Applications of Forman´s discrete Morse theory to topology visualization and mesh compression

Morse theory is a powerful tool for investigating the topology of smooth manifolds. It has been widely used by the computational topology, computer graphics, and geometric modeling communities to devise topology-based algorithms and data structures. Forman introduced a discrete version of this theory which is purely combinatorial. We aim to build, visualize, and apply the basic elements of Forman´s discrete Morse theory. We intend to use some of those concepts to visually study the topology of an object. As a basis, an algorithmic construction of optimal Forman´s discrete gradient vector fields is provided. This construction is then used to topologically analyze mesh compression schemes, such as Edgebreaker and Grow&Fold. In particular, we prove that the complexity class of the strategy optimization of Grow&Fold is MAX-SNP hard.