Surface-based computation of the Euler characteristic in the cubical grid
| Autor: | Paola Magillo, Lidija Čomić |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Surface (mathematics)
Computation Boundary (topology) 010103 numerical & computational mathematics 02 engineering and technology Digital topology Cubical grid Euler characteristic Discrete Gauss–Bonnet theorem 01 natural sciences symbols.namesake Euler characteristic 0202 electrical engineering electronic engineering information engineering Discrete Gauss–Bonnet theorem 0101 mathematics Mathematics Boundary cell Homotopy Mathematical analysis Digital topology Computer Graphics and Computer-Aided Design Manifold Vertex (geometry) Cubical grid Modeling and Simulation symbols 020201 artificial intelligence & image processing Geometry and Topology Software MathematicsofComputing_DISCRETEMATHEMATICS |
| Popis: | For well-composed (manifold) objects in the 3D cubical grid, the Euler characteristic is equal to half of the Euler characteristic of the object boundary, which in turn is equal to the number of boundary vertices minus the number of boundary faces. We extend this formula to arbitrary objects, not necessarily well-composed, by adjusting the count of boundary cells both for vertex- and for face-adjacency. We prove the correctness of our approach by constructing two well-composed polyhedral complexes homotopy equivalent to the given object with the two adjacencies. The proposed formulas for the computation of the Euler characteristic are simple, easy to implement and efficient. Experiments show that our formulas are faster to evaluate than the volume-based ones on realistic inputs, and are faster than the classical surface-based formulas. |
| Databáze: | OpenAIRE |
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