Algebraic Representation and Geometric Interpretation of Hilbert Transformed Signals

This thesis covers a fundamental problem of local phase based signal processing: the isotropic generalization of the classical one dimensional analytic signal (D. Gabor) to higher dimensional signal domains. The classical analytic signal extends a real valued one dimensional signal to a complex valued signal by means of the classical 1D Hilbert transform. This signal extension enables the complete analysis of local phase and local amplitude information for each frequency component in the sense of Fourier analysis. In case of two dimensional signal domains, e.g. for images, additional geometric information is required to characterize higher dimensional signals locally. The local geometric information is called orientation, which consists of the main orientation and apex angle for two superimposed one dimensional signals. The problem of two dimensional signal analysis is the fact that in general those signals could consist of an unlimited number of superimposed one dimensional signals with individual orientations. Local phase, amplitude and additional orientation information can be extracted by the monogenic signal (M. Felsberg and G. Sommer) which is always restricted to the subclass of intrinsically one dimensional signals, i.e. the class of signals which only make use of one degree of freedom within the embedding signal domain. In case of 2D images the monogenic signal enables the rotationally invariant analysis of lines and edges. In contrast to the 1D analytic signal the monogenic signal extends all real valued signals of dimension $n$ to a $(n1)$ - dimensional vector valued monogenic signal by means of the generalized first order Hilbert transform, which is also known as the Riesz transform. The analytic signal and the monogenic signal show that a direct relation between analytical signals and their algebraic representation exists. This fact has motivated the work and the results of this thesis, namely the extension of the 2D monogenic signal to more general 2D analytic signals, their algebraic representation, and their most geometric embedding. In case of more general 2D signals the geometric algebra will be shown to be a natural representation, and the conformal space as the geometric embedding for the signal interpretation. In this thesis we present 2D analytic signals as generalizations of the 2D monogenic signal which now extend the original 2D signal to a multi-vector valued signal in homogeneous conformal space by means of higher order Hilbert transforms, and by means of a so called hybrid matrix geometric algebra representation. The 2D analytic sign al and the more general multi-vector signal will be interpreted in conformal space which delivers a descriptive geometric interpretation and algebraic embedding of signals. In case of 2D image signals the 2D analytic signal and the multi-vector signal enable the rotationally invariant analysis of lines, edges, corners and junctions in one unified framework. Furthermore, additional local curvature can be determined by first order generalized Hilbert transforms without the need of derivatives. This so called conformal monogenic signal can be defined for any signal domain.

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