Cutting 3D freeform objects with genus-n into single boundary surfaces using topological graphs

In reverse engineering, surface reconstruction methods for freeform objects are based mainly on geometrical criteria, while topological factors are neglected. Current methods use a bottom-up approach based on local parameterization to reconstruct the object from points to a dense mesh and finally to smooth connected patches. This type of reconstruction, however, can have topological problems that might lead to parameterization difficulties, noisy surface behavior and texture anomalies. Such problems are particularly common with concave objects and shapes with complex topology of genus-n. To avoid the above problems, a new global topological approach for cutting objects with genus-n was developed and implemented. The proposed process is based on two main stages: (1) computing iso-curves on the mesh and extracting the topological graph, and (2) cutting the mesh according to the curve cutting guidelines that are calculated from the topological graph. The resulting mesh is a single boundary mesh and therefore can be flattened onto a disk. The time complexity of the algorithm is O(n log(n)). To demonstrate the feasibility of the cutting process, the mesh was also flattened. The flattened mesh can then be used for global parameterization, surface fitting and texture mapping. The robustness of the cutting process is demonstrated on several examples using sculptured freeform objects with genus-n.