Plancherel Theorems of Quaternion Hilbert Transforms Associated with Linear Canonical Transforms

The Hilbert transform has wide applications in signal analysis. The quaternion Hilbert transforms associated with the linear canonical transform are recently used to form the quaternion analytic signal. In this paper, some properties of the 2D quaternion Hilbert transforms with the two-sided quaternion linear canonical transforms are investigated, such as the Plancherel theorems, the Parseval identities and the inversion formulas of the Hilbert transforms. In particular, we define the discrete generalized quaternion Hilbert transforms and use them for the color edge detection. The proposed edge detection methods are robust to noise and can simultaneously distinguish edges from the non-edge regions very successfully.