Visualizing linear neighborhoods in non-linear vector fields

Linear approximation plays an important role in many areas employing numerical algorithms. Particularly in the field of vector field visualization, it is the basis of widely used techniques. In this paper, we introduce two methods to extract areas in two- and three-dimensional vector fields that are connected to linear flow behavior. We propose a region-growing algorithm that extracts the linear neighborhood for a certain position. The region is characterized by linear flow behavior up to a user-defined approximation threshold. While this first method computes the size of a region given the mentioned threshold, our second method computes the quality of a linear approximation given a user-defined n-ring neighborhood. The scalar field resulting from the second method is, therefore, called affine linear approximation error. Isosurfaces of this field show regions of close-to-linear and non-linear flow behavior. We demonstrate the expressiveness and discuss the properties of the extracted regions using analytical examples and several datasets from the domain of computational fluid dynamics (CFD).