Analyzing Real Vector Fields with Clifford Convolution and Clifford-Fourier Transform

Post-processing in computational fluid dynamics as well as processing of fluid flow measurements needs robust methods that can deal with scalar as well as 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., the quaternion Fourier transform of Hitzer et al. and the biquaternion Fourier transform of Sangwine et al. 1 Fluid Flow Analysis Fluid flow, especially of air and water, is usually modelled by the Navier-Stokes equations or simplifications like the Euler equations [1]. The physical fields in this model include pressure, density, velocity and internal energy [17]. These variables depend on space and often also on time. While there are mainly scalar fields, velocity is a vector field and of high importance for any analysis of numerical or physical experiments. Some numerical simulations use a discretization of the spatial domain and calculate the variables at a finite number of positions on a regular lattice (finite difference methods). Other methods split space into volume elements and assume a polynomial solution of a certain degree in each volume element (finite element methods or finite volume methods). These numerical methods create a large amount of data and its analysis, i.e. post-processing, usually uses computer graphics to creWieland Reich, Gerik Scheuermann Computer Science Institute, University of Leipzig, PO Box 100920, 04009 Leipzig, Germany, email: {reich,scheuermann}@informatik.uni-leipzig.de