Sampling formulas for 2D quaternionic signals associated with various quaternion Fourier and linear canonical transforms

The main purpose of this paper is to study different types of sampling formulas of quaternionic functions, which are bandlimited under various quaternion Fourier and linear canonical transforms. We show that the quaternionic bandlimited functions can be reconstructed from their samples as well as the samples of their derivatives and Hilbert transforms. In addition, the relationships among different types of sampling formulas under various transforms are discussed. First, if the quaternionic function is bandlimited to a rectangle that is symmetric about the origin, then the sampling formulas under various quaternion Fourier transforms are identical. If this rectangle is not symmetric about the origin, then the sampling formulas under various quaternion Fourier transforms are different from each other. Second, using the relationship between the two-sided quaternion Fourier transform and the linear canonical transform, we derive sampling formulas under various quaternion linear canonical transforms. Third, truncation errors of these sampling formulas are estimated. Finally, some simulations are provided to show how the sampling formulas can be used in applications.