Parseval wavelets on hierarchical graphs

Wavelets on graphs have been studied for the past few years, and in particular, several approaches have been proposed to design wavelet transforms on hierarchical graphs. Although such methods are computationally efficient and easy to implement, their frames are highly restricted. In this paper, we propose a general framework for the design of wavelet transforms on hierarchical graphs. Our design is guaranteed to be a Parseval tight frame, which preserves the $l_2$ norm of any input signals. To demonstrate the potential usefulness of our approach, we perform several experiments, in which we learn a wavelet frame based on our framework, and show, in inpainting experiments, that it performs better than a Haar-like hierarchical wavelet transform and a learned treelet. We also show with category theory that the algebraic properties of the proposed transform have a strong relationship with those of the hierarchical graph that represents the structure of the given data.     

Parseval frame wavelets with -dilations

We study Parseval frame wavelets in with matrix dilations of the form , where A is an arbitrary expanding n×n matrix with integer coefficients, such that |detA|=2. We show that each A-MRA admits either Parseval frame wavelets, or Parseval frame bi-wavelets. The minimal number of generators for a Parseval frame associated with an A-MRA (i.e. 1 or 2) is determined in terms of a scaling function. All Parseval frame (bi)wavelets associated with A-MRA's are described. We then introduce new classes of filter induced wavelets and bi-wavelets. It is proved that these new classes strictly contain the classes of all A-MRA Parseval frame wavelets and bi-wavelets, respectively. Finally, we demonstrate a method of constructing all filter induced Parseval frame (bi)wavelets from generalized low-pass filters.     

Further results on the connectivity of Parseval frame wavelets

In a previous paper, the authors introduced new ideas to treat the problem of connectivity of Parseval frames. With these ideas it was shown that a large set of Parseval frames is arcwise connected. In this article we exhibit a larger class of Parseval frames for which the arcwise connectivity is true. This larger class fails to include all Parseval frames.

     

A new identity for Parseval frames

In this paper we establish a surprising new identity for Parseval frames in a Hilbert space. Several variations of this result are given, including an extension to general frames. Finally, we discuss the derived results.

     

The characterization of a class of multivariate MRA and semi-orthogonal parseval frame wavelets

Let A be a d × d expansive matrix with ∣detA∣ = 2. This paper addresses Parseval frame wavelets (PFWs) in the setting of reducing subspaces of L²(R^d). We prove that all semi-orthogonal PFWs (semi-orthogonal MRA PFWs) are precisely the ones with their dimension functions being non-negative integer-valued (0 or 1). We also characterize all MRA PFWs. Some examples are provided.     

Parseval Frame Wavelet Multipliers in L2（Rd）

Let A be a d × d real expansive matrix. An A-dilation Parseval frame wavelet is a function ?? ?? L ²(? ^d ), such that the set $ \left\{ {\left| {\det A} \right|^{\frac{n} {2}} \psi \left( {A^n t - \ell } \right):n \in \mathbb{Z},\ell \in \mathbb{Z}^d } \right\} $ forms a Parseval frame for L ²(? ^d ). A measurable function f is called an A-dilation Parseval frame wavelet multiplier if the inverse Fourier transform of d??? is an A-dilation Parseval frame wavelet whenever ?? is an A-dilation Parseval frame wavelet, where ??? denotes the Fourier transform of ??. In this paper, the authors completely characterize all A-dilation Parseval frame wavelet multipliers for any integral expansive matrix A with |det(A)| = 2. As an application, the path-connectivity of the set of all A-dilation Parseval frame wavelets with a frame MRA in L ²(? ^d ) is discussed.     

关于Parseval框架小波的刻画

In this paper,we characterize all generalized low pass filters and MRA Parseval frame wavelets in L 2 （R n ） with matrix dilations of the form （Df）（x） =√ 2f（Ax）,where A is an arbitrary expanding n × n matrix with integer coefficients,such that ｜det A｜ = 2.We study the pseudo-scaling functions,generalized low pass filters and MRA Parseval frame wavelets and give some important characterizations about them.Furthermore,we give a characterization of the semiorthogonal MRA Parseval frame wavelets and provide several examples to verify our results.     

Frame representations and Parseval duals with applications to Gabor frames

Let be a frame for a Hilbert space . We investigate the conditions under which there exists a dual frame for which is also a Parseval (or tight) frame. We show that the existence of a Parseval dual is equivalent to the problem whether can be dilated to an orthonormal basis (under an oblique projection). A necessary and sufficient condition for the existence of Parseval duals is obtained in terms of the frame excess. For a frame induced by a projective unitary representation of a group , it is possible that can have a Parseval dual, but does not have a Parseval dual of the same type. The primary aim of this paper is to present a complete characterization for all the projective unitary representations such that every frame (with a necessary lower frame bound condition) has a Parseval dual of the same type. As an application of this characterization together with a result about lattice tiling, we prove that every Gabor frame (again with the same necessary lower frame bound condition) has a Parseval dual of the same type if and only if the volume of the fundamental domain of is less than or equal to .

     

具有特殊伸缩矩阵的Parseval框架小波集的结构

揭示具有特殊伸缩矩阵的Parseval框架小波集的丰富结构.借助于平移不变空间和维数函数,研究了具有特殊伸缩矩阵M的Parseval框架小波(M-PFW)、半正交M-PFW和MRA M-PFW的各种性质,探讨了M-PFW集合的各种子类,给出了这些子类的构造性算例.     

L~2(R~n)中MRA Parseval框架小波和它们的乘子

刻画了L~2(R~n)中具有扩展矩阵伸缩的广义低通滤波器和多尺度分析Parseval框架小波(缩写为MRA PFW).首先,研究了伪逆的尺度函数、广义的低通滤波器和MRA PFW,给出它们的一些刻画.接着,我们给出与MRA PFW相联系的几类乘子的一些刻画.最后,给出了一个例子来证明的结论.     

The s-elementary frame wavelets are path connected

An s-elementary frame wavelet is a function which is a frame wavelet and is defined by a Lebesgue measurable set such that . In this paper we prove that the family of s-elementary frame wavelets is a path-connected set in the -norm. This result also holds for s-elementary -dilation frame wavelets in in general. On the other hand, we prove that the path-connectedness of s-elementary frame wavelets cannot be strengthened to uniform path-connectedness. In fact, the sets of normalized tight frame wavelets and frame wavelets are not uniformly path-connected either.

     

Super‐wavelets on local fields of positive characteristic

The concept of super‐wavelet was introduced by Balan, and Han and Larson over the field of real numbers which has many applications not only in engineering branches but also in different areas of mathematics. To develop this notion on local fields having positive characteristic we obtain characterizations of super‐wavelets of finite length as well as Parseval frame multiwavelet sets of finite order in this setup. Using the group theoretical approach based on coset representatives, further we establish Shannon type multiwavelet in this perspective while providing examples of Parseval frame (multi)wavelets and (Parseval frame) super‐wavelets. In addition, we obtain necessary conditions for decomposable and extendable Parseval frame wavelets associated to Parseval frame super‐wavelets.     

A generalization of Gram–Schmidt orthogonalization generating all Parseval frames

Given an arbitrary finite sequence of vectors in a finite-dimensional Hilbert space, we describe an algorithm, which computes a Parseval frame for the subspace generated by the input vectors while preserving redundancy exactly. We further investigate several of its properties. Finally, we apply the algorithm to several numerical examples.      

On MRA Riesz wavelets

We investigate the properties of univariate MRA Riesz wavelets. In particular we obtain a generalization to semiorthogonal MRA wavelets of a well-known representation theorem for orthonormal MRA wavelets.

     

On some identities and inequalities for frames in Hilbert spaces

We give new Parseval type identities and inequalities for frames in Hilbert spaces. Our results generalize the remarkable results obtained recently by R. Balan, P.G. Casazza, D. Edidin, and G. Kutyniok.     

The road to equal-norm Parseval frames

The construction of equal-norm Parseval frames is fundamental for many applications of frame theory. We present a construction method based on a system of ordinary differential equations, which generates a flow on the set of Parseval frames that converges to equal-norm Parseval frames. We developed this method to address a question posed by Vern Paulsen: How close is a nearly equal-norm, nearly Parseval frame to an equal-norm Parseval frame? The distance estimate derived here can be used to substantiate numerically found, approximate constructions of equal-norm Parseval frames. The estimate is valid for a fairly general class of frames — requiring that the dimension of the Hilbert space and the number of frame vectors is relatively prime. In addition, we re-phrase our distance estimate to show that certain projection matrices which are nearly constant on the diagonal are close in Hilbert-Schmidt norm to ones which have a constant diagonal.