  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.
