The Quaternionic Fourier Transform and its Applications in Texture Segmentation
by
Thomas Buelow
Christian-Albrechts-Universitaet zu Kiel, Institute fuer Informatik und Praktische Matematik, Kognitive Systeme, Preusserstr. 1-9, 24105 Kiel

The quaternionic Fourier transform (QFT) is introduced as a modification of the 2D complex Fourier transform. We analyze the symmetry properties of the QFT and find that according to these properties the 2D Hartley transform,the complex Fourier transform and the QFT can be regarded as three levels of a hierarchy of 2D harmonic transforms. Based on the QFT a new definition of the analytic signal in 2D is proposed. Quaternionic Gabor filters are introduced and shown to provide an approximation to the 2D quaternionic analytic signal. The local phase of a real 2D signal is defined as the angular phase of its quaternionic Gabor filter response. We show that there is a close relation between the local intrinsically 2D structure of a signal and its local phase as defined above. Applications to the image processing task of texture segmentation are presented.

Date received: February 17, 1999