Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images

In the present work hypercomplex spectral methods of the processing and analysis of images are introduced. The thesis is divided into three main chapters. First the quaternionic Fourier transform (QFT) for 2D signals is presented and its main properties are investigated. The QFT is closely related to the 2D Fourier transform and to the 2D Hartley transform. Similarities and differences of these three transforms are investigated with special emphasis on the symmetry properties. The Clifford Fourier transform is presented as nD generalization of the QFT. Secondly the concept of the phase of a signal is considered. We distinguish the global, the local and the instantaneous phase of a signal. It is shown how these 1D concepts can be extended to 2D using the QFT. In order to extend the concept of global phase we introduce the notion of the quaternionic analytic signal of a real signal. Defining quaternionic Gabor filters leads to the definition of the local quaternionic phase. The relation between signal structure and local signal phase, which is well-known in 1D, is extended to 2D using the quaternionic phase. In the third part two application of the theory are presented. For the image processing tasks of disparity estimation and texture segmentation there exist approaches which are based on the (complex) local phase. These methods are extended to the use of the quaternionic phase. In either case the properties of the complex approaches are preserved while new features are added by using the quaternionic phase.

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