SIGGRAPH '92 Proceedings of the 19th annual conference on Computer graphics and interactive techniques
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
Surface simplification using quadric error metrics
Proceedings of the 24th annual conference on Computer graphics and interactive techniques
Constructing Hierarchies for Triangle Meshes
IEEE Transactions on Visualization and Computer Graphics
The tidy set: a minimal simplicial set for computing homology of clique complexes
Proceedings of the twenty-sixth annual symposium on Computational geometry
Reconstructing shapes with guarantees by unions of convex sets
Proceedings of the twenty-sixth annual symposium on Computational geometry
Link Conditions for Simplifying Meshes with Embedded Structures
IEEE Transactions on Visualization and Computer Graphics
Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes
Proceedings of the twenty-seventh annual symposium on Computational geometry
Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes
Proceedings of the twenty-seventh annual symposium on Computational geometry
Multinerves and helly numbers of acyclic families
Proceedings of the twenty-eighth annual symposium on Computational geometry
Linear-size approximations to the vietoris-rips filtration
Proceedings of the twenty-eighth annual symposium on Computational geometry
The simplex tree: an efficient data structure for general simplicial complexes
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes
Computational Geometry: Theory and Applications
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We study the simplification of simplicial complexes by repeated edge contractions. First, we extend to arbitrary simplicial complexes the statement that edges satisfying the link condition can be contracted while preserving the homotopy type. Our primary interest is to simplify flag complexes such as Rips complexes for which it was proved recently that they can provide topologically correct reconstructions of shapes. Flag complexes (sometimes called clique complexes) enjoy the nice property of being completely determined by the graph of their edges. But, as we simplify a flag complex by repeated edge contractions, the property that it is a flag complex is likely to be lost. Our second contribution is to propose a new representation for simplicial complexes particularly well adapted for complexes close to flag complexes. The idea is to encode a simplicial complex K by the graph G of its edges together with the inclusion-minimal simplices in the set difference G - K. We call these minimal simplices blockers. We prove that the link condition translates nicely in terms of blockers and give formulae for updating our data structure after an edge contraction. Finally, we observe in some simple cases that few blockers appear during the simplification of Rips complexes, demonstrating the efficiency of our representation in this context.