SolidMaps: Reliable Computational 3D Solid Mapping

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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.