Curvature-Based Approach to Point Correspondence Recovery in Conformal Nonrigid Motion

Abstract

This paper describes a novel method for the estimation of point correspondences on a surface undergoing conformal nonrigid motion based on changes in its Gaussian curvature. The use of Gaussian curvature in nonrigid motion analysis is justified by its invariancy towards rigid motion and the type of surface parameterization. Input to the algorithm is the set of 3D points before and after the motion. We deal with a restricted class of nonrigid motion called conformal motion. In conformal motion, the stretching is equal in all directions, but different at different points. Small motion assumption is utilized to hypothesize all possible point correspondences. Curvature changes are then computed for each hypothesis. Finally, the error between computed curvature changes and the one predicted by the conformal motion assumption is calculated. The hypothesis with the smallest error gives point correspondences between consecutive time frames. The algorithm requires calculation of the Gaussian curvature at points on surface before and after the motion. It also requires computation of the coefficients of the first fundamental form at points on surface before the motion. Estimation of point correspondences and stretching can also be refined so as to reduce the error introduced by sampling. Simulations are performed on an ellipsoidal data to illustrate performance and accuracy of derived algorithms. Then, the proposed algorithm is applied to volumetric CT data of the left ventricle (LV) of a dog′s heart. Stretching of the LV wall during its expansion and contraction phases is depicted along with the estimated point correspondences. Stretching comparisons are made between the normal and abnormal LV.