SolidMaps: Reliable Computational 3D Solid Mapping

3D对象之间的数字表示的地图(在这个词的数学意义上)是广泛的计算应用的基本成分，用于从设计到模拟到制造的工程任务；用于分析任务，例如在医学成像和计算机视觉中；用于动画和可视化。特别相关的是描述两个对象的点之间的(几何或可能的语义性质的)一一对应的连续双射，这种映射有利于涉及相关空间几何数据的处理和分析的计算任务，它们促进了对象类实例之间的信息传输，以及计算结果的传播和重用。此外，从对象到抽象域的映射是一种通用的计算工具，用于为对象配备局部坐标系(在参数化环境中)或为给定体对象生成高质量的结构化网格或网格离散。在这种情况下，重要的算法方面是这种映射的表示、构造和优化。对于更简单的相似二维情况，所有这些都是很容易理解的，并且有高效可靠的算法可用。各种表示和优化的解决方案已成功地推广到(实体、体积)三维情况。然而，对于可靠地构建3D实体地图的任务(例如，作为后续地图优化方法的有效初始化)，可以确定在本领域的主要差距。本项目的目标是缩小这一差距。它的目的是寻求一种计算方法，能够以通用和可靠的方式建立双射连续的三维立体地图。受到最近的进展的启发，但仅限于有限的特殊情况，提出了以下目标。(1)支持一大类灵活的映射域，而不是仅限于球体或立方体等基本体。(2)支持任意对象拓扑，无论是单连通的还是更高亏格的。(3)支持以对应约束的形式规定所寻找的地图需要遵守的点的子集的一般边界条件。(4)实用性：通过在自适应升级战略中组合多种备选方法，将形成理论上可靠且适合实际使用的整体有效方法。

Digitally represented maps (in the mathematical sense of the word) between 3D objects are a fundamental ingredient in a wide range of computational applications, for engineering tasks from design over simulation to fabrication; for analysis tasks, e.g., in medical imaging and computer vision; for animation and for visualization. Of particular relevance are continuous bijections, homeomorphisms, that describe a one-to-one correspondence (of geometric or possibly semantic nature) between two objects' points.Such maps benefit computational tasks that are concerned with the processing and analysis of correlated spatial geometric data, they facilitate the transfer of information between instances of an object class, and the propagation and re-use of computational results. Furthermore, maps from objects to abstract domains are used as a versatile computational tool, to equip objects with local coordinate systems (in the context of parametrization) or to generate high-quality structured mesh or grid discretizations for given volumetric objects.In this context, important algorithmic aspects are the representation, the construction, and the optimization of such maps. For the simpler analogous 2D case, all these are well-understood and efficient reliable algorithms are available. Various solutions for representation and optimization have been successfully generalized to the (solid, volumetric) 3D case. However, regarding the task of reliable construction of 3D solid maps in the first place (for instance as valid initialization for subsequent map optimization methods), a major gap can be identified in the state of the art.This project’s goal is to close this gap. It aims for a computational method that is able to establish bijective continuous 3D solid maps in a general and reliable manner. Inspired by recent advances, that however are limited to restricted special cases only, the following objectives are addressed. (1) Support for a large, flexible class of mapping domains, instead of being restricted to primitives such as spheres or cubes. (2) Support for arbitrary object topology, whether simply-connected or of higher genus. (3) Support for the prescription of general boundary conditions in the form of correspondence constraints for a subset of points, that the sought map is required to respect. (4) Practical applicability: through a combination of a spectrum of multiple alternative approaches in an adaptive escalation strategy an overall efficient method will be formed that is theoretically solid and reliable while being fit and justifiable for practical use.