Tight Wavelet Frames Research Articles

Frame wavelet sets in [formula omitted

In this paper, we discuss the characterization of frame wavelet sets. We extend some results obtained earlier in the one-dimensional case. More specifically, we completely characterize tight frame wavelet sets in higher dimensions and obtain some necessary conditions and sufficient conditions for a set E to be a frame wavelet set in R d . Several examples are presented and compared with those in the one-dimensional case. Using our results, one can easily construct various frame wavelet sets.

Tight Frame Wavelets, their Dimension Functions, MRA Tight Frame Wavelets and Connectivity Properties

This is a continuation of our study of generalized low pass filters and MRA frame wavelets. In this first study we concentrated on the construction of such functions. Here we are particularly interested in the role played by the dimension function. In particular we characterize all semi-orthogonal Tight Frame Wavelets (TFW) by showing that they correspond precisely to those for which the dimension function is non-negative integer-valued. We also show that a TFW arises from our MRA construction if and only if the dimension of a particular linear space is either zero or one. We present many examples. In addition we obtain a result concerning the connectivity of TFW's that are MSF tight frame wavelets.

Construction for a class of smooth wavelet tight frames

From the inequality |P(z)|2 + |P(−z)|2 ⩽ 1, assuming that both of the low-pass filters and high-pass filters are unknown, we design compactly supported wavelet tight frames. The unknowing of low-pass filters allows the design more freedom, and both the low-pass filters and high-pass filters have symmetries or anti-symmetries. We give the algorithm for filters with odd and even lengths separately, some concrete examples of wavelet tight frames with the length 4, 5, 6, 7, and at last we give the result of decomposing Lena image with them.

Framelets: MRA-based constructions of wavelet frames

We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spline, pseudo-spline tight frames, and symmetric bi-frames with short supports and high approximation orders. Several explicit examples are discussed. The connection of these frames with multiresolution analysis guarantees the existence of fast implementation algorithms, which we discuss briefly as well.

Characterization of General Tight Wavelet Frames with Matrix Dilations and Tightness Preserving Oversampling

Characterization of General Tight Wavelet Frames with Matrix Dilations and Tightness Preserving Oversampling

Subspaces with normalized tight frame wavelets in ℝ

In this paper we investigate the subspaces of L 2 ( R ) L^2(\mathbb R) which have normalized tight frame wavelets that are defined by set functions on some measurable subsets of R \mathbb R called Bessel sets. We show that a subspace admitting such a normalized tight frame wavelet falls into a class of subspaces called reducing subspaces. We also consider the subspaces of L 2 ( R ) L^2(\mathbb R) that are generated by a Bessel set E E in a special way. We present some results concerning the relation between a Bessel set E E and the corresponding subspace of L 2 ( R ) L^2(\mathbb R) which either has a normalized tight frame wavelet defined by the set function on E E or is generated by E E .

Generalized low pass filters and MRA frame wavelets

A tight frame wavelet ψ is an L 2(ℝ) function such that {ψ jk(x)} = {2j/2 ψ(2 j x −k), j, k ∈ ℤ},is a tight frame for L 2 (ℝ).We introduce a class of “generalized low pass filters” that allows us to define (and construct) the subclass of MRA tight frame wavelets. This leads us to an associated class of “generalized scaling functions” that are not necessarily obtained from a multiresolution analysis. We study several properties of these classes of “generalized” wavelets, scaling functions and filters (such as their multipliers and their connectivity). We also compare our approach with those recently obtained by other authors.

Tight compactly supported wavelet frames of arbitrarily high smoothness

Based on Ron and Shen’s new method for constructing tight wave- let frames, we show that one can construct, for any dilation matrix, and in any spatial dimension, tight wavelet frames generated by compactly supported functions with arbitrarily high smoothness.

On Dual Wavelet Tight Frames

A characterization of multivariate dual wavelet tight frames for any general dilation matrix is presented in this paper. As an application, Lawton's result on wavelet tight frames inL2(R) is generalized to then-dimensional case. Two ways of constructing certain dual wavelet tight frames inL2(Rn) are suggested. Finally, examples of smooth wavelet tight frames inL2(R) andH2(R) are provided. In particular, an example is given to demonstrate that there is a function ψ whose Fourier transform is positive, compactly supported, and infinitely differentiable which generates a non-MRA wavelet tight frame inH2(R).

Some Results in the Theory Modulated Filter Banks and Modulated Wavelet Tight Frames

Perfect reconstruction (PR) FIR filter banks, obtained by modulation of a linear-phase, lowpass, prototype filter and of length 2Mm are well known. Recently, PR modulated filter banks (MFBs) with the analysis and synthesis banks obtained from different prototypes have been reported. This paper describes a general form of modulation that includes modulations used in the literature. This modulation depends on an integer parameter, the modulation phase. The PR property is characterized for MFBs with finite and infinite impulse response filters. The MFB PR problem reduces to roughly M/2 two-channel PR problems. A natural dichotomy in the PR conditions leads us to the concepts of Type 1 and Type 2 MFBs. Unitary MFBs are characterized by the M/2 two-channel PR filter banks also being unitary (for FIR filters of length N = 2Mm, these results are given in (Malvar, Electr Lett. 26, June 1990, 906–907; Koilpillai and Vaidyanathan, IEEE Trans. SP40, No. 4, Apr. 1992, 770–783)). We also give a necessary and sufficient condition for a large class (including FIR) unitary MFB prototypes to have symmetric (even or odd) prototype filters, and exhibit unitary MFBs without symmetric prototypes. A parameterization of all FIR unitary MFBs is also given. An efficient design procedure for FIR unitary MFBs is developed. It turns out that MFBs can be implemented efficiently using Type III and Type IV DCTs. Compactly supported modulated wavelet tight frames are shown to exist and completely parameterized. K-regular modulated WTFs are designed numerically and analytically by solving a set of non-linear equations over the parameters. Design of optimal modulated WTFs for the representation of any given signal is described with examples, and this is used to design smooth modulated WTFs.

On cosine-modulated wavelet orthonormal bases

Multiplicity M, K-regular, orthonormal wavelet bases (that have implications in transform coding applications) have previously been constructed by several authors. The paper describes and parameterizes the cosine-modulated class of multiplicity M wavelet tight frames (WTFs). In these WTFs, the scaling function uniquely determines the wavelets. This is in contrast to the general multiplicity M case, where one has to, for any given application, design the scaling function and the wavelets. Several design techniques for the design of K regular cosine-modulated WTFs are described and their relative merits discussed. Wavelets in K-regular WTFs may or may not be smooth, Since coding applications use WTFs with short length scaling and wavelet vectors (since long filters produce ringing artifacts, which is undesirable in, say, image coding), many smooth designs of K regular WTFs of short lengths are presented. In some cases, analytical formulas for the scaling and wavelet vectors are also given. In many applications, smoothness of the wavelets is more important than K regularity. The authors define smoothness of filter banks and WTFs using the concept of total variation and give several useful designs based on this smoothness criterion. Optimal design of cosine-modulated WTFs for signal representation is also described. All WTFs constructed in the paper are orthonormal bases.

Unitary FIR filter banks and symmetry

Recently, unitary FIR filter banks with linear-phase have been completely parameterized by exploiting the eigenstructure of the exchange matrix. This correspondence gives new characterizations of polyphase matrices of filter banks with various types of symmetry on the filters. Using matrix extensions of the well-known hyperbolic and orthogonal lattices, we give a alternative proof for the parameterization of linear-phase unitary filter banks. A complete parameterization of FIR unitary filter banks with each of the different types of symmetries considered (not just linear-phase) is also given. These results can also be used to generate non-unitary filter banks with symmetries, though no completeness results can be obtained. In some cases implicit, and in others explicit parameterization of wavelet tight frames associated with these filter banks are also given. This paper only considers filter banks with an even number of channels. A similar theory can be developed if the number of channels is an odd integer. >

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation