Wigner Distribution and associated uncertainty principles in the framework of octonion Linear Canonical Transform

TL;DR

This paper introduces the Wigner distribution in the octonion linear canonical transform domain, extending its properties and establishing uncertainty principles relevant for advanced signal and image analysis.

Contribution

It defines and explores the properties of Wigner distribution in octonion LCT domain, including new integral transforms and uncertainty principles.

Findings

Defined 1D and 3D WDOL and established their properties.

Linked WDOL with existing transforms like OLCT and quaternion LCT.

Developed uncertainty principles in the WDOL framework.

Abstract

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