Convolution and Correlation Theorems for Wigner-Ville Distribution Associated with the Quaternion Offset Linear Canonical Transform

TL;DR

This paper introduces the Wigner-Ville Distribution associated with the quaternion offset linear canonical transform (WVD-QOLCT), establishing its properties and related convolution and correlation theorems for advanced signal and image analysis.

Contribution

It defines WVD-QOLCT, derives its key properties, and develops new convolution and correlation theorems, extending previous results for QWVD and WVD-QLCT.

Findings

WVD-QOLCT properties such as nonlinearity and orthogonality are established.

New convolution and correlation operators for WVD-QOLCT are proposed.

Theorems generalize existing results for QWVD and WVD-QLCT.

Abstract

The quaternion offset linear canonical transform(QOLCT) has gained much popularity in recent years because of its applications in many areas, including color image and signal processing. At the same time the applications of Wigner-Ville distribution (WVD) in signal analysis and image processing can not be excluded. In this paper we investigate the Winger-Ville Distribution associated with quaternion offset linear canonical transform (WVD-QOLCT). Firstly, we propose the definition of the WVD-QOLCT, and then several important properties of newly defined WVD-QOLCT, such as nonlinearity, bounded, reconstruction formula, orthogonality relation and Plancherel formula are derived. Secondly a novel canonical convolution operator and a related correlation operator for WVD-QOLCT are proposed. Moreover, based on the proposed operators, the corresponding generalized convolution, correlation theorems are studied.We also show that the convolution and correlation theorems of the QWVD and WVD-QLCT can be looked as a special case of our achieved results.