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We present the first general scheme to describe all four types of characteristic curves of flow fields – stream, path, streak, and time lines – as tangent curves of a derived vector field. Thus, all these lines can be obtained by a simple integration of an autonomous ODE system. Our approach draws on the principal ideas of the recently introduced tangent curve description of streak lines. We provide the first description of time lines as tangent curves of a derived vector field, which could previously only be constructed in a geometric manner. Furthermore, our scheme gives rise to new types of curves. In particular, we introduce advected stream lines as a parameter-free variant of the time line metaphor. With our novel mathematical description of characteristic curves, a large number of feature extraction and analysis tools becomes available for all types of characteristic curves, which were previously only available for stream and path lines. We will highlight some of these possible applications including the computation of time line curvature fields and the extraction of cores of swirling advected stream lines.
We propose a combinatorial algorithm to track critical points of 2D
time-dependent scalar fields. Existing tracking algorithms such as Feature
Flow Fields apply numerical schemes utilizing derivatives of the data,
which makes them prone to noise and involve a large number of computational
parameters. In contrast, our method is robust against noise
since it does not require derivatives, interpolation, and numerical integration.
Furthermore, we propose an importance measure that combines the
spatial persistence of a critical point with its temporal evolution. This
leads to a time-aware feature hierarchy, which allows us to discriminate
important from spurious features. Our method requires only a single,
easy-to-tune computational parameter and is naturally formulated in an
out-of-core fashion, which enables the analysis of large data sets. We apply
our method to a number of data sets and compare it to the stabilized
continuous Feature Flow Field tracking algorithm.