Moment Invariants for 2D Flow Fields via Normalization in Detail

The analysis of 2D flow data is often guided by the search for characteristic structures with semantic meaning. One way to approach this question is to identify structures of interest by a human observer, with the goal of finding similar structures in the same or other datasets. The major challenges related to this task are to specify the notion of similarity and define respective pattern descriptors. While the descriptors should be invariant to certain transformations, such as rotation and scaling, they should provide a similarity measure with respect to other transformations, such as deformations. In this paper, we propose to use moment invariants as pattern descriptors for flow fields. Moment invariants are one of the most popular techniques for the description of objects in the field of image recognition. They have recently also been applied to identify 2D vector patterns limited to the directional properties of flow fields. Moreover, we discuss which transformations should be considered for the application to flow analysis. In contrast to previous work, we follow the intuitive approach of moment normalization, which results in a complete and independent set of translation, rotation, and scaling invariant flow field descriptors. They also allow to distinguish flow features with different velocity profiles. We apply the moment invariants in a pattern recognition algorithm to a real world dataset and show that the theoretical results can be extended to discrete functions in a robust way.