On Two-Dimensional Quaternion Wigner-Ville Distribution

We present the two-dimensional quaternion Wigner-Ville distribution (QWVD). The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the classical Wigner-Ville distribution definition. Based on the properties of quaternions and the QFT kernel we obtain three types of the QWVD. We discuss some useful properties of various definitions for the QWVD, which are extensions of the classical Wigner-Ville distribution properties. 1. Introduction The classical Wigner-Ville distribution (WVD) or Wigner-Ville transform (WVT) is an important tool in the time-frequency signal analysis. It was first introduced by Eugene Wigner in his calculation of the quantum corrections of classical statistical mechanics. It was independently derived again by J. Ville in 1948 as a quadratic representation of the local time-frequency energy of a signal. In [1–3], the authors introduced the WVT and established some important properties of the WVT. The transform is then extended to the linear canonical transform (LCT) domain by replacing the kernel of the classical Fourier transform (FT) with the kernel of the LCT in the WVD domain [4]. As a generalization of the real and complex Fourier transform (FT), the quaternion Fourier transform (QFT) has been of interest to researchers for some years. A number of useful properties of the QFT have been found including shift, modulation, convolution, correlation, differentiation, energy conservation, uncertainty principle, and so on. Due to the noncommutative property of quaternion multiplication, there are three different types of two-dimensional QFTs. These three QFTs are so-called a left-sided QFT, a right-sided QFT, and a two-sided QFT, respectively (see, e.g., [5–9]). In [10, 11], special properties of the asymptotic behaviour of the right-sided QFT are discussed and generalization of the classical Bohner-Millos theorems to the framework of quaternion analysis is established. Many generalized transforms are closely related to the QFTs, for example, the quaternion wavelet transform, fractional quaternion Fourier transform, quaternion linear canonical transform, and quaternionic windowed Fourier transform [12–18]. Based on the QFTs, one also may extend the WVD to the quaternion algebra while enjoying similar properties as in the classical case. Therefore, the main purpose of this paper is to propose a generalization of the classical WVD to quaternion algebra, which we call the quaternion Wigner-Ville distribution (QWVD). Our generalization is constructed by substituting the