Elastic symmetry analysis of anatomical structures

We introduce a framework for analyzing symmetry of 3D anatomical structures using elastic deformations of their boundaries (surfaces). The basic idea is to define a space of parameterized surfaces and to compute geodesic paths between the objects and their arbitrary reflections using a Riemannian structure. Elastic matching, based on optimal (non-linear) re-parameterizations (grid deformations) of surfaces, provides a better registration of points across shapes, as compared to the commonly-used linear registrations. A crucial step of orientation alignment, akin to finding planes of symmetry, is performed as a search for shortest geodesic paths. This framework is fully automatic and provides a measure of symmetry, the nearest symmetric shape and the optimal deformation to make an object symmetric. We demonstrate this framework through multiple toy examples on simple and complicated surfaces. We also explore the use of symmetry analysis in differentiating between healthy and subjects with Attention Deficit Hyper-activity Disorder.