Wavelet transform and radon transform on the quaternion Heisenberg group

Let Q be the quaternion Heisenberg group,and let P be the affine automorphism group of Q.We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on L2(Q).A class of radial wavelets is constructed.The inverse wavelet transform is simplified by using radial wavelets.Then we investigate the Radon transform on Q.A Semyanistyi–Lizorkin space is introduced,on which the Radon transform is a bijection.We deal with the Radon transform on Q both by the Euclidean Fourier transform and the group Fourier transform.These two treatments are essentially equivalent.We also give an inversion formula by using wavelets,which does not require the smoothness of functions if the wavelet is smooth.In addition,we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on Q.     

Wavelet transform associated with linear canonical Hankel transform

The main goal of this paper is to study about the continuous as well as discrete wavelet transform in terms of linear canonical Hankel transform (LCH‐transform) and discuss some of its basic properties. Parseval's relation and reconstruction formula of continuous linear canonical Hankel wavelet transform (CLCH‐wavelet transform) is obtained. Moreover, semidiscrete and discrete LCH‐wavelet transform are also discussed.     

Mexican hat wavelet transform of distributions

Theory of Weierstrass transform is exploited to derive many interesting new properties of the Mexican hat wavelet transform. A real inversion formula in the differential operator form for the Mexican hat wavelet transform is established. Mexican hat wavelet transform of distributions is defined and its properties are studied. An approximation property of the distributional wavelet transform is investigated which is supported by a nice example.     

The quadratic-phase Fourier wavelet transform

In this paper, we define the quadratic-phase Fourier wavelet transform (QPFWT) and discuss its basic properties including convolution for QPFWT. Further, inversion formula and the Parseval relation of QPFWT are also discussed. Continuity of QPFWT on some function spaces are studied. Moreover, some applications of quadratic-phase Fourier transform (QPFT) to solve the boundary value problems of generalized partial differential equations.     

Continuity of the fractional Hankel wavelet transform on the spaces of type S

In this present article, we study the fractional Hankel transform and its inverse on certain Gel'fand‐Shilov spaces of type S. The continuous fractional wavelet transform is defined involving the fractional Hankel transform. The continuity of fractional Hankel wavelet transform is discussed on Gel'fand‐Shilov spaces of type S. This article goes further to discuss the continuity property of fractional Hankel transform and fractional Hankel wavelet transform on the ultradifferentiable function spaces.     

一个二进小波变换像空间的再生核函数

1 引言小波分析是结合泛函分析、应用数学、逼近论、调和分析、广义函数论等数学知识的结晶,具有深刻的理论意义和广泛的应用范围,被称为"数学显微镜".基于其多分辨分析的特点以及在时、频两域都具有表征信号局部特征的功能,应用它可以解决许多Fourier变换不能解决的难题,为工程应用提供了一种新的、更有效的分析工具[1].     

高维广义积分子波变换及其性质

在Soblev空间上定义了一种新的广义积分子波变换,它包括了通常意义下的子波变换,付氏变换,R-变换,同时改进和延伸了朱在文献[1]中引入的广义积分子波变换等.并且对这种新的子波变换的基本性质进行了研究.     

一维连续小波变换

1引言小波分析是近年来迅速发展起来的一门新兴学科,小波分析最显著的特征是频域和时域具有良好局部化特性,可以观察函数的任意细节,被誉为数学的显微镜．它不仅理论深刻,且理论与应用的发展交织在一起,它成功地应用于信噪分离、图像编码、图像的边缘检测、数据压缩、计算机视觉中的多分辨率分析等领域．     

指数型Radon变换的一些结果

该论文主要研究在平面情形下指数型 Radon 变换的连续性,得到了它的近似反演公式，并对近似反演的数值解法加以改进.借助一些技巧,该文从理论上还建立了精确反演公式，从而推广了古典 Radon 变换的相应结果.     

一种利用多通道小波变换去噪的算法

通过对多通道小波和小波变换以及奇异信号的小波变换特性的讨论和分析,提出了一种利用M道小波变换去噪的算法,并利用此算法对加噪信号进行了模拟实验,实验结果表明,该算法简便,易于编程实现且去噪效果理想。     

About asymptotic formulas for the inverse Radon transform

We give a proof and generalizations of the Gelfand-Graev asymptotic formula (formulated in 1962 for the odd-dimensional inverse Radon transform).     

A new complex approach towards approximate inverse and generalized Radon transform

In this paper, we first consider the framework of Sobolev spaces and derive analytically a reconstruction algorithm by means of the residue theorem of complex analysis, the approximate inverse, Gaussian mollifier and integral equations. And we successfully extend Natterer’s results to the generalized Radon transform with non-uniform attenuation. Finally, we investigate the smoothing properties of the generalized Radon transform.     

Uncertainty principles for the right-sided multivariate continuous quaternion wavelet transform

ABSTRACT

In this paper, we present some new elements of harmonic analysis related to the right-sided multivariate continuous quaternion wavelet transform. The main objective of this article is to introduce the concept of the right-sided multivariate continuous quaternion wavelet transform and investigate its different properties using the machinery of multivariate quaternion Fourier transform. Last, we have proven a number of uncertainty principles for the right-sided multivariate continuous quaternion wavelet transform.     

A characterization of inverse Radon transform on the Laguerre hypergroup

Let K=[0,∞)×R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this note we give another characterization for a subspace of S(K) (Schwartz space) such that the Radon transform Rα on K is a bijection. We show that this characterization is equivalent to that in [M.M. Nessibi, K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997) 337-363]. In addition, we establish an inversion formula of the Radon transform Rα in the weak sense.     

Image inpainting using complex 2-D dual-tree wavelet transform

The dual-tree complex wavelet transform is a useful tool in signal and image processing. In this paper, we propose a dual-tree complex wavelet transform (CWT) based algorithm for image inpainting problem. Our approach is based on Cai, Chan, Shen and Shen’s framelet-based algorithm. The complex wavelet transform outperforms the standard real wavelet transform in the sense of shift-invariance, directionality and anti-aliasing. Numerical results illustrate the good performance of our algorithm.     

Cauchy wavelet transform of ultra-distributions in tube domains

In this paper, Cauchy wavelet transform of ultra-distributions in tube domains is defined and its various properties are studied using the theory of Cauchy integrals and Poisson integrals of ultra-distributions.     

Uncertainty principles for the continuous Hankel Wavelet transform

The aim of this paper is to prove Heisenberg-type uncertainty principles for the continuous Hankel wavelet transform. We also analyse the concentration of this transform on sets of finite measure. Benedicks-type uncertainty principle is given.     

Continuous and discrete Bessel wavelet transforms

Using the theory of Hankel convolution, continuous and discrete Bessel wavelet transforms are defined. Certain boundedness results and inversion formula for the continuous Bessel wavelet transform are obtained. Important properties of the discrete Bessel wavelet transform are given.     

A NOTE ON SINGULAR VALUE DECOMPOSITION FOR RADON TRANSFORM IN R~n

The singular value decomposition is derived when the Radon transform is restricted to functions which are square integrable on the unit ball in Rn with respect to the weight Wλ(x). It fulfilles mainly by means of the projection-slice theorem.The range of the Radon transform is spanned by products of Gegenbauer polynomials and spherical harmonics. The inverse transform of the those basis functions are given. This immediately leads to an inversion formula by series expansion and range characterizations.     

On the boundedness result of wavelet transform associated with fractional Hankel transform

This article is concerned with the study of the continuity of wavelet transform involving fractional Hankel transform on certain function spaces. The n-dimensional boundedness property of the fractional wavelet transform is also discussed on Sobolev type space. Particular cases are also considered.