Summary: A GENUS BOUND FOR DIGITAL IMAGE BOUNDARIES
LOWELL ABRAMS AND DONNIELL E. FISHKIND
Abstract. Shattuck and Leahy  conjectured--and Abrams, Fishkind, and Priebe ,
proved--that the boundary of a digital image is topologically equivalent to a sphere if and only
if certain related foreground and background graphs are both trees. In this manuscript we extend
this result by proving upper and lower bounds on digital image boundary genus in terms of the
foreground and background graphs, and we show that these bounds are best possible. Our results
have current application to topology correction in medical imaging.
Key words. digital image, digital topology, combinatorial topology, surface.
AMS subject classifications. 05C10, 57M15.
1. Overview. Digital topology is an area of great theoretical interest having the
additional bonus of significant application in imaging science and related areas. Our
results are mathematical--the notation and setting are detailed in Section 2--but we
begin with a brief description of a current application.
The human cerebral cortex, when viewed as closed at the brain stem, is topo-
logically like a sphere. Magnetic resonance imaging (MRI) can differentiate between
tissue that is interior to the cerebral cortex and tissue that is exterior to the cere-
bral cortex. Because of the finiteness of resolution, what is generated by MRI is a
3-dimensional array of cubes, each cube classified by MRI as "foreground" (tissue in-
terior to the cerebral cortex) or "background" (tissue exterior to the cerebral cortex),