The shape of an image: A study of mapper on images

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.

Downloads

Download the Paper Download the BiBTeX

Citation

Alejandro Robles, Mustafa Hajij, and Paul Rosen. The shape of an image: A study of mapper on images. In International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications, 2018.

Bibtex


@inproceedings{Robles.2018.VISAPP,
  title = {The shape of an image: A study of mapper on images},
  author = {Alejandro Robles and Mustafa Hajij and Paul Rosen},
  booktitle = {International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications},
  year = {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.}
}