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Vector Field Editing and Periodic Orbit Extraction Using Morse Decomposition

Guoning Chen, Konstantin Mischaikow, Robert S. Laramee, Pawel Pilarczyk, and Eugene Zhang
IEEE Transactions on Visualization and Computer Graphics, 2007, Vol 13(1), pp 769-785.

Paper (PDF, 3.94 Mb).

This material is based upon work supported by the National Science Foundation under Grant No. CCF-0546881.

Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).


Design and control of vector fields is critical for many visualization and graphics tasks such as vector field visualization, fluid simulation, and texture synthesis. The fundamental qualitative structures associated with vector fields are fixed points, periodic orbits, and separatrices. In this paper we provide a new technique that allows for the systematic creation and
cancellation of fixed points and periodic orbits. This technique enables vector field design and editing on the plane and surfaces with desired qualitative properties.

The technique is based on Conley theory which provides a unified framework that supports the cancellation of fixed points
and periodic orbits. We also introduce a novel periodic orbit extraction and visualization algorithm that detects, for the first
time, periodic orbits on surfaces. Furthermore, we describe the application of our periodic orbit detection and vector field
simplification algorithm to engine simulation data demonstrating the utility of the approach.

We apply our design system to vector field visualization by creating datasets containing periodic orbits. This helps us
understand the effectiveness of existing visualization techniques. Finally, we propose a new streamline-based technique that allows vector field topology to be easily identified.


1. Two vector fields (top) and their corresponding Entity Connection graphs (bottom). The field in the left contains sources (green dots), sinks (red dots), saddles (blue dots), repelling periodic orbits (green loops), and attacting periodic orbits (red loops). The field in the right is obtained by performing flow smoothing inside the region indicated by the white loop.


2. CFD simulation of the tumble motion inside a combustion chamber (left). Through user-guided smoothing, the major features remain while noises are removed (right).


3. Example fields created using our design system. Our novel streamline-based visualization technique allows periodic orbits and separatrices to be maintained in the display.