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 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 , 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.
Unable to display preview. Download preview PDF.
- 1.Th. Bülow. Hypercomplex Spectral Signal Representations for Image Processing and Analysis. PhD thesis, University of Kiel, 1999. to appear.Google Scholar
- 2.Th. Bülow and G. Sommer. Das Konzept einer zweidimensionalen Phase unter Verwendung einer algebraisch erweiterten Signalrepräsentation. In E. Paulus, F. M. Wahl (Eds.), 19. DAGM Symposium Mustererkennung, Braunschweig, pages 351–358. Springer-Verlag, 1997.Google Scholar
- 3.Th. Bülow and G. Sommer. Multi-dimensional signal processing using an algebraically extended signal representation. In G. Sommer, J.J. Koenderink (Eds.), Proc. Int. Workshop on Algebraic Frames for the Perception-Action Cycle, Kiel, LNCS vol. 1315, pages 148–163. Springer-Verlag, 1997.CrossRefGoogle Scholar
- 4.Th. Bülow and G. Sommer. Quaternionic Gabor filters for local structure classification. In 14th International Conference on Pattern Recognition, August 17—20 1998, Brisbane, Australia, 1998. 808–810.Google Scholar
- 5.T.A. Ell. Quaternion Fourier transforms for analysis of 2-dimensional linear timeinvariant partial-differential systems. In Proc. 32nd IEEE Conf. on Decision and Control, San Antonio, TX, USA, 15—17 Dec., pages 1830–1841, 1993.Google Scholar
- 6.D. Gabor. Theory of communication. Journal of the IEE, 93:429–457, 1946.Google Scholar
- 7.G.H. Granlund and H. Knutsson. Signal Processing for Computer Vision. Kluwer Academic Publishers, 1995.Google Scholar