A new extrinsic sample mean in the shape space with applications to the boundaries of anatomical structures

Shape analysis is of great importance in many fields of medical imaging and computational biology. In this paper, we consider the shape space as the set of smooth planar immersed curves in R(2) (parameterized curves) and, using the property of being isometric to a classical manifold immersed in a Euclidean space, we introduce a new extrinsic sample mean and a new extrinsic variance for a finite set of shapes, which are not necessarily star shaped. This is a fundamental tool in medical image analysis, for instance, to assess uncertainties that arise in locating anatomical structures such as the prostate and the bladder. We apply it to a dataset consisting of parallel planar axial CT sections of human prostate, in order to study the variability between boundaries that have been manually delineated by several observers.