A survey of topology and geometry-constrained segmentation methods in weakly supervised settings

Incorporating prior knowledge into a segmentation process— whether it be geometrical constraints such as landmarks to overcome the issue of weak boundary definition, shape prior knowledge or volume/area penalization, or topological prescriptions in order for the segmented shape to be homeomorphic to the initial one or to preserve the contextual relations between objects— proves to achieve more accurate results, while limiting human intervention. In this contribution, we intend to give an exhaustive overview of these so-called weakly/semi-supervised segmentation methods, following three main angles of inquiry: inclusion of geometrical constraints (landmarks, shape prior knowledge, volume/area penalization, etc.), incorporation of topological constraints (topology preservation enforcement, prescription of the number of connected components/holes, regularity enforcement on the evolving front, etc.), and, lastly, joint treatment of segmentation and registration that can be viewed as a special case of cosegmentation.