  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, for $\text{H}_1$-persistent homology with $\mathbb{Z}_2$
coefficients. 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.
