Constrained conformal/quasi-conformal map and its applications - Zhang, Minqi (2014)
With the rapid development of 3D-capture and modeling technologies, a vast number of 3D surface models have been created. More analysis needs to be done on these models, especially for the data from some subjects such as medical imaging. One of the key tools for analyzing these 3D surface models is conformal map. Discrete conformal map is an important parametrization for computer graphics. The surface parametrization refers to the process of mapping the surface onto canonical domains such as 2D plane. However, area or angle distortion would be inevitably introduced in the course of this procedure. The discrete conformal map provides a special parametrization which can minimize angle distortion. From a practical point of view, the conformal map offers a powerful tool to handle a number of geometric problems in various fields. In this thesis, we achieve various breakthroughs for conformal map in both theorem and applications. In the theorem, we propose a new way to compute the extremal quasiconformal map; and in practice, we greatly increase the flexibility and the scope of the conformal map by applying the conformal map to a wide range of applications such as medical images, networking routing, the Escher-Droste effect, and so on. Computing the extremal quasi-conformal map on general surface: The conformal map has desirable properties, but does not always exist; When the conformal map does not exist, the Teichm¨uller mapping (T-map) is a special extremal quasi-conformal map, which is a perfect substitution for the conformal map. For the first time, we present a simple yet effective technique to compute extremal quasi-conformal maps on general surfaces. Our method is linear, local, easy to implement and has a very natural parallel structure. Furthermore it requires neither the numerical solver nor the global coordinate system. Surface registration for brainstem by Ricci flow: The morphometry of the brainstem surface is of utmost importance for the disease Adolescent Idiopathic Scoliosis (AIS). We propose an effective method to accurately compute the registration between brainstem surfaces with consistent landmark features by using the conformal map and extremal quasi-conformal map. We also show the efficacy of the proposed registration algorithm through the experiments on 15 AIS models and 15 normal models. Surface registration for vestibular system by holomorphic 1-form: Another important system related to the AIS is the vestibular system (VS). The non-trivial topology of the VS poses great technical challenges for geometric analysis. We present an effective and practical solution to register vestibular systems by using the holomorphic 1-form and extremal quasi-conformal map techniques. Our method is robust, automatic and efficient. Greedy routing through distributed parametrization: The greedy routing problem is an important topic for wireless sensor networks. We propose a solution by applying a parametrization similar to the conformal map to build the virtual coordinate system first. The new parametrization, which is modified from the holomorphic 1-form, can embed the network domain to a canonical domain, which allows greedy routing to have guaranteed delivery. Generalized Escher-Droste effect: The Escher-Droste effect is a special type of recursive picture, which depicts fascinating visual effects. We generalize the classical Escher-Droste effect, such that the self-invariant mappings of the images include general conformal mappings. We regard the planar conformal mappings as the basic image editing tools. The algorithm is simple, fast, flexible, and is capable of generating the Escher-Droste effects for a broad category of images. Our contribution here is two-fold: on one hand, we make great headway in applying the conformal map as a basic tool, especially in the situation where the conformal map does not exist and the setting that requires delicate change for the traditional conformal algorithm. On the other hand, we introduce the conformal map to the other realm, which also opens a new gate for that realm and brings unexpected progress.