sampling points

Summary We study the stability of Gabor frames with arbitrary sampling points in the time-frequency plane, in several aspects. We prove that a Gabor frame generated by a window function in the Segal algebra S₀(R^d) remains a frame even if (possibly) all the sampling points undergo an arbitrary perturbation, as long as this is uniformly small. We give explicit stability bounds when the window function is nice enough, showing that the allowed perturbation depends only on the lower frame bound of the original family and some qualitative parameters of the window under consideration. For the perturbation of window functions we show that a Gabor frame generated by any window function with arbitrary sampling points remains a frame when the window function has a small perturbation in S₀(R^d) sense. We also study the stability of dual frames, which is useful in practice but has not found much attention in the literature. We give some general results on this topic and explain consequences to Gabor frames.     

Symmetric orthogonal filters and wavelets with linear-phase moments

In this paper we study symmetric orthogonal filters with linear-phase moments, which are of interest in wavelet analysis and its applications. We investigate relations and connections among the linear-phase moments, sum rules, and symmetry of an orthogonal filter. As one of the results, we show that if a real-valued orthogonal filter a is symmetric about a point, then a has sum rules of order m if and only if it has linear-phase moments of order 2m. These connections among the linear-phase moments, sum rules, and symmetry help us to reduce the computational complexity of constructing symmetric real-valued orthogonal filters, and to understand better symmetric complex-valued orthogonal filters with linear-phase moments. To illustrate the results in the paper, we provide many examples of univariate symmetric orthogonal filters with linear-phase moments. In particular, we obtain an example of symmetric real-valued 4-orthogonal filters whose associated orthogonal 4-refinable function lies in C²(R).     

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

This paper generalizes the mixed extension principle in L ₂(ℝ ^d ) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces H ^s (ℝ ^d ) and H ^−s (ℝ ^d ). In terms of masks for φ,ψ ¹,…,ψ ^L ∈H ^s (ℝ ^d ) and , simple sufficient conditions are given to ensure that (X ^s (φ;ψ ¹,…,ψ ^L ), forms a pair of dual wavelet frames in (H ^s (ℝ ^d ),H ^−s (ℝ ^d )), where

Construction of nonseparable dual -wavelet frames in

Suppose and are two pairs of dual M-wavelet frames and N-wavelet frames in and , respectively, where M and N are s₁×s₁ and s₂×s₂ dilation matrices with a₁(|det(M)|-1) and (2a₂+1)(|det(N)|-1). Moreover, their mask symbols both satisfy mixed extension principle (MEP). Based on the mask symbols, a family of nonseparable dual Ω-wavelet frames in are constructed, where s=s₁+s₂, and with Θ and M^-1Θ both being integer matrices. Then a convolution scheme for improving regularity of wavelet frames is given. From the nonseparable dual Ω-wavelet frames, nonseparable Ω-wavelet frames with high regularity can be constructed easily. We give an algorithm for constructing nonseparable dual symmetric or antisymmetric wavelet frames in . From the dual Ω-wavelet frames, nonseparable dual Ω-wavelet frames with symmetry can be obtained easily. In the end, two examples are given.     

Flexible Gabor-wavelet atomic decompositions for L 2-Sobolev spaces

In this paper we present a general construction of frames, which allows one to ensure that certain families of functions (atoms) obtained by a suitable combination of translation, modulation, and dilation will form Banach frames for the family of L²-Sobolev spaces on ℝ of any order. In this construction a parameter α∈[0,1) governs the dependence of the dilation factor on the frequency parameter. The well-known Gabor and wavelet frames (also valid for the same scale of Hilbert spaces) using suitable Schwartz functions as building blocks arise as special cases (α=0) and a limiting case (α→1), respectively. In contrast to those limiting cases, it is no longer possible to use group-theoretical arguments. Nevertheless, we will show how to constructively ensure that for Schwartz analyzing atoms and any sufficiently dense but discrete and well-structured family of parameters one can guarantee the frame property. As a consequence of this novel constructive technique, one can generate quasicoherent dual frames by an iterative algorithm. As will be shown in a subsequent paper, the new frames introduced here generate Banach frames for corresponding families of α-modulation spaces. Mathematics Subject Classification (2000) 42C15, 46S30, 49M27, 65T60     

Constructing pairs of dual bandlimited framelets with desired time localization

For sufficiently small translation parameters, we prove that any bandlimited function ψ, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame with a dual frame also having the wavelet structure. This dual frame is generated by a finite linear combination of dilations of ψ with explicitly given coefficients. The result allows a simple construction procedure for pairs of dual wavelet frames whose generators have compact support in the Fourier domain and desired time localization. The construction is based on characterizing equations for dual wavelet frames and relies on a technical condition. We exhibit a general class of function satisfying this condition; in particular, we construct piecewise polynomial functions satisfying the condition.      

Supports of Locally Linearly Independent M-Refinable Functions,Attractors of Iterated Function Systems and Tilings

This paper is devoted to a study of supports of locally linearly independent M-refinable functions by means of attractors of iterated function systems, where M is an integer greater than (or equal to) 2. For this purpose, the local linear independence of shifts of M-refinable functions is required. So we give a complete characterization for this local linear independence property by finite matrix products, strictly in terms of the mask. We do this in a more general setting, the vector refinement equations. A connection between self-affine tilings and L ₂ solutions of refinement equations without satisfying the basic sum rule is pointed out, which leads to many further problems. Several examples are provided to illustrate the general theory.     

Periodic wavelet frames

In this paper, starting from any two functions satisfying some simple conditions, using a periodization method, we construct a dual pair of periodic wavelet frames and show their optimal bounds. The obtained periodic wavelet frames possess trigonometric polynomial expressions. Finally, we present two examples to explain our theory.     

Characterization of Sobolev spaces of arbitrary smoothness using nonstationary tight wavelet frames

In this paper we shall characterize Sobolev spaces of an arbitrary order of smoothness using nonstationary tight wavelet frames for L ₂(ℝ). In particular, we show that a Sobolev space of an arbitrary fixed order of smoothness can be characterized in terms of the weighted ℓ₂-norm of the analysis wavelet coefficient sequences using a fixed compactly supported nonstationary tight wavelet frame in L ₂(ℝ) derived from masks of pseudosplines in [15]. This implies that any compactly supported nonstationary tight wavelet frame of L ₂(ℝ) in [15] can be properly normalized into a pair of dual frames in the corresponding pair of dual Sobolev spaces of an arbitrary fixed order of smoothness. Research supported in part by NSERC Canada under Grant RGP 228051. Research supported in part by Grant R-146-000-060-112 at the National University of Singapore.     

Wavelet bi‐frames with uniform symmetry

Wavelet bi‐frames with uniform symmetry are discussed in this paper. Every refinable function in the bi‐frame system is symmetric, which is very useful in the image processing and curve and surface multiresolution processing. By the aid of the lifting scheme, bi‐frame multiresolution algorithms can be divided into several iterative steps, and each step can be shown by a symmetric template. The template‐based procedure is established for constructing bi‐frames with uniform symmetry and N > 2 generators. In particular, we take the bi‐frame with three generators as an example to provide a clearer picture of the template‐based procedure for constructing bi‐frames. Three types of bi‐frames with three generators are studied, and some examples with certain smoothness are constructed. These examples include some bi‐frames with interpolating property. Copyright © 2016 John Wiley & Sons, Ltd.     

Supports of fourier transforms of refinable frame functions and their applications to FMRA

Let M be a d × d expansive matrix, and FL ²(??) be a reducing subspace of L ²(? ^d ). This paper characterizes bounded measurable sets in ? ^d which are the supports of Fourier transforms of M-refinable frame functions. As applications, we derive the characterization of bounded measurable sets as the supports of Fourier transforms of FMRA (W-type FMRA) frame scaling functions and MRA (W-type MRA) scaling functions for FL ²(??), respectively. Some examples are also provided.     

Construction of multiwavelets with high approximation order and symmetry

In this paper, based on existing symmetric multiwavelets, we give an explicit algorithm for constructing multiwavelets with high approximation order and symmetry. Concretely, suppose Φ(x) := (φ1(x), . . . , φr(x))T is a symmetric refinable function vectors with approximation order m. For an arbitrary nonnegative integer n, a new symmetric refinable function vector Φnew(x) := (φn1 ew(x), . . . , φrn ew(x))T with approximation order m + n can be constructed through the algorithm mentioned above. Additionally,...     

Symmetric orthogonal filters and wavelets with linear-phase moments

In this paper we study symmetric orthogonal filters with linear-phase moments, which are of interest in wavelet analysis and its applications. We investigate relations and connections among the linear-phase moments, sum rules, and symmetry of an orthogonal filter. As one of the results, we show that if a real-valued orthogonal filter $a$ is symmetric about a point, then $a$ has sum rules of order $m$ if and only if it has linear-phase moments of order $2m$. These connections among the linear-phase moments, sum rules, and symmetry help us to reduce the computational complexity of constructing symmetric real-valued orthogonal filters, and to understand better symmetric complex-valued orthogonal filters with linear-phase moments. To illustrate the results in the paper, we provide many examples of univariate symmetric orthogonal filters with linear-phase moments. In particular, we obtain an example of symmetric real-valued 4-orthogonal filters whose associated orthogonal 4-refinable function lies in $C^2\left(\mathbb{R}\right)$.     

Symmetric Framelets

We study tight wavelet frames associated with symmetric compactly supported refinable functions, which are obtained with the unitary extension principle . We give a criterion for the existence of two symmetric or antisymmetric compactly supported framelets. All refinable masks of length up to 6 satisfying this criterion are found.     

Two-direction refinable functions and two-direction wavelets with high approximation order and regularity

The concept of two-direction refinable functions and two-direction wavelets is introduced. We investigate the existence of distributional(or L~2-stable) solutions of the two-direction refinement equation: (?)(x)=(?)p_k~ (?)(mx-k) (?)p_k~-(?)(k-mx), where m≥2 is an integer.Based on the positive mask {p_k~ } and negative mask {p_k~-},the conditions that guarantee the above equation has compactly distributional solutions or L~2-stable solutions are established.Furthermore,the condition that the L~2-stable solution of the above equation can generate a two-direction MRA is given.The support interval of (?)(x) is discussed amply.The definition of orthogonal two-direction refinable function and orthogonal two-direction wavelets is presented,and the orthogonality criteria for two-direction refinable functions are established.An algorithm for construct- ing orthogonal two-direction refinable functions and their two-direction wavelets is presented.Another construction algorithm for two-direction L~2-refinable functions,which have nonnegative symbol masks and possess high approximation order and regularity,is presented.Finally,two construction examples are given.