Uncertainty principle for the quaternion Fourier transform

The two-sided quaternion Fourier transform was introduced for the analysis of 2D linear time-invariant partial-differential systems. It has been shown to be a powerful tool in image processing. In this paper, several uncertainty inequalities for the two-sided quaternion Fourier transform are given with optimal constants, including the Pitt's inequality, logarithmic uncertainty inequality, Hausdorff-Young inequality, Hirschman's entropy inequality, generalized Heisenberg inequality, local uncertainty principle and qualitative uncertainty principle.