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Persistence diagrams, which summarize the birth and death of homologicalfeatures extracted from data, are employed as stable signatures forapplications in image analysis and other areas. Besides simply considering themultiset of intervals included in a persistence diagram, some applications needto find representative cycles for the intervals. In this paper, we address theproblem of computing these representative cycles, termed as persistent1-cycles, for $\text{H}_1$-persistent homology with $\mathbb{Z}_2$coefficients. The definition of persistent cycles is based on the intervalmodule decomposition of persistence modules, which reveals the structure ofpersistent homology. After showing that the computation of the optimalpersistent 1-cycles is NP-hard, we propose an alternative set of meaningfulpersistent 1-cycles that can be computed with an efficient polynomial timealgorithm. We also inspect the stability issues of the optimal persistent1-cycles and the persistent 1-cycles computed by our algorithm with theobservation that the perturbations of both cannot be properly bounded. Wedesign a software which applies our algorithm to various datasets. Experimentson 3D point clouds, mineral structures, and images show the effectiveness ofour algorithm in practice.

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