CRII: III: Topology-Driven, Multi-Scale Tensor Field Visualization

多维建模已经并正在使科学、工程和艺术等许多研究领域取得突破，并且不断产生需要分析、挖掘和可视化的高维数据。数据的大小通常很大，因为建模的精细分辨率保证了所开发理论的准确性。流体和固体力学中的应力和应变张量，医学成像中的扩散张量，形状建模中的曲率张量以及地震工程中的变形梯度张量只是其中的几个例子。这些张量以数据数组的形式出现，并且具有可视化拓扑结构及其动态的能力，可以潜在地为领域科学家提供新的见解，并可能对物理和生物现象有新的基本理解。除了在科学界分享通过这项研究创造的知识和工具外，研究者还将把这项工作纳入本科生和研究生的教材以及K-12学生的暑期课程中。由于数据的复杂性和广泛性，任何可视化都必须注意遮挡和杂乱。特别是，该项目将研究支持拓扑驱动的理论和算法，三维多尺度可视化，二阶张量场。一个重点将是这些张量从一个尺度到另一个尺度的平滑过渡，以确保拓扑结构的一致性和拓扑特征的保存。另一个重点是定义所有可能的拓扑和开发可视化算法，在一个尺度上表示场拓扑的动态。

Multi-dimensional modeling has enabled and is enabling breakthroughs in many research fields such as science, engineering and art, and is constantly producing high dimensional data that requires analyzing, mining and visualization. The size of the data is often big as the fine resolution of the modeling warrants accuracy in the theories developed. The stress and strain tensors in fluid and solid mechanics, the diffusion tensor in medical imaging, the curvature tensor in shape modeling, and the deformation gradient tensor in earthquake engineering are just a few examples. These tensors appear in the form of data arrays, and having the capability of visualizing the topological structures and their dynamics can potentially empower domain scientists with new insights and possibly new fundamental understandings of the physical and biological phenomena. In addition to sharing the knowledge and tools created through this research among scientific communities, the investigator will incorporate the work into teaching materials for undergraduate and graduate students and into summer programs for K-12 students.Due to the complexity and extensiveness of the data, any visualization has to be mindful of occlusion and cluttering. In particular, this project will investigate theories and algorithms that support topology-driven, multi-scale visualization of three-dimensional, second-order tensor fields. One focus will be on the smooth transition of these tensor quantities from one scale to another to ensure consistency in the topological structures and preservation of topological features. The other focus is on defining all possible topologies and developing visualization algorithms that represent the dynamics of the field topology at one scale.