CHS: Small: High-Dimensional Euclidean Embedding for 4D Volumetric Shape with Multi-Tensor Fields

这项研究的总体目标是开发一个严格的计算系统，使人体的内部工作更容易理解和分析。现实世界中许多复杂的四维（时空）动态物体具有异质性和各向异性（沿不同轴方向不均匀）的特性，通常可以通过多模态成像设备（如4D- ct /MRI/超声/DTI）捕获，迫切需要对这些物体进行建模和分析。例如，在心脏病学中，心脏器官和组织在心脏周期的不同阶段具有复杂的特性、形状几何、运动和变形的4D可变形体积的实时高保真建模和处理对于建立一个有效和统一的工具非常重要，医生可以使用这个工具来准确地可视化、跟踪和诊断。在肺癌治疗、前列腺癌治疗等方面也存在类似的应用。该项目还将为本科生和研究生提供一些教育活动，并向当地中学生提供服务。本项目围绕一个高维欧氏几何嵌入框架，将黎曼度量、张量场和纳什嵌入理论集成在一起，使得从一个新的角度有效、高效地表示和处理四维黎曼体积形状成为可能。高维嵌入的计算实现将从三维/四维异构数据特征/属性空间中获得的具有任意度量张量场的三维/四维形状转换为保留所有固有几何特征并集成其他多模态属性的新型高维形状等距空间。通过统一黎曼度量张量场的广义几何嵌入空间允许对数据信息学中涉及的形状优化、处理和测量中的几何可扩展性和可变性进行形式化和多样化的研究。在高维嵌入空间中，优化、重构、比较和分析等复杂的黎曼度量计算将被各向同性度量下简单高效的欧几里得计算所取代。通过使用四维形状张量重建和分析对框架进行验证，将有可能为医学成像和生物医学界提供一种准确、稳健和严格的方法，用于对不同物体的多异质特征和属性进行几何推理和定量评估。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The overall objective of this research is to develop a rigorous computing system to make the internal workings of the human body easier to understand and analyze. Many complex real-world 4D (space-time) dynamic objects have both heterogenous and anisotropic (unequal along different axes) properties, which can often be captured by multi-modality imaging devices (e.g., 4D-CT/MRI/Ultrasound/DTI), and there is a pressing need to model and analyze these objects. For example, in cardiology, high-fidelity modeling and processing of 4D deformable volumes of cardiac organs and tissues with complex properties, shape geometry, motion and deformation at different phases of the cardiac cycle in real-time becomes important for building an effective and unified tool which doctors can then use to accurately visualize, track, and diagnose. Similar applications also exist in lung cancer treatment, prostate cancer treatment, and so on. This project will also provide several educational activities for undergraduate and graduate students, as well as outreach to local middle school students. This project centers around a high-d Euclidean geometric embedding framework that integrates Riemannian metric, tensor field, and Nash embedding theory, making it possible to effectively and efficiently represent and process the 4D Riemannian volumetric shapes from a new perspective. The computational realization of the high-d embedding will transform a 3D/4D shape with arbitrary metric tensor fields obtained from 3D/4D heterogenous data feature/property space into a novel high-d shape isometric space which preserves all intrinsic geometric characteristics as well as integrating other multi-modality properties. The generalized geometric embedding space through the unified Riemannian metric tensor fields allows formal and diverse study of geometry scalability and variability in shape optimization, processing and measurement involved in data informatics. In the high-d embedding space, complicated Riemannian metric computations in optimization, reconstruction, comparison and analysis will be replaced with simple and efficient Euclidean computations under the isotropic metric. Through the validation of the framework using 4D shape-tensor reconstruction and analysis, it will be possible to offer medical imaging and biomedicine communities an accurate, robust, and rigorous approach for geometric reasoning and quantitative assessment of multi-heterogenous features and properties across different objects.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.