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：Elaine Cohen，联合pi Richard F. riesenfeld犹他大学计算学院摘要：许多重要类型的问题可以用数学形式提出，然而许多现实世界的应用程序并没有以传统数学求解器可以使用的形式提供数据。例如，医学模型可以由点样本的三维体积组成，而不是由数学求解器所需的光滑模型组成。本研究涉及通过使用基于鲁棒几何的方法来解决问题，将这些表示现实世界数据的多种重要方法与数学求解器的功能统一起来。研究人员将把这些结果应用于机械系统的计算机原型和基于计算机的外科训练。本研究将为医学领域中肿瘤成像的虚拟卡尺、与复杂医学数据的力反馈交互、工程领域中CAD建模机械部件与激光扫描真实部件的配合检验等提供工具。本研究提出使用分层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。此类应用的例子包括工程和医学中的实际应用。由于这种方法依赖于固有几何属性的边界，而不是直接操作底层表示，因此该方法还允许单个应用程序问题将几个不同的模型表示混合在一起。这在复杂的问题中出现，其中一些数据可能是分析的，一些是获得的，而一些只是粗略的近似值。

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.