A Novel Approach to the 2D Analytic Signal

Abstract

The analytic signal of a real signal is a standard concept in 1D signal processing. However, the definition of an analytic signal for real 2D signals is not possible by a straightforward extension of the 1D definition. There rather occur several different approaches in the literature. We review the main approaches and propose a new definition which is based on the recently introduced quaternionic Fourier transform. The approach most closely related to ours is the one by Hahn [8], which defines the analytic signal, which he calls complex signal, to have a single quadrant spectrum. This approach suffers form the fact that the original real signal is not reconstructible from its complex signal. We show that this drawback is cured by replacing the complex frequency domain by the quaternionic frequency domain defined by the quaternionic Fourier transform. It is shown how the new definition comprises all the older ones. Experimental results demonstrate that the new definition of the analytic signal in 2D is superior to the older approaches.

This work was supported by the Studienstiftung des deutschen Volkes (Th.B.) and by the DFG (So-320/2-1) (G.S.).