Computational Differential Geometry Tools for Surface Interrogation, Fairing, and Design

Abstract

This thesis presents a set of new mesh processing methods which are based on computational differential geometry techniques. The underlying idea of the methods consists of using proper discrete approximations of differential surface properties. The methods developed in the thesis contribute to the areas of curvature feature detection, mesh parameterization, fair mesh generation, mesh denoising, and free-form and variational mesh deformations. Comparisons of the developed methods with several state-of-the-art techniques and algorithms are done. The results of numerous numerical experiments demonstrate significant advantages of the proposed methods over conventional techniques. Applications of the methods are discussed and demonstrated. The main contributions of the thesis are as follows: Similarity-based Mesh Denoising. A new, powerful, and high quality feature preserving mesh/soup denoising technique and a new scheme for comparing different mesh/soup smoothing methods are proposed. The technique is based on a similarity-weighted averaging procedure and a new and robust similarity measuring scheme. Fair Mesh Generation via Elastica. A new numerical scheme for generating fair meshes is developed. Applications to shape restoration are considered. The scheme is build upon a discrete approximation of Willmore flow. A tangent speed component is introduced to the discrete Willmore flow in order to improve the quality of the evolving mesh and to increase computational stability. Fast and Robust Detection of Feature Lines on Meshes. A new, fast, and robust crest line detection method is developed. Applications to feature-adaptive mesh simplification and segmentation are considered. A novel thresholding scheme and a simple new formula for computing directional curvature derivatives are also introduced. Fast Low-Stretch Mesh Parameterization. A new, fast, simple, and valid low-stretch mesh parameterization scheme and its application for effcient remeshing are proposed by using a moving mesh approach. The scheme is based on a weighted quasi-conformal parameterization which equalizes the local stretch distribution. Particularly, the scheme does not generate regions of undesirable high anisotropic stretch. Free-Form Skeleton-driven Mesh Deformations. A new and powerful approach for generating natural-looking large-scale mesh deformations is proposed. An interesting feature of the approach consists of preserving original shape thickness. New self-intersection fairing schemes are also developed. Multiresolutional and variational extensions of the approach are considered.