The Shape of an Image: A Study of Mapper on Images

TL;DR

This paper explores the application of the Mapper topological construction to images, offering a robust method that generalizes traditional contour trees by only requiring continuity, and provides algorithms and simplification techniques.

Contribution

It introduces a customized Mapper construction for images, along with a fast algorithm and methods to simplify the structure, extending topological analysis beyond Morse functions.

Findings

Mapper can handle a broader class of images than contour trees.

The paper provides a fast algorithm for Mapper computation on images.

It guarantees Mapper's equivalence to contour, join, and split trees for simply connected domains.

Abstract

We study the topological construction called Mapper in the context of simply connected domains, in particular on images. The Mapper construction can be considered as a generalization for contour, split, and joint trees on simply connected domains. A contour tree on an image domain assumes the height function to be a piecewise linear Morse function. This is a rather restrictive class of functions and does not allow us to explore the topology for most real world images. The Mapper construction avoids this limitation by assuming only continuity on the height function allowing this construction to robustly deal with a significant larger set of images. We provide a customized construction for Mapper on images, give a fast algorithm to compute it, and show how to simplify the Mapper structure in this case. Finally, we provide a simple procedure that guarantees the equivalence of Mapper to contour, join, and split trees on a simply connected domain.