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

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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.