Hierarchical Solving of Symbolic Problems in Heterogeneous Geometry Environments

异质几何环境中符号问题的分层求解PI：Elaine Cohen，共同PI理查德F.Riesenfield计算学院，犹他州大学摘要许多重要类型的问题可以用数学形式提出，但许多现实世界的应用程序不能以传统数学解算器可以使用的形式提供数据。例如，医学模型可以由3D体积的点样本组成，而不是由数学解算器所需的平滑模型组成。这项研究涉及通过使用基于稳健几何的方法来解决问题，将这些表示真实世界数据的多种重要方法与数学解算器的能力统一起来。研究人员将把这些结果用于机械系统的计算机原型设计和基于计算机的外科培训。这项研究应该能够为医学领域中成像的肿瘤或与复杂医学数据的力反馈交互提供虚拟卡尺，以及在工程领域中检查CAD建模的机械零件与激光扫描的真实零件的匹配。本研究提出使用分层的几何表示来计算符号问题的解，从而将符号求解器的适用性扩展到数据可能不是方程形式的领域。这类应用的例子包括在工程和医学方面的真实应用。由于这种方法依赖于内在几何属性的界限，而不是直接操作底层表示，因此该方法还将允许单个应用问题将几个不同的模型表示混合在一起。这出现在一些复杂的问题中，其中一些数据可能是分析性的，一些是获得的，而一些只是粗略的近似。

Hierarchical Solving of Symbolic Problems in Heterogeneous Geometry EnvironmentsPI: Elaine Cohen, Co-PI Richard F. RiesenfeldSchool of Computing, University of UtahAbstractMany important types of problems can be posed in a mathematical form, yet many real-world applications do not provide data in a form that can be used by traditional mathematical solvers. For example, medical models may be made of a 3D volume of point samples rather than the smooth models needed by the mathematical solvers. This research involves unifying these multiple, important methods of representing real-world data with the power of mathematical solvers by using methods based on robust geometry to solve problems. The investigators will use these results in applications for computer prototyping of mechanical systems and for computer-based surgical training. This research should be able to provide such tools as virtual calipers for imaged tumors or force-feedback interaction with complex medical data in the medical realm, and checking the fit of a CAD modeled mechanical part with a laser-scanned real part in the engineering realm.This research proposes to use hierarchical, geometric representations to compute solutions of symbolic problems, thereby expanding the applicability of symbolic solvers to areas where the data may not be in the form of equations. Examples of such applications include real-world applications in engineering and medicine. Because this approach relies on bounds to intrinsic geometric properties, rather than manipulating only the underlying representations directly, the approach also will allow a single application problem to have several different model representations mixed together. This arises in complex problems where some data may be analytical, some acquired, and some just rough approximations.