Kambhamettu and Goldgof [ 9 ] and Benayoun et al [ 1 ] both propose methods of surface correspondence based on the minimisation of a cost function which involves the difference in the curvature of the surfaces. Both methods are directed at solving the same problem - that of tracking the non-rigid motion of the cardiac ventricular wall in a time sequence of images. As pointed out by Tagare et al [ 14 ], curvature is a rigid invariant of shape and its applicability to general non-rigid correspondence is problematic.
The methods of Scott and Longuet-Higgins [ 11 ] and Shapiro and Brady [ 12 ], which are closely related to each other, describe methods of correspondence between two sets of points, the connectivity of which is not specified. Although the authors present results for correspondence of 2D shape, these methods are extendible to the 3D case. However, the lack of connectivity between points means that illegal (crossing) correspondences between the sets of points are not excluded as solutions. Therefore, these two methods are better suited to the determination of the correspondences arising from a rigid transformation of one pointset onto the other.
We have previously described a method of non-rigid correpondnece in 2D between two closed, pixellated boundaries [ 7 ] [ 6 ]. The method was based on generating sparse polygonal approximations for each shape; no curvature estimation of either boundary was required. Results were presented which demonstrated the ability of this algorithm to provide accurate, non-rigid correpondences. The pair-wise corresponder was used within a framework for automatic landmark generation which demonstrated that landmarks similar to those identified manually were produced by this approach.
Alan Brett