Ph.D Thesis


Ph.D StudentNaitsat Alexander
SubjectMappings of Deformable Geometric Data and their
Applications in Two and Three Dimensions
DepartmentDepartment of Electrical and Computers Engineering
Supervisors PROFESSOR EMERITUS Yehoshua Zeevi
PROF. Emil Saucan
Full Thesis textFull thesis text - English Version


Abstract

 This thesis addresses the problem of how to map two and three dimensional objects under certain geometric constraints, while incurring minimal shape distortion. We refer to it as to the optimal mapping problem. Our goal is to analyze the optimal mapping problem, find efficient approaches to this problem for scenarios in which previously-available methods fail, and to present new practical applications of our algorithms.

First, we present a new algorithm for optimizing geometric energies and computing positively oriented simplicial mappings. Our major improvements over the existing methods are: introduction of new energies for repairing inverted and collapsed simplices, adaptive partitioning of vertices into coordinate blocks with the blended local-global strategy for more efficient optimization and introduction of an improved convergence criteria. Our algorithm achieves state-of-the-art results in distortion minimization, even under hard positional constraints and highly distorted invalid initializations that contain thousands of collapsed and inverted elements. We show that, over a wide range of 2D and 3D problems, our algorithm is more robust than existing techniques for locally injective mapping. In particular, initializing our algorithm with novel initialization schemes outperforms existing surface parameterization methods.

We present also a new quantitative method for detecting changes in 3D models. The dissimilarity between shapes is quantified as a measure of the effort it takes to deform one 3D region into another. In this case, our main tool is an assessment of conformal and isometric distortions of mappings between volumes. We demonstrate how our method can be employed for detecting and quantifying changes in volumetric medical images.

Next, we present a local approach to optimizing a rich family of geometry-based energies defined on planes, surfaces and over volumetric domains. This approach is based on the concept of first order distortion measures and on the steepest descent optimization, employed to induce globally injective optimal deformations of triangular and tetrahedral meshes. The proposed techniques can be employed to devise as-close-to-being-conformal as possible mappings and other deformations

that are nearly optimal with respect to related distortion measures, such as the isometric distortion and the distortion of a local volume.

Finally, we present a theoretical analysis of the optimal mapping problem. The aim of this analysis is to close the gap between practical applications and mathematical studies by means of a formal definition of deformations and complementary geometric measures, employed in practice.