Time-Dependent Flow seen through Approximate Observer Killing Fields

Markus Hadwiger, Matej Mlejnek, Thomas Theußl, Peter Rautek

Research output: Contribution to journalArticlepeer-review

24 Scopus citations


Flow fields are usually visualized relative to a global observer, i.e., a single frame of reference. However, often no global frame can depict all flow features equally well. Likewise, objective criteria for detecting features such as vortices often use either a global reference frame, or compute a separate frame for each point in space and time. We propose the first general framework that enables choosing a smooth trade-off between these two extremes. Using global optimization to minimize specific differential geometric properties, we compute a time-dependent observer velocity field that describes the motion of a continuous field of observers adapted to the input flow. This requires developing the novel notion of an observed time derivative. While individual observers are restricted to rigid motions, overall we compute an approximate Killing field, corresponding to almost-rigid motion. This enables continuous transitions between different observers. Instead of focusing only on flow features, we furthermore develop a novel general notion of visualizing how all observers jointly perceive the input field. This in fact requires introducing the concept of an observation time, with respect to which a visualization is computed. We develop the corresponding notions of observed stream, path, streak, and time lines. For efficiency, these characteristic curves can be computed using standard approaches, by first transforming the input field accordingly. Finally, we prove that the input flow perceived by the observer field is objective. This makes derived flow features, such as vortices, objective as well.

Original languageEnglish (US)
Article number8440037
Pages (from-to)1257-1266
Number of pages10
JournalIEEE Transactions on Visualization and Computer Graphics
Issue number1
StatePublished - Jan 2019


  • Flow visualization
  • Killing vector fields
  • Lie derivatives
  • infinitesimal isometries
  • objectivity
  • observer frames of reference

ASJC Scopus subject areas

  • Software
  • Signal Processing
  • Computer Vision and Pattern Recognition
  • Computer Graphics and Computer-Aided Design


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