The tracking of boundaries in multidimensional medical images

We became interested in the problems associated with boundary tracking in multidimensional spaces due to our involvement in medical imaging. We give as background our motivation from medicine. We define objects and their surfaces in discrete three-dimensional space and the state of the art in three-dimensional boundary tracking. Discussed are generalizations, conjectures, and open problems. Finally, we discuss what seems to us a suitable mathematical framework for future research in this area.