Computing minimum area homologies

TL;DR

This paper introduces an efficient, provably-guaranteed similarity measure for homologous cycles on surfaces based on minimum area homology, expanding the scope beyond previous homotopy-based measures.

Contribution

The authors develop a novel, computationally efficient method to calculate minimum area homologies between cycles, with broader applicability and provable guarantees.

Findings

The algorithm effectively computes minimum area homologies on various inputs.

It demonstrates broader applicability than previous homotopy-based measures.

The implementation is efficient, leveraging linear algebra tools.

Abstract

Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a similarity measure for any two cycles that are homologous, where we calculate the minimum area of any homology (or connected bounding chain) between the two cycles. The minimum area homology exists for broader classes of cycles than previous measures which are based on homotopy. It is also much easier to compute than previously defined measures, yielding an efficient implementation that is based on linear algebra tools. We demonstrate our algorithm on a range of inputs, showing examples which highlight the feasibility of this similarity measure.