Simplification and visual effectiveness of Morse connection graphs

Abstract

Vector field visualization has been a popular research topic in the past few decades. Vector field features have traditionally been defined as trajectories with certain special properties. Examples of such features include stationary points, periodic trajectories and separatrices. However, such features are generally fragile with respect to perturbation of the data, or the choice of a numerical method used to approximate the trajectories. Because of this, algorithms to compute them are not robust enough to be practical. An alternative approach uses Morse sets, regions of close to circulating flow, as topological features. Morse sets can be computed using graph algorithms from a finite directed graph representing all trajectories. Algorithms for computing Morse sets are generally more robust since the resulting features are sets of points with properties that can be rigorously verified by an automatic algorithm rather than individual trajectories that can only be approximated. The Morse Connection Graph, whose nodes correspond to Morse sets and arcs represent trajectories that connect them, provides a concise summary of the flow structure. One shortcoming of Morse Connection Graphs (MCG) is the lack of ability to convey information on high-level flow structure in a visually effective manner. This paper describes a method to simplify Morse decompositions by iteratively merging pairs of Morse sets that are adjacent in the Morse Connection Graph (MCG). When Morse sets A and B are merged, they are replaced by a single Morse set, that can be thought of as the union of A, B and all trajectories connecting A and B. Cost criteria for the possible Morse set mergers are evaluated based on resulting size, complexity as well as user input factors. Visual representation of the MCG is improved using a triangulation of the Morse space along with a specialized trajectory visualization algorithm. Smoothing techniques are discussed to make the resulting visualization more effective at conveying information to the user.