# Wigner-Ville distribution associated with the quaternion offset linear canonical transforms

## TL;DR

This paper introduces a new Wigner-Ville distribution associated with the quaternion offset linear canonical transform, combining features of both transforms for advanced signal and image analysis.

## Contribution

It defines the WVD-QOLCT and derives key properties, including inversion, Plancherel, Heisenberg inequality, Lieb's theorem, and Poisson summation formula.

## Key findings

- Derived inversion and Plancherel formulas
- Established Heisenberg inequality and Lieb's theorem
- Provided Poisson summation formula for WVD-QOLCT

## Abstract

The Wigner-Ville distribution (WVD) and quaternion offset linear canonical transform (QOLCT) are a useful tools in signal analysis and image processing. The purpose of this paper is to define the Wigner-Ville distribution associated with quaternionic offset linear canonical transform (WVD-QOLCT). Actually, this transform combines both the results and flexibility of the two transform WVD and QOLCT. We derive some important properties of this transform such as inversion and Plancherel formulas, we establish a version of Heisenberg inequality, Lieb's theorem and we give the Poisson summation formula for the WVD-QOLCT.