A fast persistence-based segmentation of noisy 2D clouds with provable guarantees

•A 2D cloud of points is automatically segmented without any extra input parameters.•The output is a hierarchy of segmentations ordered by their 1D topological persistence.•The running time is O(nlog n) for a cloud C⊂R2 of n points with any real coordinates.•For any ε-sample of a graph G⊂R2, the boundaries of all regions are 2ε-close to G.•The publicly available C++ code can automatically segment any real-life images. [Display omitted] We design a new fast algorithm to automatically segment a 2D cloud of points into persistent regions. The only input is a dotted image without any extra parameters, say a scanned black-and-white map with almost closed curves or any image with detected edge points. The output is a hierarchy of segmentations into regions whose boundaries have a long enough life span (persistence) in a sequence of nested neighborhoods of the input points. We give conditions on a noisy sample of a graph, when the boundaries of resulting regions are geometrically close to original cycles in the unknown graph.