The Shape Of An Image – A Study Of Mapper On Images

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.

Alejandro Robles, Mustafa Hajij, and Paul Rosen. The Shape Of An Image – A Study Of Mapper On Images. International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications, 2018.

@inproceedings{robles2018shape,
  title = {The Shape of an Image - A Study of Mapper on Images},
  author = {Robles, Alejandro and Hajij, Mustafa and Rosen, Paul},
  booktitle = {International Joint Conference on Computer Vision, Imaging and Computer
    Graphics Theory and Applications},
  series = {VISAPP},
  volume = {4},
  pages = {339--347},
  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.}
}