Extensions of differential-geometric algorithms for estimation of three-dimensional non-rigid motion and correspondence

The dissertation addresses the problem of non-rigid motion and correspondence estimation in 3D images. It is assumed that no prior information, other than the 3D data itself, is available. The algorithms are further required to be able to perform estimation at arbitrary points on 3D surfaces.; A coherent framework is developed for differential-geometric motion and correspondence estimation under the given assumptions. Our initial experimental evaluation of a number of algorithms suitable for the task led us to the conclusion that the most promising tool for achieving our objectives is the unit normal algorithm. The major effort in the dissertation was put into advancing the algorithmic and mathematical concepts of the unit normal algorithm to new applications, as well as into applying its ideas to alternative strategies.; The following extensions were developed. The new least-squares affine Gaussian curvature algorithm is capable of recovering the full affine motion model at a computational price comparable to that of the unit normal algorithm. The kernel unit normal algorithm extends the previous (affine) algorithm to non-linear models, by utilizing user-supplied kernel functions that allow construction of virtually any motion model for which respective kernels are available. The kernel Gaussian curvature algorithm carries out a similar extension of the affine Gaussian curvature algorithm. The hybrid unit normal/Gaussian curvature algorithm combines the objective functions of the (affine) unit normal and Gaussian curvature algorithms in a weighted fashion. In addition, the technique was developed for enforcing the orthogonal parametrization of surfaces used in the differential-geometric algorithms.; Experimental evaluation of the algorithms was performed on a number of artificial and real shapes. All algorithms are able to recover the synthetic motion models to within 2.5% accuracy.; The least-squares Gaussian curvature algorithm outperforms the unit normal algorithm on artificial shapes with relatively high curvedness; on real shapes the Gaussian curvature algorithm is currently less accurate due to numerical errors in estimation of differential-geometric parameters. The hybrid algorithm exhibits consistent improvement over pure unit normal and Gaussian curvature algorithms. The kernel unit normal algorithm yields higher correspondence estimation accuracy on simple homogeneous polynomial motion but somewhat lower accuracy on real shapes.; Various future extensions of the developed algorithms can lead to significant advances in non rigid motion estimation and other areas in computer vision and kernel methods of machine learning.