The Φ-Operator: A Rotation Operator For Flow Analysis and Visualization

A prevalent strategy in vector field data analysis is to extract and classify the integral curves derived from flow data. With the exception of topology-based methods, many of these classifications depend on the choice of algorithm thresholds. However, topology-based methods have yet to be fully developed for time-varying vector field analysis. In this paper, we introduce a novel descriptor, called the Φ−operator for flow data analysis that is based on the signed curvature of an integral curve. With this descriptor, the vector-valued data analysis is reduced to a scalar field analysis problem. We present the definition and computation of the Φ field and show how to utilize it to achieve a flow domain partitioning based on the behaviors of the integral curves seeded at different spatial locations. A unified framework for the computation of the Φ field and its gradient field, |∇Φ|, is introduced. This framework is easy to implement and can be applied to the characterization of streamlines, pathlines, and streaklines based on their different geometric attributes. A new flow feature, referred to as cycloid boundary curves, corresponding to the ridges of the |∇Φ| field is revealed that classifies the flow domain into regions with different rotational behavior. We apply this framework to the analysis and visualization of a number of synthetic and simulation flow data sets. Finally, we provide a detailed comparison of this new structure with a number of wellknown flow features, including vector field topology, vortex regions, FTLE ridges, and singularity paths. Our comparison shows that collectively the Φ and |∇Φ| fields encode more flow information than previous individual feature descriptors alone. ∗This work was supported by the National Science Foundation, IIS-1352722. †Guoning Chen and Lei Zhang are with University of Houston ‡ David Thompson and Adrian Sescu are with Mississippi State University §Robert S. Laramee is with the Swansea University, UK.