共查询到20条相似文献,搜索用时 78 毫秒
1.
In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets. 相似文献
2.
Kit Ian Kou Yan Yang Cuiming Zou 《Mathematical Methods in the Applied Sciences》2017,40(11):3892-3900
The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform cannot both be sharply localized. It plays an important role in signal processing and physics. This paper generalizes the uncertainty principle for measurable sets from complex domain to hypercomplex domain using quaternion algebras, associated with the quaternion Fourier transform. The performance is then evaluated in signal recovery problems where there is an interplay of missing and time‐limiting data. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
3.
Yan Yang 《Integral Transforms and Special Functions》2016,27(3):213-226
The quaternion Fourier transform has been widely employed in the colour image processing. The use of quaternions allow the analysis of colour images as vector fields. In this paper, the right-sided quaternion Fourier transform and its properties are reviewed. Using the polar form of quaternions, two novel uncertainty principles associated with covariance are established. They prescribe the lower bounds with covariances on the products of the effective widths of quaternionic signals in the space and frequency domains. The results generalize the Heisenberg's uncertainty principle to the 2D quaternionic space. 相似文献
4.
Emna Tefjeni 《Integral Transforms and Special Functions》2020,31(8):669-684
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. 相似文献
5.
Mawardi Bahri 《Applied mathematics and computation》2010,216(8):2366-21
In this paper, we generalize the classical windowed Fourier transform (WFT) to quaternion-valued signals, called the quaternionic windowed Fourier transform (QWFT). Using the spectral representation of the quaternionic Fourier transform (QFT), we derive several important properties such as reconstruction formula, reproducing kernel, isometry, and orthogonality relation. Taking the Gaussian function as window function we obtain quaternionic Gabor filters which play the role of coefficient functions when decomposing the signal in the quaternionic Gabor basis. We apply the QWFT properties and the (right-sided) QFT to establish a Heisenberg type uncertainty principle for the QWFT. Finally, we briefly introduce an application of the QWFT to a linear time-varying system. 相似文献
6.
Kit Ian Kou Jianyu Ou Joao Morais 《Mathematical Methods in the Applied Sciences》2016,39(10):2722-2736
In the present paper, we generalize the linear canonical transform (LCT) to quaternion‐valued signals, known as the quaternionic LCT (QLCT). Using the properties of the LCT, we establish an uncertainty principle for the two‐sided QLCT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion‐valued signals in the spatial and frequency domains. It is shown that only a Gaussian quaternionic signal minimizes the uncertainty. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
7.
The uncertainty principle for quaternionic linear operators in quaternionic Hilbert spaces is established, which generalizes the result of Goh-Micchelli. It turns out that there appears an additional term given by a commutator that reflects the feature of quaternions. The result is further strengthened when one operator is self-adjoint, which extends under weaker conditions the uncertainty principle of Dang-Deng-Qian from complex numbers to quaternions. In particular, our results are applied to concrete settings related to quaternionic Fock spaces, quaternionic periodic functions, quaternion Fourier transforms, quaternion linear canonical transforms, and nonharmonic quaternion Fourier transforms. 相似文献
8.
The quaternion Fourier transform (QFT) is one of the key tools in studying color image processing. Indeed, a deep understanding of the QFT has created the color images to be transformed as whole, rather than as color separated component. In addition, understanding the QFT paves the way for understanding other integral transform, such as the quaternion fractional Fourier transform, quaternion linear canonical transform, and quaternion Wigner–Ville distribution. The aim of this paper is twofold: first to provide some of the theoretical background regarding the quaternion bound variation function. We then apply it to derive the quaternion Fourier and linear canonical inversion formulas. Secondly, to provide some in tuition for how the quaternion Fourier and linear canonical inversion theorems work on the absolutely integrable function space. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
9.
Kit Ian Kou Ming‐Sheng Liu Shu‐Zhen Tao 《Mathematical Methods in the Applied Sciences》2016,39(18):5607-5618
The present paper first introduces the notion of quaternion infinite series of positive term and establishes its several tests. Next, we give the definitions of the positive‐definite quaternion sequence and the positive semi‐definite quaternion function, and we extend the classical Herglotz's theorem to the quaternion linear canonical transform setting. Then we investigate the properties of the two‐sided quaternion linear canonical transform, such as time shift characteristics and differential characteristics. Finally, we derive its several basic properties of the quaternion linear canonical transform of a probability measure, in particular, and establish the Bochner–Minlos theorem. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
10.
Mawardi Bahri 《Advances in Applied Clifford Algebras》2016,26(2):513-527
This paper proposes a new uncertainty principle for the two-sided quaternion Fourier transform. This uncertainty principle describes that the spread of a quaternion-valued function and its two-sided quaternion Fourier transform (QFT) are inversely proportional. We obtain a tighter lower bound about the product of the spread of quaternion signal in the QFT domain. As a consequence, we show that the quaternionic Gabor filters minimize the uncertainty. 相似文献
11.
12.
LIU Jianming 《数学年刊B辑(英文版)》2000,21(3):343-350
61. IntroductionLet G be a connected noncompart semisimple Lie group with finite center and K amaimal compact subgroup of G, and X = G/K the associated Riemannian symmetricspace of noncompact type.Let (Vr, f) be an irreducible unitary representation of K, and E' be the homogeneousvector bundle over G/K associated with the given representation f. It is well krmwn that across section j 6 F(E") mad be identified with a vector-vained function f: G - V. whichi. right-K-cowiaat of type TI i.… 相似文献
13.
Eduardo Bayro-Corrochano 《Numerical Algorithms》2005,39(1-3):35-55
This paper presents the theory and practicalities of the quaternion wavelet transform. The contribution of this work is to generalize the real and complex wavelet transforms and to derive for the first time a quaternionic wavelet pyramid for multi-resolution analysis using the quaternion phase concept. The three quaternion phase components of the detail wavelet filters together with a confidence mask are used for the computation of a denser image velocity field which is updated through various levels of a multi-resolution pyramid. Our local model computes the motion by the linear evaluation of the disparity equations involving the three phases of the quaternion detail high-pass filters. A confidence measure singles out those regions where horizontal and vertical displacement can reliably be estimated simultaneously. The paper is useful for researchers and practitioners interested in the theory and applications of the quaternion wavelet transform. 相似文献
14.
We study the windowed Fourier transform in the framework of Clifford analysis, which we call the Clifford windowed Fourier
transform (CWFT). Based on the spectral representation of the Clifford Fourier transform (CFT), we derive several important
properties such as shift, modulation, reconstruction formula, orthogonality relation, isometry, and reproducing kernel. We
also present an example to show the differences between the classical windowed Fourier transform (WFT) and the CWFT. Finally,
as an application we establish a Heisenberg type uncertainty principle for the CWFT. 相似文献
15.
四元数矩阵实表示的基本性质及应用 总被引:2,自引:0,他引:2
郑福 《数学的实践与认识》2009,39(4)
在四元数实矩阵表示的基础上,给出了四元数矩阵的相同表示,利用友向量的概念,给出了这种实表示的性质,并进一步研究了四元数力学中的系列数值计算问题. 相似文献
16.
C. Baccar 《Integral Transforms and Special Functions》2016,27(6):413-429
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. 相似文献
17.
Fethi SOLTANI 《数学研究及应用》2017,37(5):563-576
We recall some properties of the Segal-Bargmann transform; and we establish for this transform qualitative uncertainty principles: local uncertainty principle, Heisenberg uncertainty principle, Donoho-Stark''s uncertainty principle and Matolcsi-Sz\"ucs uncertainty principle. 相似文献
18.
19.
As a time‐shifted and frequency‐modulated version of the linear canonical transform (LCT), the offset linear canonical transform (OLCT) provides a more general framework of most existing linear integral transforms in signal processing and optics. To study simultaneous localization of a signal and its OLCT, the classical Heisenberg's uncertainty principle has been recently generalized for the OLCT. In this paper, we complement it by presenting another two uncertainty principles, ie, Donoho‐Stark's uncertainty principle and Amrein‐Berthier‐Benedicks's uncertainty principle, for the OLCT. Moreover, we generalize the short‐time LCT to the short‐time OLCT. We likewise present Lieb's uncertainty principle for the short‐time OLCT and give a lower bound for its essential support. 相似文献
20.