Authors:
Alejandro Robles
;
Mustafa Hajij
and
Paul Rosen
Affiliation:
University of South Florida, United States
Keyword(s):
Mapper, Contour Tree, Topological Data Analysis.
Related
Ontology
Subjects/Areas/Topics:
Computer Vision, Visualization and Computer Graphics
;
Image and Video Analysis
;
Shape Representation and Matching
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.