Shape transformation and matching plays a fundamental role in engineering and biomedicine. Shape metric with powerfully discriminative ability is critical and highly desired in practice for machine learning in big geometric data. This project aims to develop the rigorous computational framework for finding the intrinsic mapping between shapes and the intrinsic shape metric among general surfaces based on Teichmüller theory. This proposed research will open a new paradigm for geometric analysis, lay down the theoretical foundation and develop a new class of algorithms and software tools for shape transformation and matching. It is expected to greatly increase the applicability of mining and learning technologies for the emerging ubiquitous large-scale geometric data. Its success will significantly advance computational/digital geometry and will enhance the abilities of geometry and topology theories to solve real-world shape analysis problems. The resulting algorithms will have a broad range of applications in the fields of science, engineering and biomedicine. Potential applications include morphometry analysis, cancer detection and abnormality prediction in medical imaging, motion/deformation tracking and dynamics analysis in graphics and vision, and facial recognition, expression analysis and information. This research aims to develop the rigorous computational framework of Teichmüller theory for the Teichmüller Map, which is unique and has the minimal angle distortion in its homotopy class, and the derived Teichmüller Distance among general surfaces. The computation approach is based on the insight from the quasiconformal Teichmüller theory and the variational principle. It first solves the computation of the holomorphic quadratic differentials and then computes the Teichmüller maps. The discrete Teichmüller theory will be established, and the existence and uniqueness of the solution is guaranteed. The results are applied for shape analysis and deep learning in big data, including facial recognition, expression analysis, brain study, etc. The new paradigm of discrete Teichmüller theory will make major impacts on computer science, including geometric modeling, computer graphics, visualization, vision, networking, and medical imaging. The methodology will also have impacts on pure sciences, engineering and biomedicine, and potential benefits for homeland security and national defense. The algorithms developed during the research will be made freely available on the world wide web. The PI will encourage minority groups at FIU to pursue research in geometry, and make the research accessible to more audience through the seminars, course development, workshops and conferences.
