Discrete uncertainty principle in quaternion setting and application in signal reconstruction

In this paper, a novel discrete uncertainty principle associated with discrete Quaternion Fourier transform is established to give the relationship between the nonzero numbers of the discrete quaternion-valued signals and their Quaternion Fourier transforms. We obtain that the product of the numbers of nonzero elements of a sequence f(t,s), (t = 0, 1, ..., M - 1, s = 0, 1, ..., N = 1) and its Quaternion Fourier transform is no less than MN and the result is sharp. Then we extend the uncertainty principle of discrete signals proved by Donoho and Starkin to two dimensional case. It suggests how sparsity helps in the recovery of missing frequency. The experimental results on the recovery of Lena also demonstrate the necessity of the two-dimensional discrete uncertainty principle associated with Quaternion Fourier transforms.