TY - JOUR
T1 - Reconstructing embedded graphs from persistence diagrams
AU - Belton, Robin Lynne
AU - Fasy, Brittany Terese
AU - Mertz, Rostik
AU - Micka, Samuel
AU - Millman, David L.
AU - Salinas, Daniel
AU - Schenfisch, Anna
AU - Schupbach, Jordan
AU - Williams, Lucia
N1 - Publisher Copyright:
© 2020 Elsevier B.V.
PY - 2020/10
Y1 - 2020/10
N2 - The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space. Recent work has shown that well-chosen (finite) sets of PDs can differentiate between geometric simplicial complexes, providing a method for representing complex shapes using a finite set of descriptors. A related inverse problem is the following: given a set of PDs (or an oracle we can query for persistence diagrams), what is underlying geometric simplicial complex? In this paper, we present an algorithm for reconstructing embedded graphs in Rd (plane graphs in R2) with n vertices from n2−n+d+1 directional (augmented) PDs. Additionally, we empirically validate the correctness and time-complexity of our algorithm in R2 on randomly generated plane graphs using our implementation, and explain the numerical limitations of implementing our algorithm.
AB - The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space. Recent work has shown that well-chosen (finite) sets of PDs can differentiate between geometric simplicial complexes, providing a method for representing complex shapes using a finite set of descriptors. A related inverse problem is the following: given a set of PDs (or an oracle we can query for persistence diagrams), what is underlying geometric simplicial complex? In this paper, we present an algorithm for reconstructing embedded graphs in Rd (plane graphs in R2) with n vertices from n2−n+d+1 directional (augmented) PDs. Additionally, we empirically validate the correctness and time-complexity of our algorithm in R2 on randomly generated plane graphs using our implementation, and explain the numerical limitations of implementing our algorithm.
KW - Persistent homology
KW - Shape reconstruction
KW - Topological descriptors
UR - https://www.scopus.com/pages/publications/85084389503
U2 - 10.1016/j.comgeo.2020.101658
DO - 10.1016/j.comgeo.2020.101658
M3 - Article
AN - SCOPUS:85084389503
SN - 0925-7721
VL - 90
JO - Computational Geometry: Theory and Applications
JF - Computational Geometry: Theory and Applications
M1 - 101658
ER -