Hypercomplex signals - A novel extension of the analytic signal to the multidimensional case

The construction of Gabor's complex signal-which is also known as the analy tic signal-provides direct access to a real one-dimensional (I-D) signal's local amplitude and phase. The complex signal is built from a real signal b y adding its Hilbert transform-which is a phase-shifted version of the sign al-as an imaginary part to the signal. Since its introduction, the complex signal has become an important tool in signal processing, with applications , for example, in narrowband communication. Different approaches to an n-D analytic or complex signal have been proposed in the past. We review these approaches and propose the hypercomplex signal as a novel extension of the complex signal to n-D. This extension leads to a new definition of local ph ase, which reveals information on the intrinsic dimensionality of the signa l. The different approaches are unified by expressing all of them as combin ations of the signal and its partial and total Hilbert transforms. Examples that clarify how the approaches differ in their definitions of local phase and amplitude are shown. An example is provided for the two-dimensional (2 -D) hypercomplex signal, which shows how the novel phase concept can be use d in texture segmentation.