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。 许多复杂的真实世界4D（时空）动态对象具有异质和各向异性（沿着不同轴不相等）性质，其通常可以由多模态成像设备（例如，4D-CT/MRI/超声/DTI），并且迫切需要对这些对象进行建模和分析。例如，在心脏病学中，实时地对心脏器官和组织的4D可变形体积进行高保真度建模和处理，这些心脏器官和组织具有复杂的属性、形状几何结构、心动周期的不同阶段的运动和变形，对于构建有效且统一的工具变得重要，然后医生可以使用该工具来准确地可视化、跟踪和诊断。 类似的应用还存在于肺癌治疗、前列腺癌治疗等方面，该项目还将为本科生和研究生提供多项教育活动，并推广到当地中学生。该项目围绕一个高维欧几里德几何嵌入框架，该框架集成了黎曼度量，张量场和纳什嵌入理论，使其能够从一个新的角度有效地表示和处理4D黎曼体积形状。 高维嵌入的计算实现将从3D/4D异质数据特征/属性空间获得的具有任意度量张量场的3D/4D形状变换到新的高维形状等距空间中，该等距空间保留所有内在几何特征以及集成其他多模态属性。 通过统一的黎曼度量张量场的广义几何嵌入空间允许在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的正式和多样的研究。 在高维嵌入空间中，优化、重构、比较和分析中复杂的黎曼度量计算将被各向同性度量下简单有效的欧氏计算所取代。通过使用4D形状张量重建和分析的框架的验证，将有可能为医学成像和生物医学社区提供准确，鲁棒，和严格的方法进行几何推理和定量评估的多，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值进行评估来支持和更广泛的影响审查标准。

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.