Hypercomplex signals-a novel extension of the analytic signal tothe multidimensional case
Bulow, T.; Sommer, G.
Signal Processing, IEEE Transactions on
Volume 49, Issue 11, Nov 2001 Page(s):2844 - 2852
Digital Object Identifier 10.1109/78.960432
Summary:The construction of Gabor's (1946) complex signal-which is also
known as the analytic signal-provides direct access to a real
one-dimensional (1-D) signal's local amplitude and phase. The complex
signal is built from a real signal by adding its Hilbert transform-which
is a phase-shifted version of the signal-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 phase, which reveals information on the intrinsic dimensionality
of the signal. The different approaches are unified by expressing all of
them as combinations 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 used in texture segmentation
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