🍩 Database of Original & Non-Theoretical Uses of Topology

(found 2 matches in 0.008716s)

  1. Topology-Preserving Terrain Simplification (2020)

    Ulderico Fugacci, Michael Kerber, Hugo Manet
    Abstract We give necessary and sufficient criteria for elementary operations in a two-dimensional terrain to preserve the persistent homology induced by the height function. These operations are edge flips and removals of interior vertices, re-triangulating the link of the removed vertex. This problem is motivated by topological terrain simplification, which means removing as many critical vertices of a terrain as possible while maintaining geometric closeness to the original surface. Existing methods manage to reduce the maximal possible number of critical vertices, but increase thereby the number of regular vertices. Our method can be used to post-process a simplified terrain, drastically reducing its size and preserving its favorable properties.

  2. Homological Scaffold via Minimal Homology Bases (2021)

    Marco Guerra, Alessandro De Gregorio, Ulderico Fugacci, Giovanni Petri, Francesco Vaccarino
    Abstract The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global features onto individual network components, unless one provides a principled way to make such a choice. In this paper, we apply recent advances in the computation of minimal homology bases to introduce a quasi-canonical version of the scaffold, called minimal, and employ it to analyze data both real and in silico. At the same time, we verify that, statistically, the standard scaffold is a good proxy of the minimal one for sufficiently complex networks.