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

Alejandro Robles, Mustafa Hajij, Paul Rosen

2018

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 significantly 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.

Paper Citation

in Harvard Style

Robles A., Hajij M. and Rosen P. (2018). The Shape of an Image - A Study of Mapper on Images. In Proceedings of the 13th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2018) - Volume 4: VISAPP; ISBN 978-989-758-290-5, SciTePress, pages 339-347. DOI: 10.5220/0006574803390347

in Bibtex Style

@conference{visapp18,
author={Alejandro Robles and Mustafa Hajij and Paul Rosen},
title={The Shape of an Image - A Study of Mapper on Images},
booktitle={Proceedings of the 13th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2018) - Volume 4: VISAPP},
year={2018},
pages={339-347},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0006574803390347},
isbn={978-989-758-290-5},
}

in EndNote Style

TY - CONF

JO - Proceedings of the 13th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2018) - Volume 4: VISAPP
TI - The Shape of an Image - A Study of Mapper on Images
SN - 978-989-758-290-5
AU - Robles A.
AU - Hajij M.
AU - Rosen P.
PY - 2018
SP - 339
EP - 347
DO - 10.5220/0006574803390347
PB - SciTePress