The Shape in Medical Imaging (ShapeMI) workshop is part of the world's leading conference on Medical Image Computing and Computer Assisted Interventions (MICCAI). It focuses on advanced methods for shape analysis and geometric learning in medical imaging. This year's Best Paper Award was presented to Martin Hanik, Hans-Christian Hege and Christoph von Tycowicz for their paper "Bi-invariant Two-Sample Tests in Lie Groups for Shape Analysis".
Lie groups play an enormous role in modern geometry and in many different applications. One example is shape analysis: if a geometric shape is defined by a transformation of a reference shape, each shape corresponds to one element of a Lie group. However, Lie groups are curved manifolds. If you want to characterize a set of such elements statistically, you have to use non-euclidean statistics. Corresponding methods for Riemannian manifolds have been developed in recent years. However, they can only be applied to Lie groups if a bi-invariant metric exists - a special case that often does not exist. This means that, in general, these statistical methods do not respect the group structure. In statistical shape analysis, for example, this leads to the fact that the results depend on the choice of the reference shape. This award-winning publication proposes generalizations of statistical tests (two-sample tests) that are compatible with the structure of Lie groups, even in cases where there is no bi-invariant metric. The proposed generalizations are consistent with known expressions in the specific case of flat vector spaces (which was not the case for earlier generalizations for Riemannian manifolds).
The newly invented group test was demonstrated using the example of a morphometric analysis of pathological changes that occur at an intermediate stage of the development of Alzheimer type dementia.
The picture shows:
Group test for differences between mean values of right hippocampi for cognitively normal and impaired subjects; the colors represent p-values (FDR corrected).
