Clifford Fourier Transform on Vector Fields Abstract — Image

processing and computer vision have a lot of useful tools for the analysis of scalar fields. There are robust methods for feature extraction and the computation of derivatives. Furthermore, interpolation and the resulting effects for the application of filter can be analyzed in detail. It seems to be sensible to try to apply these methods to vector fields, too, in order to get a solid theoretical base for feature extraction. Recently, we introduced the Clifford convolution which is an extension of the classical convolution on scalar fields. Clifford convolution is a unified notation for the convolution of scalar and vector fields. It has nice geometric properties which were used for pattern matching on vector fields. In image processing, convolution and Fourier transform are closely connected by the convolution theorem. In this paper, we extend the Fourier transform to general elements of the Clifford Algebra, called multivectors, which include scalars and vectors. The resulting convolution theorems for the Clifford convolution and the derivation theorems are extensions of the theorems for convolution and Fourier transform on scalar fields. The Clifford Fourier transform allows a frequency analysis of vector fields and of the behavior of vector valued filters. In frequency space, vectors are transformed into general mulitvectors of the Clifford Algebra. Many basic vector valued pattern such as source, sink, saddle points and potential vortices can be described by a few multivectors in frequency space. Index Terms — Flow visualization, vector fields, Fourier transform, convolution, Clifford algebr