Local Hypercomplex Signal Representations and Applications

Abstract

The concept of the analytic signal is an important concept in one-dimensional signal theory since it makes the instantaneous amplitude and phase of a real signal directly accessible. Regrettably, there is no straightforward extension of this concept to multidimensional signals, yet. There are rather different approaches to an extension which have different drawbacks. In the first part of this chapter we will review the main approaches and introduce a new one which overcomes some of the problems of the older approaches. The new definition is easily described in the frequency domain. However, in contrast to the 1-D analytic signal we will use the quaternionic frequency domain instead of the complex Fourier domain. Based on the so defined quaternionic analytic signal [36] the instantaneous amplitude and quaternionic phase of a 2-D signal can be defined [34].

This work has been supported by German National Merit Foundation and by DFG Grants So-320-2-1 and So-320-2-2.