A challenge in modern data science is how to integrate information from 3-dimensional shapes into statistical models. Examples of applications where this challenge is central is associating the shape of a tumor to molecular processes or associating the shape of the roots of a rice plant to crop yield. In graphics and anatomy, there is the related question of how to warp one object into another, such as the molar of a child to a molar of an adult. This project seeks to develop methodology to transform these shape data, such as meshes or 3-dimensional images, into representations for which standard statistical models are available. These methods are crucial to advancing data-enabled science in extracting, conceptualizing, interpreting, and visualizing information residing in datasets comprising complex objects such 3D shapes. Cutting edge ideas in mathematics, specifically from the field of geometry, will be used to address the fundamental problems of (i) modeling structural variation in diverse collections of shapes and (ii) modeling transformations between shapes. Solutions to these problems are crucial to many practical applications and disciplines including biology, medicine, social sciences, and ecology. As such, an important component of the project is validation of methods and tools through applications in radiology and anthropology
Geometric concepts beyond Riemannian geometry and smooth manifolds will be leveraged to develop novel statistical methodology to address complex challenges in data analysis. Methods and tools, grounded on solid mathematical foundations, will be developed for modeling complex objects (such as shapes and surfaces) and complex relations within data (such as multi-commodity flow or aligning shapes). The research will address two statistical challenges using geometric tools: (1) representing surfaces and shapes via integral geometry, and (2) learning group actions or transformations between pairs of objects using the geometry of fiber bundles. Addressing the first challenge results in a framework for parametric and non-parametric statistical models for collections of shapes and surfaces, without the requirement of landmark points and without requiring the shapes to be isomorphic. Addressing the second challenge provides a statistical framework for alignment problems ranging from aligning a collection of shapes such as teeth, to optimization problems on networks such as multi-commodity flow. The solutions proposed will impact statistics and geometry, and the methods may catalyze transformative advances in practical applications and scientific disciplines as diverse as biology, medicine, social sciences, and ecology.
