An Interplay of Wigner-Ville Distribution and 2D Hyper-Complex Quadratic-Phase Fourier Transform

Two-dimensional hyper-complex (Quaternion) quadratic-phase Fourier transforms (Q-QPFT) have gained much popularity in recent years because of their 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 cannot be ruled out. In this paper, we study the two-dimensional hyper-complex (Quaternion) Wigner-Ville distribution associated with the quadratic-phase Fourier transform (WVD-QQPFT) by employing the advantages of quaternion quadratic-phase Fourier transforms (Q-QPFT) and Wigner-Ville distribution (WVD). First, we propose the definition of the WVD-QQPFT and its relationship with the classical Wigner-Ville distribution in the quaternion setting. Next, we investigate the general properties of the newly defined WVD-QQPFT, including complex conjugate, symmetry-conjugation, nonlinearity, boundedness, reconstruction formula, Moyal's formula, and Plancherel formula. Finally, we propose the convolution and correlation theorems associated with WVD-QQPFT.