A Novel Approach to the 2D Analytic Signal

The analytic signal of a real signal is a standard concept in 1D signal processing. However, the deenition of an analytic signal for real 2D signals is not possible by a straightforward extension of the 1D deenition. There rather occur several diierent approaches in the literature. We review the main approaches and propose a new deenition which is based on the recently introduced quaternionic Fourier transform. The approach most closely related to ours is the one by Hahn 8], which de-nes the analytic signal, which he calls complex signal, to have a single quadrant spectrum. This approach suuers 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 deened by the quaternionic Fourier transform. It is shown how the new deenition comprises all the older ones. Experimental results demonstrate that the new deenition of the analytic signal in 2D is superior to the older approaches.