Non-rigid Registration by Geometry-Constrained Diffusion

Assume that only partial knowledge about a non-rigid registration is given: certain points, curves or surfaces in one 3D image are known to map to certain points, curves or surfaces in another 3D image. In trying to identify the non-rigid displacement field, we face a generalized aperture problem since along the curves and surfaces, point correspondences are not given. We will advocate the viewpoint that the aperture and the 3D interpolation problem may be solved simultaneously by finding the simplest displacement field. This is obtained by a geometry-constrained diffusion, which in a precise sense yields the simplest displacement field. The point registration obtained may be used for segmentation, growth modeling, shape analysis or kinematic interpolation. The algorithm applies to geometrical objects of any dimensionality. We may thus keep any number of fiducial points, curves and/or surfaces fixed while finding the simplest registration. Examples of inferred point correspondences in a synthetic example and a longitudinal growth study of the human mandible are given.