Analyzing real vector fields with clifford convolution and clifford-fourier transform

Abstract

Postprocessing in computational fluid dynamics and processing of fluid flow measurements need robust methods that can deal with scalar and vector fields. While image processing of scalar data is a well-established discipline, there is a lack of similar methods for vector data. This paper surveys a particular approach defining convolution operators on vector fields using geometric algebra. This includes a corresponding Clifford-Fourier transform including a convolution theorem. Finally, a comparison is tried with related approaches for a Fourier transform of spatial vector or multivector data. In particular, we analyze the Fourier series based on quaternion holomorphic functions of Gürlebeck et al. (Funktionentheorie in der Ebene und im Raum, Birkhäuser, Basel, 2006), the quaternion Fourier transform of Hitzer (Proceedings of Function Theories in Higher Dimensions, 2006) and the biquaternion Fourier transform of Sangwine et al. (IEEE Trans. Signal Process. 56(4),1522-1531, 2007). © 2010 Springer-Verlag London Limited.