Clifford Fourier Transform on Vector Fields

Abstract

Image processing and computer vision have robust methods for feature extraction and the computation of derivatives of scalar fields. Furthermore, interpolation and the effects of applying a filter can be analyzed in detail and can be advantages when applying these methods to vector fields to obtain a solid theoretical basis for feature extraction. We recently introduced the Clifford convolution, which is an extension of the classical convolution on scalar fields and provides a unified notation for the convolution of scalar and vector fields. It has attractive geometric properties that allow pattern matching on vector fields. In image processing, the convolution and the Fourier transform operators are closely related by the convolution theorem and, in this paper, we extend the Fourier transform to include general elements of Clifford Algebra, called multivectors, including scalars and vectors. The resulting convolution and derivative theorems are extensions of those for convolution and the Fourier transform on scalar fields. The Clifford Fourier transform allows a frequency analysis of vector fields and the behavior of vector-valued filters. In frequency space, vectors are transformed into general multivectors of the Clifford Algebra. Many basic vector-valued patterns, such as source, sink, saddle points, and potential vortices, can be described by a few multivectors in frequency space.

Hilbert Transforms in Clifford Analysis

“…It becomes apparent, also from the given examples, that the Clifford analysis framework is most appropriate to develop these multidimensional Hilbert transforms. That Clifford analysis could be a powerful tool in multidimensional signal analysis became already clear during the last decade from the several constructions of multidimensional Fourier transforms with quaternionic or Clifford algebra valued kernels with direct applications in signal analysis and pattern recognition, see [20,21,24,[32][33][34]39] and also the review paper [23] wherein the relations between the different approaches are established. In view of the fact that in the underly-ing paper the interaction of the Clifford-Hilbert transforms with only the standard Fourier transform was considered, their interplay with the various Clifford-Fourier transforms remains an intriguing and promising topic for further research.…”

Quaternion and Clifford Fourier Transforms and Wavelets

“…Together with Bülow and Sommer, Felsberg applied these CFTs to image stucture processing (key-notion: structure multivector) [50,24]. Ebling and Scheuermann [44,43] consequently applied to vector signal processing in two-and three dimensions, respectively, the following twodimensional CFT [78,63,78] to define their Clifford-Fourier transform of threedimensional multivector signals: that means, they researched the properties of F 3 {g}(ω) of (2.1) in detail when applied to full multivector signals g : R 3 → C 3,0 . This included an investigation of the uncertainty inequality for this type of CFT.…”

“…In this context the Clifford Fourier transformations by Felsberg [50] for one-and two-dimensional signals, by Ebling and Scheuermann for twoand three-dimensional vector signal processing [44,43], and by Mawardi and Hitzer for general multivector signals in C 3,0 [78,63,78], and their respective kernels, as already reviewed in Section 2.1, should also be considered.…”

Moment Invariants for the Analysis of 2D Flow Fields

“…To find patterns of different size and orientation one has to perform the filtering multiple times using adjusted filter masks. To speed up the time-intensive convolution step, Ebling and Scheuermann introduced a vector Fourier transform based on Clifford algebra [6]. This reduces the filter operation to a multiplication in frequency domain.…”