Wavelet Frames Research Articles

Approximation of reconstruction formula for continuous wavelet and Weyl-Heisenberg frames

As for an orthonormal basis, a frame allows each element in the underlying Hilbert space to be written as an unconditionally convergent infinite linear combination of the frame elements. The coefficients are called frame coefficients. Peter G. Casazza and Ole Christensen introduced some methods to approximate frame coefficients. In this article, we investigate some of these results for a continuous frame. As a consequence, approximation of the solution to a moment problem is also discussed. We also apply the results to wavelet frames and Weyl-Heisenberg frames.

The Nice Feature of a Sort of Orthogonal Finitely Supported Five-Variant Wavelet Packages

In this article, the notion of orthogonal nonseparable five-variant wavelet packages, whi- ch is the generalization of orthogonal univariate wavelet packages, is introduced. A new approach for constructing the wavelet packages is presented by iteration method as well wavelet frames. The orthogonality properties for five-dimensional wavelet packages are discussed. Three orthogonality formulas concerning these wavelet packages are estabished. A necessary and sufficient condition for the existence of the pyramid decomposition scheme of space is presented.

Some Smooth Compactly Supported Tight Framelets

For any dilation matrix with integer entries, we construct a family of smooth compactly supported tight wavelet frames in $L^2(\mathbb{R}^d)$, $d\ge 1$. Estimates for the degrees of smoothness of these framelets are given. Our construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stockler.

$A$-transform of Wavelet Frames

In this paper, we introduced the concept of $A$-transform $A = (a_{p,q,j,k})$ and study the action of $A$ on $f\in L^2(\mathbb{R}^+)$ and on its wavelet coefficients. Further, we also establish the Parseval frame condition for $A$-transform of $f \in L^2(\mathbb{R}^+)$ whose wavelet series expansion is known.

The Characters of Multiple Tight Multivariate Wavelet Frames and Application in Quantum Physics

Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. This paper is devoted to the study and construction of finitely supported tight multivariate frames of multivariate multi-wavelets. Inparticular, a necessary conditi- on for their existence is obtained to present some feasible idea for designing such MRA tight frames. The characteristics of binary multiscale pseudoframes for subspaces is investigated. The constructi- on of a GMS of Paley-Wiener subspace of is studied. A constructive method for affine multivariate frames based on such a GMS is established.

The Research of Bivariate Minimum-Energy Frames and Frames of Subspace and Application in Particle Physics

Materials science is an applied science concerned with the relationship between the struc- ture and properties of materials. Frames have become the focus of active research field, both in the- ory and in applications. In the article, the binary minimum-energy wavelet frames and frame multi- resolution are introduced. A precise exist-ence criterion for minimum-energy frames in terms of an ineqity condition on the Laurent poly-nomial symbols of the filter functions is provided. An explicit formula for designing minimum-energy frames is also established. The sufficient con-dition for the existence of affine pseudoframes is obtained by virtue of a generalized multiresolution analysis. The pyramid decomposition scheme is established based on such a generalized multiresolution structure.

The Characters of Multiple Tight Multivariate Wavelet Frames and Application in Quantum Physics

Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. This paper is devoted to the study and construction of finitely supported tight multivariate frames of multivariate multi-wavelets. Inparticular, a necessary conditi- on for their existence is obtained to present some feasible idea for designing such MRA tight frames. The characteristics of binary multiscale pseudoframes for subspaces is investigated. The constructi- on of a GMS of Paley-Wiener subspace of is studied. A constructive method for affine multivariate frames based on such a GMS is established.

Homogeneous Besov Spaces on Stratified Lie Groups and Their Wavelet Characterization

We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov spaceB˙p,qsin terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spacesB˙p,qswith1≤p,q<∞ands∈ℝ.

The Research of Bivariate Minimum-Energy Frames and Frames of Subspace and Application in Particle Physics

Materials science is an applied science concerned with the relationship between the struc- ture and properties of materials. Frames have become the focus of active research field, both in the- ory and in applications. In the article, the binary minimum-energy wavelet frames and frame multi- resolution are introduced. A precise exist-ence criterion for minimum-energy frames in terms of an ineqity condition on the Laurent poly-nomial symbols of the filter functions is provided. An explicit formula for designing minimum-energy frames is also established. The sufficient con-dition for the existence of affine pseudoframes is obtained by virtue of a generalized multiresolution analysis. The pyramid decomposition scheme is established based on such a generalized multiresolution structure.

Unified conditions for wavelet frames

Unified conditions for wavelet frames

P-Wavelet frame packets on a half-line using the Walsh–Fourier transform

This paper deals with a construction of wavelet frame packets on the positive half-line ℝ+ using the splitting trick for frames. It is shown that, as long as finitely many splitting steps are applied, the resulting sequence of functions is a frame of L 2(ℝ+). If the matrix M(ξ) associated with the splitting is unitary, then the splitting can be applied infinitely many times to prove the existence of frame with the frame bounds as shown in Theorem 3.3.

Testing the isotropy of high energy cosmic rays using spherical needlets

For many decades, ultrahigh energy charged particles of unknown origin that can be observed from the ground have been a puzzle for particle physicists and astrophysicists. As an attempt to discriminate among several possible production scenarios, astrophysicists try to test the statistical isotropy of the directions of arrival of these cosmic rays. At the highest energies, they are supposed to point toward their sources with good accuracy. However, the observations are so rare that testing the distribution of such samples of directional data on the sphere is nontrivial. In this paper, we choose a nonparametric framework that makes weak hypotheses on the alternative distributions and allows in turn to detect various and possibly unexpected forms of anisotropy. We explore two particular procedures. Both are derived from fitting the empirical distribution with wavelet expansions of densities. We use the wavelet frame introduced by [SIAM J. Math. Anal. 38 (2006b) 574-594 (electronic)], the so-called needlets. The expansions are truncated at scale indices no larger than some ${J^{\star}}$, and the $L^p$ distances between those estimates and the null density are computed. One family of tests (called Multiple) is based on the idea of testing the distance from the null for each choice of $J=1,\ldots,{J^{\star}}$, whereas the so-called PlugIn approach is based on the single full ${J^{\star}}$ expansion, but with thresholded wavelet coefficients. We describe the practical implementation of these two procedures and compare them to other methods in the literature. As alternatives to isotropy, we consider both very simple toy models and more realistic nonisotropic models based on Physics-inspired simulations. The Monte Carlo study shows good performance of the Multiple test, even at moderate sample size, for a wide sample of alternative hypotheses and for different choices of the parameter ${J^{\star}}$. On the 69 most energetic events published by the Pierre Auger Collaboration, the needlet-based procedures suggest statistical evidence for anisotropy. Using several values for the parameters of the methods, our procedures yield $p$-values below 1%, but with uncontrolled multiplicity issues. The flexibility of this method and the possibility to modify it to take into account a large variety of extensions of the problem make it an interesting option for future investigation of the origin of ultrahigh energy cosmic rays.

CONSTRUCTION OF MRA E-TIGHT FRAME WAVELETS, MULTIPLIERS AND CONNECTIVITY PROPERTIES

A method that constructs an MRA E-tight frame wavelet by using a generalized low pass E-filter is given, and it is showed that all of MRA E-tight frame wavelets can be obtained via the method, where the dilation matrix E is the quincunx matrix or the matrix consists of (0, 1)T and (2, 0)T. Furthermore, the properties of MRA E-tight frame wavelet multipliers as well as E-pseudoscaling function multipliers and generalized low pass E-filter multipliers are characterized. In addition, as an application of these multipliers, we discuss the connectivity of the set of all MRA E-tight frame wavelets in L2(R2).

Framelet-Based Blind Motion Deblurring From a Single Image

How to recover a clear image from a single motion-blurred image has long been a challenging open problem in digital imaging. In this paper, we focus on how to recover a motion-blurred image due to camera shake. A regularization-based approach is proposed to remove motion blurring from the image by regularizing the sparsity of both the original image and the motion-blur kernel under tight wavelet frame systems. Furthermore, an adapted version of the split Bregman method is proposed to efficiently solve the resulting minimization problem. The experiments on both synthesized images and real images show that our algorithm can effectively remove complex motion blurring from natural images without requiring any prior information of the motion-blur kernel.

Wavelet frame based surface reconstruction from unorganized points

Applications of wavelet frames to image restoration problems (e.g. image deblurring and inpainting) have been successful due to their redundancy and capability of sparsely approximating piecewise smooth functions like images (see e.g. [13,14,8]). However, wavelet frames have not yet been used for surface reconstruction problems. Recently in [6], connections between one of the wavelet frame based image restoration model [8,28,48] and variational models (e.g. the ROF model [45]) were rigorously established. Such connections not only grant new insights to wavelet frame based image restorations, it also case a geometric explanation to wavelet frame based approaches. This leads us to a wavelet frame based model, as well as a fast algorithm, to reconstruct implicit surfaces from unorganized point sets in R^3. We will demonstrate the effectiveness of the proposed model using several commonly used examples.

On s-Elementary Super Frame Wavelets and Their Path-Connectedness

A super wavelet of length n is an n-tuple (? 1,? 2,?,? n ) in the product space $\prod_{j=1}^{n} L^{2}(\mathbb{R})$ , such that the coordinated dilates of all its coordinated translates form an orthonormal basis for $\prod_{j=1}^{n} L^{2} (\mathbb{R})$ . This concept is generalized to the so-called super frame wavelets, super tight frame wavelets and super normalized tight frame wavelets (or super Parseval frame wavelets), namely an n-tuple (? 1,? 2,?,? n ) in $\prod_{j=1}^{n}L^{2} (\mathbb{R})$ such that the coordinated dilates of all its coordinated translates form a frame, a tight frame, or a normalized tight frame for $\prod_{j=1}^{n} L^{2}(\mathbb{R})$ . In this paper, we study the super frame wavelets and the super tight frame wavelets whose Fourier transforms are defined by set theoretical functions (called s-elementary frame wavelets). An m-tuple of sets (E 1,E 2,?,E m ) is said to be ?-disjoint if the E j 's are pair-wise disjoint under the 2?-translations. We prove that a ?-disjoint m-tuple (E 1,E 2,?,E m ) of frame sets (i.e., ? j defined by $\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}$ is a frame wavelet for L 2(?) for each j) lead to a super frame wavelet (? 1,? 2,?,? m ) for $\prod_{j=1}^{m} L^{2} (\mathbb{R})$ where $\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}$ . In the case of super tight frame wavelets, we prove that (? 1,? 2,?,? m ), defined by $\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}$ , is a super tight frame wavelet for ?1?j?m L 2(?) with frame bound k 0 if and only if each ? j is a tight frame wavelet for L 2(?) with frame bound k 0 and that (E 1,E 2,?,E m ) is ?-disjoint. Denote the set of all ?-disjoint s-elementary super frame wavelets for ?1?j?m L 2(?) by $\mathfrak{S}(m)$ and the set of all s-elementary super tight frame wavelets (with the same frame bound k 0) for ?1?j?m L 2(?) by $\mathfrak{S}^{k_{0}}(m)$ . We further prove that $\mathfrak{S}(m)$ and $\mathfrak{S}^{k_{0}}(m)$ are both path-connected under the ?1?j?m L 2(?) norm, for any given positive integers m and k 0.

Constructing pairs of dual bandlimited frame wavelets in [formula omitted

Given a real, expansive dilation matrix we prove that any bandlimited function ψ∈L2(Rn), for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame for certain translation lattices. Moreover, there exists a dual wavelet frame 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 relies on a technical condition on ψ, and we exhibit a general class of function satisfying this condition.

Dyadic wavelet frames on a half-line using the Walsh–Fourier transform

This paper deals with a construction of dyadic wavelet frames on a positive half-line ℝ+ using the Walsh–Fourier transform. We prove necessary and sufficient conditions for the system to be a wavelet frame in L 2(ℝ+). These conditions are found to be better than those of Daubechies [Ten Lecture on Wavelets, NSF-CBMS Regional Conferences in Applied Mathematics, Vol. 61, SIAM, Philadelphia, PA, 1992], Chui and Shi [Inequalities of Littlewood–Paley type for frames and wavelets, SIAM J. Math. Anal. 24 (1993), pp. 263–277], Casazza and Christenson [Weyl–Heisenberg frames for subspaces of L 2(ℝ), Proc. Amer. Math. Soc. 129 (2001), pp. 145–154] and Christenson [Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bull. Amer. Math. Soc. 38 (2001), pp. 273–291; An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003].

Wavelet frame based blind image inpainting

Image inpainting has been widely used in practice to repair damaged/missing pixels of given images. Most of the existing inpainting techniques require knowing beforehand where those damaged pixels are, either given as a priori or detected by some pre-processing. However, in certain applications, such information neither is available nor can be reliably pre-detected, e.g. removing random-valued impulse noise from images or removing certain scratches from archived photographs. This paper introduces a blind inpainting model to solve this type of problems, i.e., a model of simultaneously identifying and recovering damaged pixels of the given image. A tight frame based regularization approach is developed in this paper for such blind inpainting problems, and the resulted minimization problem is solved by the split Bregman algorithm first proposed by Goldstein and Osher (2009) [1]. The proposed blind inpainting method is applied to various challenging image restoration tasks, including recovering images that are blurry and damaged by scratches and removing image noise mixed with both Gaussian and random-valued impulse noise. The experiments show that our method is compared favorably against many available two-staged methods in these applications.

TU‐G‐110‐09: Spectral CT (SCT) Reconstruction Using Tight Wavelet Frames

Purpose: X‐ray CT has been widely applied in a number of clinical and preclinical applications. However, the images are often insufficient to distinguish diagnostically crucial shading differences due to poor tissue contrast and inherent quantum noise. Recently, Spectral CT (SCT) base on photon counting technology has attracted a lot of attention due to its ability of providing full energy resolved tomography images. Currently, SCT is often reconstructed by using the conventional filtered backprojection (FBP) algorithms on each energy channel independently. Yet, it is expected that the anatomical features between energy channels are highly correlated. In this work, we propose an correlated iterative SCT reconstruction algorithm using tight wavelet frame based regularizations. Methods: In our study, we propose a model that reconstructs SCT images via solving an optimization problem. The objective function contains both a data fidelity term, requiring the consistency between reconstructed data and the measurement in each energy channel, and a regularization term applied along both the spatial dimension and the energy dimension. Bregman iteration method is used to solve the optimization model efficiently. To validate our algorithm, we synthesis a set of energy resolved x‐ray projections in fan‐beam scanning geometry on an NCAT digital phantom. Reconstruction results from our algorithm are compared with those from the conventional FBP algorithm. Results: It is found that our algorithm has lead to visually better image quality than the FBP algorithm. Signal‐to‐Noise Ratio (SNR) is used to evaluate the reconstruction quality. Our algorithm can enhance the SNR by a factor of 4–5 compared with the FBP algorithm. Conclusions: We have designed a wavelet frame based regularization method for SCT image reconstruction. Higher quality SCT images than that from FBP algorithms have been achieved.