Gradient-Preserving Cuts for Scalar Representations of Vector Fields
Final Report Abstract
We introduced an approach to represent vector fields (usually resulting from flow simulation or flow measurement approaches) as the (co-)gradient of a scalar field. Since it is known that in general this is impossible for smooth scalar fields, we introduced the concept of gradient-preserving cuts in scalar fields. In particular, we introduced a representation of a 2D steady vector field v by two scalar fields a, b, such that the isolines of a correspond to stream lines of v, and b increases with constant speed under integration of v. This way, we get a direct encoding of stream lines, i.e., a numerical integration of v can be replaced by a local isoline extraction of a. To guarantee a solution in every case, gradientpreserving cuts were introduced such that the scalar fields are allowed to be discontinuous in the values but continuous in the gradient. Along with a piecewise linear discretization and a proper placement of the cuts, the fields a and b can be computed. We showed several evaluations on non-trivial vector fields. Further, we introduced new approaches to visually represent uncertainty in flow ensembles. FTLE (Finite Time Lyapunov Exponent) computation is one of the standard approaches to Lagrangian flow analysis. The main features of interest in FTLE fields are ridges that represent hyperbolic Lagrangian Coherent Structures. FTLE ridges tend to become sharp and crisp with increasing integration time, where the sharpness of the ridges is an indicator of the strength of separation. The additional consideration of uncertainty in flows leads to more blurred ridges in the FTLE fields. There are multiple causes for such blurred ridges: either the locations of the ridges are uncertain, or the strength of the ridges is uncertain, or there is low uncertainty but weak separation. Existing approaches for uncertain FTLE computation are unable to distinguish these different sources of uncertainty in the ridges. We introduced a new approach to define and visualize FTLE fields for flow ensembles. Before computing and comparing FTLE fields for the ensemble members, we compute optimal displacements of the domains to mutually align the ridges of the ensemble members as much as possible. We did so in a way that an explicit geometry extraction and alignment of the ridges is not necessary. The additional consideration of these displacements allows for a visual distinction between uncertainty in ridge location, ridge sharpness, and separation strength. We applied the approach to several synthetic and real ensemble data sets.
Publications
-
Flow Map Processing by Space-Time Deformation Proc. of ISVC (International Symposium on Visual Computing), Springer
T. Wilde, C. Rössl & H. Theisel
-
Lagrangian Q-criterion and Transport of Salt and Temperature. IEEE Scientific Visualization Contest
S. Wolligandt, J. Zimmermann, T. Wilde, M. Motejat & H. Theisel
-
Uncertain Stream Lines. Proc. VMV (Vision, Modeling and Visualization) 2023. The Eurographics Association
J. Zimmermann, M. Motejat C. Rössl & H. Theisel
-
FTLE for Flow Ensembles by Optimal Domain Displacement.
J. Zimmermann, M. Motejat, C. Rössl & H. Theisel
-
Scalar Representation of 2D Steady Vector Fields.
H. Theisel, M. Motejat, J. Zimmermann & C. Rössl
