He, Jiwen2013-12-022013-12-02August 2012013-08http://hdl.handle.net/10657/489In medical 3D-imaging, one of the main goals of image registration is to accurately compare two observed 3D-shapes. In this dissertation, we consider optimal matching of surfaces by a variational approach based on Hilbert spaces of diffeomorphic transformations. We first formulate, in an abstract setting, the optimal matching as an optimal control problem, where a vector field flow is sought to minimize a cost functional that consists of the kinetic energy and the matching quality. To make the problem computationally accessible, we then incorporate reproducing kernel Hilbert spaces with the Gaussian kernels and weighted sums of Dirac measures. We propose a second order method based the Bellman's optimality principle and develop a dynamic programming algorithm. We apply successfully the second order method to diffeomorphic matching of anterior leaflet and posterior leaflet snapshots. We obtain a quadratic convergence for data sets consisting of hundreds of points. To further enhance the computational efficiency for large data sets, we introduce new representations of shapes and develop a multi-scale method. Finally, we incorporate a stretching fraction in the cost function to explore the elastic model and provide a computationally feasible algorithm including the elasticity energy. The performance of the algorithm is illustrated by numerical results for examples from medical 3D-imaging of the mitral valve to reduce excessive contraction and stretching.application/pdfengThe author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).Diffeomorphic image matchingDeformable surfacesReproducing kernel Hilbert spacesDirac measuresSecond order methodNewton descentDynamic programmingApplied mathematicsA Second Order Variational Approach For Diffeomorphic Matching Of 3D Surfaces2013-12-02Thesisborn digital