Persistence-based Handle and Tunnel Loops Computation Revisited for Speed Up Tamal K. Dey ∗ ,a , Kuiyu Li a a Department of Computer Science and Engineering, The Ohio State University 2015 Neil Ave, Columbus, OH 43210, USA Abstract Loops in surfaces associated with topological features such as handles and tunnels are important entities in many applications including surface parameterization, feature identification, and topological simplification. Recently, a persistent homology based algorithm has been proposed to compute them. The algorithm has several advantages including its simplicity, combinatorial nature and independence from computing other extra structures. In this paper, we propose changes to this loop computation algorithm based on some novel observations. These changes reduce the computation time of the algorithm dramatically. In particular, our experimental results show that the suggested changes achieve considerable speed up for large data sets without sacrificing loop qualities. Key words: Shape analysis, topology, loops in surfaces, homology, persistent homology, topological persistence 1. Introduction Computations of meaningful non-trivial loops in surfaces is a fundamental problem that crops up in various applica- tions such as surface parameterization [2, 16, 17, 22], feature identification [1, 4, 9, 20], topological repair and simplifica- tion [5, 18, 24, 25]. As a result, considerable amount of re- search has been devoted in recent years to compute such loops [9, 13, 14, 15, 27]. In most applications, the loops should be linked to the topol- ogy of the surface and be small in size. To this goal, Dey, Li, Sun, and Cohen-Steiner [10] recently proposed a persistence- based algorithm to compute a special class of loops called han- dle and tunnel loops. This class of loops introduced in [9] cap- tures the intuitive notion of ‘handles’ and ‘tunnels’ in a shape. To be more precise, let M denote a connected closed surface sitting in R 3 . The handle and tunnel loops in M are defined in terms of the first homology group of M and its embeddings in R 3 . One can intuitively think that a loop is a handle loop if it bounds a disk in the interior of M whereas it is a tunnel loop if it bounds a disk in the exterior of M. Figure 1 shows such loops in a 3D model Kitten and an iso-surface Atom. Persistent homology introduced by Edelsbrunner, Letscher, and Zomorodian [12] perfectly fits the definition of handle and tunnel loops. Consequently, they could be computed with the persistence algorithm easily. Dey et al. [10] brought geome- try into the loop computation by incorporating geodesic dis- tances into the persistence algorithm. Of course, the output loops may not be optimal geometrically, but they are small as ∗ Phone: 614-292-3563, Fax: 614-292-2911 Email addresses: tamaldey@cse.ohio-state.edu (Tamal K. Dey), liku@cse.ohio-state.edu (Kuiyu Li) Figure 1: Handle (green) and tunnel (red) loops in 3D model Kitten and iso- surface Atom. empirical results confirm. One advantage of the persistence- based algorithm is that it is mostly combinatorial in nature and thereby avoids costly, error-prone numerical computations. Furthermore, unlike many previous methods, this algorithm does not require computing any extra data structures such as Reeb graphs [4, 6, 21], medial axes [25], or curve skeletons [9]. In the persistence algorithm, a core component is a sim- plex pairing algorithm which pairs simplices from an ordered sequence of simplices called filtration of a simplicial com- plex [12, 26]. This Pairing algorithm is used in [10] to compute handle and tunnel loops. A simplex is called either positive if it creates a cycle or negative if it destroys a cycle when it is added according to the order in the filtration. A negative p-simplex σ is always paired with a unique positive ( p - 1)-simplex σ ′ where σ kills a ( p - 1)-cycle created by σ ′ . Suppose that a surface complex M along with complexes that tessellate its in- terior and exterior volumes are given. Denote the union of the surface and volume complexes as K. A handle or tunnel loop is obtained when a volume triangle in K \ M is paired with a pos- Preprint submitted to Computers and Graphics March 17, 2009