ITR: Distances and Generalized Geodesics for High-Dimensional Implicit and Point Cloud Surfaces:Theory, Computational Framework, and Applications in Information Sciences and Eng.

几乎所有的信息科学和工程学科都需要计算事件和测地线之间的距离或它们之间的最佳路径。 测地线用于机器人的路径规划、计算神经科学中的大脑分析、图像分割以及在表面上寻找基本几何对象。在高维度中，它们是数据挖掘和信息发现问题的构建块，应用范围从金融到生物学到图像和视频处理。在一般空间中，广义测地线也有许多应用，从脑成像到彩色图像增强到计算机图形学。由于距离函数和测地线是解释和利用数据以及嵌入其中的知识和信息的基础，因此重要的是要对广义距离函数和测地线有一个很好的理论理解，并开发有效的技术来精确和快速地计算它们。这项工作需要在高维和不同类型的数据表示，包括隐式超曲面，无组织的点集，如从示例中学习高维信息时获得的流形样本，以及来自流行的3D范围扫描仪的点云。该项目解决了这些理论和计算问题，并集中在上述领域的具体重要应用。这些应用程序可以有效地接近，只有一旦这里研究的理论和计算框架已经derived.in教育竞技场，目标是继续目前的努力，在高等教育的跨学科方面，并在分发和普及开发的技术，包括使软件公开。

Almost all disciplines in information sciences and engineering use and need to compute the distance between events and geodesics or optimal paths between them. Geodesics are used for example for pathplanning in robotics, brain analysis in computational neuroscience, image segmentation, and finding fundamental geometric objects on surfaces. In high dimensions, they are the building blocks for problems in data mining and information discovery, with applications ranging from finance to biology to image and video processing. In general spaces, generalized geodesics also have numerous applications, from brain imaging to color image enhancement to computer graphicsSince distance functions and geodesics are fundamental for the interpretation and utilization ofdata and the knowledge and information embedded in it, it is important to have a good theoretical understanding of generalized distance functions and geodesics, and to develop efficient techniques for their accurate and fast computation. The work is needed in high dimensions and for diverse types of data representations, including implicit hyper-surfaces, sets of unorganized points such as manifold samples obtained when learning high dimensional information from examples, and point clouds from popular 3D range scanners. This project addresses these theoretical and computational issues, and concentrates on specific important applications in the areas mentioned above. These applications can be efficiently approached only once the theoretical and computational frameworks here studied have been derived.In the educational arena, the goal is to continue with current efforts in the interdisciplinary aspects of advanced education and in the distribution and popularization of the developed techniques,including making software publicly available.