3D nonrigid motion analysis under small deformations

Recovery of motion parameters and point correspondences is a fundamental problem in computer vision. Although a great deal of research has been done solving rigid motion, nonrigid motion analysis has only recently been addressed, and is gaining interest due to its wide range of applications. This thesis introduces a novel method for estimating motion parameters and point correspondences between 2D curves and 3D surfaces under small nonrigid deformations. We use curvature to track point correspondences and motion parameters between 2D curves under small deformations. Curvature is invariant to the type of parameterization used, and to rigid transformations. These properties help us to deal with purely nonrigid motion, as curvature is not affected by translation/rotation of a given point. The motion parameter in this analysis is a point function, called the displacement function. In this work, it was defined as a polynomial function of a point on the curve under motion. Simulations are performed for illustrating accuracy of the algorithm's estimation by generating deformations on different curves. Real data experiments are also performed, using MR grid data, breast contours from mammogram images and contours of the left ventricle of heart. Three approaches are presented in this thesis for tracking point correspondences, and estimating motion parameters in 3D nonrigid motion. Unlike the 2D case, the motion parameter used here is a vector point function (not scalar), also called displacement function. It it represents both direction and magnitude of deformation at a given point. The first approach involves the use of changes in the "discriminant" for nonrigid motion analysis. Discriminant is defined as the infinitesimal area of surface at a given point. The second approach involves the use of changes in the unit normal at any given point on a surface under deformation. The third approach uses the challenges in Gaussian curvature along with the changes in unit normal direction. In all these approaches, the displacement function can be any point function representing the deformation involved. In our work, we have chosen a polynomial vector function to approximate the deformation. Simulations are performed for each approach by generating nonrigid motion on an ellipsoidal data set to illustrate performance and accuracy of the derived algorithms. Then, the algorithm is tested on a sequence of facial range images. A total of 16 sets of images were used in this experiment, where eight persons have generated two expressions each. We demonstrate the correct correspondence recovery by tracking points on the face during each facial expression. The correspondences estimation is also evaluated by comparing the algorithm's output against manual tracking of feature points by different users. In addition, nonrigid motion segmentation is successfully performed on these images. The motion parameters generated by the algorithm have also been used to demonstrate the abnormality in cardiac motion from CT images.