八元数线性正则变换框架下的维格纳分布及其测不准原理
Wigner Distribution and associated uncertainty principles in the framework of octonion Linear Canonical Transform
摘要：
八元数傅里叶变换（OFT）的最新推广是八元数线性正则变换（OLCT），由于其在彩色图像和信号处理中的应用而在当今时代变得流行。另一方面，也不能排除Wigner分布在信号和图像分析中的应用。本文在八元数线性正则变换域（WDOL）中引入了一种新的积分变换，即Wigner分布。首先提出了一维WDOL（1DWDOL）的定义，并将其与1DOLCT和1DOFT的关系进行了推广。然后探讨了1DWDOL的几个重要性质，如重构公式、Raylestern定理。其次，我们介绍了三维WDOL（3DWDOL）的定义，并建立了它与四元数LCT域WD（WDQLCT）和LCT域3DWD（3DWDLCT）的关系。然后研究了与3DWDOL相关的重构公式、Raylestern定理和Riemann Lebesgue引理等性质。本文的关键在于发展著名的测不准原理，包括与WDOL相关的HeisenbergsUP、LogarithmicUP和HausdorffYoung不等式
英文摘要：
The most recent generalization of octonion Fourier transform (OFT) is the octonion linear canonical transform (OLCT) that has become popular in present era due to its applications in color image and signal processing. On the other hand the applications of Wigner distribution (WD) in signal and image analysis cannot be excluded. In this paper, we introduce novel integral transform coined as the Wigner distribution in the octonion linear canonical transform domain (WDOL). We first propose the definition of the one dimensional WDOL (1DWDOL), we extend its relationship with 1DOLCT and 1DOFT. Then explore several important properties of 1DWDOL, such as reconstruction formula, Rayleighs theorem. Second, we introduce the definition of three dimensional WDOL (3DWDOL) and establish its relationships with the WD associated with quaternion LCT (WDQLCT) and 3DWD in LCT domain (3DWDLCT). Then we study properties like reconstruction formula, Rayleighs theorem and Riemann Lebesgue Lemma associated with 3DWDOL. The crux of this paper lies in developing well known uncertainty principles (UPs) including Heisenbergs UP, Logarithmic UP and Hausdorff Young inequality associated with WDOL
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