An efficient computation of handle and tunnel loops via Reeb graphs

Abstract

A special family of non-trivial loops on a surface called handle and tunnel loops associates closely to geometric features of "handles" and "tunnels" respectively in a 3D model. The identification of these handle and tunnel loops can benefit a broad range of applications from topology simplification / repair, and surface parameterization, to feature and shape recognition. Many of the existing efficient algorithms for computing non-trivial loops cannot be used to compute these special type of loops. The two algorithms known for computing handle and tunnel loops provably have a serious drawback that they both require a tessellation of the interior and exterior spaces bounded by the surface. Computing such a tessellation of three dimensional space around the surface is a non-trivial task and can be quite expensive. Furthermore, such a tessellation may need to refine the surface mesh, thus causing the undesirable side-effect of outputting the loops on an altered surface mesh.In this paper, we present an efficient algorithm to compute a basis for handle and tunnel loops without requiring any 3D tessellation. This saves time considerably for large meshes making the algorithm scalable while computing the loops on the original input mesh and not on some refined version of it. We use the concept of the Reeb graph which together with several key theoretical insights on linking number provide an initial set of loops that provably constitute a handle and a tunnel basis. We further develop a novel strategy to tighten these handle and tunnel basis loops to make them geometrically relevant. We demonstrate the efficiency and effectiveness of our algorithm as well as show its robustness against noise, and other anomalies in the input.

Voting for Distortion Points in Geometric Processing

“…For high-genus surfaces, we first convert them into genus-zero surfaces and then find the cut using the above procedure for genus-zero surfaces. The conversion from high genus to zero genus is as follows: 1) Compute the handles of the high-genus surface using [41]; 2) Randomly perturb the generated handles by adding a random offset in terms of the geodesic distance; 3) Cut the high-genus surface along these perturbed handles to create some holes; 4) Fill in the holes to generate a genus-zero surface. Fig.…”

Geometry Aware Tori Decomposition

“…So Dey et al [DFW13] make use of the Reeb graphs, which can be computed efficiently, to avoid the tetrahedralization. By perturbing the cycles inside and outside, [DFW13] uses the winding number between the perturbed cycles to decide if a specific cycle is a tunnel or handle cycle. In this paper, we use the same method for classifying the fundamental cycles computed using our algorithm into tunnels and handles.…”

“…But such methods rely on persistence filtration, which is computation-intensive and inefficient when the input model is large in size. A Reeb graph based method [DFW13] is more efficient than the other methods, as the Reeb graph captures the topological features of the surface mesh while greatly reduces complexity. However, the quality of its results depends on which Morse function is used to construct the Reeb graph.…”

Two Measures for the Homology Groups of Binary Volumes

“…Erickson and Whittlesey founded an algorithm in [EW05] that computes the shortest base for the first homology group for oriented 2-manifolds. Dey et al [DFW13] developed a similar work, but also classifying the 1-holes into tunnels and handles. Chen and Freedman [CF10] measured the 1-holes of a complex by the "length" of the homology generators and they also gave an algorithm to compute the smallest set of homology generators.…”