Stable Morse Decompositions for Piecewise Constant Vector Fields on Surfaces

Abstract

Numerical simulations and experimental observations are inherently imprecise. Therefore, most vector fields of interest in scientific visualization are known only up to an error. In such cases, some topological features, especially those not stable enough, may be artifacts of the imprecision of the input. This paper introduces a technique to compute topological features of user-prescribed stability with respect to perturbation of the input vector field. In order to make our approach simple and efficient, we develop our algorithms for the case of piecewise constant (PC) vector fields. Our approach is based on a super-transition graph, a common graph representation of all PC vector fields whose vector value in a mesh triangle is contained in a convex set of vectors associated with that triangle. The graph is used to compute a Morse decomposition that is coarse enough to be correct for all vector fields satisfying the constraint. Apart from computing stable Morse decompositions, our technique can also be used to estimate the stability of Morse sets with respect to perturbation of the vector field or to compute topological features of continuous vector fields using the PC framework.