Discrete Quaternion Offset Linear Canonical Transform and Its Application

The offset linear canonical transform (OLCT) extends the linear canonical transform (LCT) by introducing additional offset parameters. These parameters allow the OLCT to perform more flexible operations such as shifts in both time and frequency domains. In this paper, we propose the discrete quaternion offset linear canonical transform (DQOLCT) using quaternion algebra. We explore the fundamental properties of 2D DQOLCT, including the reconstruction formula, the Parseval theorem, the convolution theorem, and the correlation theorem. We also derive an efficient computational method for the 2D DQOLCT. Finally, we demonstrate a new image encryption scheme based on the DQOLCT, which combines with the double random phase encoding (DRPE) and the generalized Arnold transform. Experimental results and security analysis demonstrate the method's feasibility and robustness.