CGV: Small: Towards a Mathematics of Terrain

在这个项目中，地形被定义为地球表面高于参考大地水准面的高度。在过去的几十年里，越来越多的（x, y）和z的高精度地形数据已经可用。改进的海底测深数据也已收集，其他行星及其卫星的高程数据现在也可获得（用于“地形”的广义定义）。PI在这个项目中的目标是开发和验证一种新的地形数学表示，它将更接近地形形成的物理原理，并且比不现实的地形更容易被设计成合法的现实地形。除了本身构成了一个有趣的更深层次数学应用之外，这种地质上合理的地形表示基础将使设计诸如压缩和定位之类的运算符成为可能。这项工作将推广和扩展PI之前成功的地形表示和算法工作，如ODETLAP。新的地形表示将是一系列不同类别的参数化转换，灵感来自地形形成的物理原理。对真实世界建模，转换将是非线性的（例如，真实的河谷不能叠加和添加）。非线性是强大的，但很难研究。第一类转换，称为舀取，将模拟河谷的形成，并保证只产生水文上有效的地形。还将研究侵蚀、沉积和丘陵形成的变化。每一类转换都有许多设计选项；例如，应该使用更少但更强大的转换，还是使用更多但不那么强大的转换？PI的目标是将地形编码为尽可能少的比特，同时除了RMS误差之外，还要满足更丰富的、依赖于应用程序的指标，例如多观察者定位以最大化视野，然后进行路径规划以避开这些观察者。水文精度和视觉可识别性是其他指标。该项目延续了PI与巴西维科萨联邦大学Marcus Andrade教授的合作。项目成果将通过对真实地形数据库的广泛测试加以验证。更广泛的影响：这项工作最简单的含义将是更紧凑的地形压缩算法。因此，这项研究将允许消费者使用便携式产品（如GPS导航仪）访问和处理更大的地形数据集。更容易访问大型地形数据库将有助于在可能的现实地形上进行概率分布，这反过来将允许优化操作，例如多观察者选址和路径规划（前者的应用范围从手机塔选址到监视，而后者对于运输过程中的节能很重要）。更好的大型地形数据的水文应用包括洪泛区规划（20世纪90年代美国的洪水损失达500亿美元）。通过研究生参与PI的研究和他的研究生课程，该项目还将有助于增加基础学科中受过良好教育的劳动力，这对美国的生产力和未来的经济繁荣至关重要。

Terrain, in this project, is defined as the elevation of the earth's surface above some reference geoid. Over the last few decades, ever larger quantities of terrain data with higher accuracy in (x, y) and z have become available. Improved bathymetry data of the sea floor has also been collected, and elevation data for other planets and their satellites is now available (for a generalized definition of "terrain"). The PI's goal in this project is to develop and validate a new mathematical representation of terrain, which will be closer to the physics of how terrain is formed and be designed to represent legal realistic terrain more easily than unrealistic terrain. Aside from constituting an interesting application of deeper mathematics in its own right, such a foundation for terrain representation that is geologically sound will enable the design of operators such as compression and siting from first principles. This work will generalize and extend the PI's previous successful terrain representation and algorithms work, such as ODETLAP. The new terrain representation will be a sequence of parameterized transformations of various classes inspired by the physics of how terrain is formed. Modeling the real world, the transformations will be nonlinear (e.g., real river valleys cannot be superimposed and added). Nonlinearity is powerful, but difficult to study. The first class of transformations, called scooping, will model how river valleys form, and will guarantee to produce only hydrologically valid terrain. Erosion, deposition and hill creation transformations will also be studied. Each class of transformation has many design options; for example, should fewer and more powerful, rather than many but less powerful, transformations be used? The PI's goal is to encode the terrain in as few bits as possible while satisfying, in addition to RMS error, richer, application-dependent, metrics such as multi-observer siting to maximize viewshed, and then path planning to avoid those observers. Hydrological accuracy and visual recognizability are other metrics. This project continues the PI's collaboration with Professor Marcus Andrade at the Federal University of Vicosa in Brazil. Project outcomes will be validated by means of extensive tests on real terrain databases.Broader Impacts: The simplest implication of this work will be more compact terrain compression algorithms. Thus, this research will allow larger terrain datasets to be accessed and processed by consumers in portable products such as GPS navigators. Easier access to large terrain databases will facilitate a probability distribution over possible realistic terrain, which in turn will allow optimizing operations such as multi-observer siting and path planning (the former has applications ranging from cell phone tower siting to surveillance, while the latter is important for energy conservation during transportation). Hydrological applications of better large terrain data include floodplain planning (flood damage in the US amounted to $50,000,000,000 during the 1990s). Through involvement of graduate students in the PI's research and through his graduate courses, this project will also help to increase the educated workforce in a foundational discipline that is important to American productivity and future economic prosperity.