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Persistent 1-Cycles: Definition, Computation, and Its Application

  • Tamal K. Dey
  • Tao HouEmail author
  • Sayan Mandal
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11382)

Abstract

Persistence diagrams, which summarize the birth and death of homological features extracted from data, are employed as stable signatures for applications in image analysis and other areas. Besides simply considering the multiset of intervals included in a persistence diagram, some applications need to find representative cycles for the intervals. In this paper, we address the problem of computing these representative cycles, termed as persistent 1-cycles. The definition of persistent cycles is based on the interval module decomposition of persistence modules, which reveals the structure of persistent homology. After showing that the computation of the optimal persistent 1-cycles is NP-hard, we propose an alternative set of meaningful persistent 1-cycles that can be computed with an efficient polynomial time algorithm. We also inspect the stability issues of the optimal persistent 1-cycles and the persistent 1-cycles computed by our algorithm with the observation that the perturbations of both cannot be properly bounded. We design a software which applies our algorithm to various datasets. Experiments on 3D point clouds, mineral structures, and images show the effectiveness of our algorithm in practice.

Keywords

Persistent homology Persistent cycle Minimal cycle NP-hardness Image segmentation 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringThe Ohio State UniversityColumbusUSA

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