Wigner distribution and associated uncertainty principles in the framework of octonion linear canonical transform

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 (1D-WDOL), we extend its relationship with 1D-OLCT and 1D-OFT. Then explore several important properties of 1D-WDOL, such as reconstruction formula, Rayleigh’s theorem. Second, we introduce the definition of three dimensional WDOL (3D-WDOL) and establish its relationships with the WD associated with quaternion LCT (WD-QLCT) and 3D-WD in LCT domain (3D-WDLCT). Then we study properties like reconstruction formula, Rayleigh’s theorem and Riemann–Lebesgue Lemma associated with 3D-WDOL. The crux of this paper lies in developing well known uncertainty principles (UPs) including Heisenberg’s UP, Logarithmic UP and Hausdorff–Young inequality associated with WDOL.