A Clifford Fourier Transform for Vector Field Analysis and Visualization

Vector fields arise in many areas of computational science and engineering. For effective visualization of vector fields it is necessary to identify and extract important features inherent in the data, defined by filters that characterize certain patterns. Our prior approach for vector field analysis used the Clifford Fourier transform for efficient pattern recognition for vector field data defined on regular grids [1, 2]. Using the frequency domain, correlation and convolution of vectors can be computed as a Clifford multiplication, enabling us to determine similarity between a vector field and a pre-defined pattern mask (e.g., for critical points). Moreover, compression and spectral analysis of vector fields is possible using this method. Our approach in its current form only applies to rectilinear grids. We combine this approach with a fast Fourier transform to handle scalar data on arbitrary grids [3]. Our extension enables us to provide a feature-based visualization of vector field data defined on arbitrary grids, or completely scattered data. Besides providing the theory of Clifford Fourier transform for unstructured vector data, we explain how efficient pattern matching and visualization of various selectable features can be performed efficiently.