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.

Surface-bounded growth modeling applied to human mandibles.

From a set of longitudinal three-dimensional scans of the same anatomical structure, we have accurately modeled the temporal shape and size changes using a linear shape model. On a total of 31 computed tomography scans of the mandible from six patients, 14,851 semilandmarks are found automatically using shape features and a new algorithm called geometry-constrained diffusion. The semilandmarks are mapped into Procrustes space. Principal component analysis extracts a one-dimensional subspace, which is used to construct a linear growth model. The worst case mean modeling error in a cross validation study is 3.7 mm.