SolidMaps: Reliable Computational 3D Solid Mapping

3D对象之间的数字表示的映射（在数学意义上）是广泛的计算应用中的基本成分，用于从设计到模拟到制造的工程任务;用于分析任务，例如，用于医学成像和计算机视觉;用于动画和可视化。特别相关的是连续双射，同胚，描述两个对象的点之间的一对一的对应关系（几何或可能语义性质）。这样的地图有利于与相关空间几何数据的处理和分析有关的计算任务，它们促进了对象类实例之间的信息传递，以及计算结果的传播和重用。此外，从对象到抽象域的映射被用作通用的计算工具，为对象配备局部坐标系（在参数化的上下文中）或为给定的体积对象生成高质量的结构化网格或网格离散化。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。对于更简单的类似2D的情况下，所有这些都是很好理解的，有效的可靠的算法是可用的。各种解决方案的代表性和优化已成功地推广到（固体，体积）三维的情况下。然而，关于首先可靠地构建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），可以在现有技术中确定一个主要差距。该项目的目标是缩小这一差距。它的目的是一种计算方法，能够建立双射连续的三维立体地图，在一个通用的和可靠的方式。受最近进展的启发，但仅限于有限的特殊情况，解决了以下目标。(1)支持一个大的，灵活的映射域类，而不是被限制为原语，如球体或立方体。(2)支持任意对象拓扑，无论是简单连接还是更高的亏格。(3)支持以点子集的对应约束的形式规定一般边界条件，所寻求的地图需要遵守这些条件。(4)实用性：通过在自适应升级策略中组合一系列多个备选方法，将形成一种总体有效的方法，该方法在理论上是坚实和可靠的，同时对于实际使用是适合和合理的。

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.