A practical approach to two-dimensional scalar topology

Computing and analyzing the topology of scalar fields has proven to be a powerful tool in a wide variety of applications. In recent years the field has evolved from computing contour trees of two-dimensional functions to Reeb graphs of general two-manifolds, analyzing the topology of time-dependent volumes, and finally to creating Morse-Smale complexes of two and three dimensional functions. However, apart from theoretical advances practical applications depend on the development of robust and easy to implement algorithms. The progression from initial to practical algorithms is evident, for example, in the contour tree computation where the latest algorithms consist of no more than a couple of dozens lines of pseudo-code. In this paper we describe a similarly simple approach to compute progressive Morse-Smale complexes of functions over two-manifolds. We discuss compact and transparent data-structures used to compute and store Morse-Smale complexes and demonstrate how they can be used to implement interactive topology based simplification. In particular, we show how special cases arising, for example, from manifolds with boundaries or highly quantized functions are handled effectively. Overall the new algorithm is easier to implement and more efficient both run-time and storage wise than previous approaches by avoiding to refine a given triangulation.