AF: Small: Applied and Theoretical Algorithm Problems in Computational Geometry

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This project aims to study an array of important computational geometry problems in several applied areas such as medicine, biology, biomedical imaging, and data mining, and to develop new algorithmic solutions for these problems in biomedical applications. In addition, the project also seeks to investigate a set of fundamental theoretical geometric problems and to forge new geometric computing techniques.Emerging biomedical imaging technologies and modalities have been revolutionizing the field of disease diagnosis and prognosis, pushing a paradigm shift in diagnosis and prognosis study and practice from qualitative to quantitative and from tissue/structure level to molecular/cellular level. Molecular/cellular imaging holds the promise of transforming modern diagnosis and prognosis, and offers numerous advantages over the traditional practice. This project will apply geometric computing techniques and data mining methods to develop new algorithms for vital cell identification and analysis problems in microscopy images, such as computing and analyzing the architectural structures of dendritic cells and other types of cells in multi-spectral microscopy images of tumor-draining lymph nodes for prognosis of breast cancer, and detecting and classifying cells in histology images of joint tissue for diagnosis of rheumatoid arthritis and other autoimmune diseases. Radiation therapy/surgery is a major modality for modern cancer treatment. This project will design new algorithms for an intriguing type of geometric motion planning problems that seek a set of paths to cover target tumor regions under special constraints and criteria. These problems arise in dynamic Gamma Knife radiosurgery and are at the core of a novel radiosurgery approach for breast cancer treatment. Besides, the project will develop new algorithmic techniques for solving a number of theoretical problems that are among the most fundamental tasks in computational geometry, such as computing optimal paths, visibility, Voronoi diagrams, geodesic diameters and centers, geometric clustering, and shape approximation. The research plan of the project includes a crucial component of algorithm implementation, experimentation, evaluation, software development, and practical applications. This research will integrate and enhance the power of computer algorithms and modern biomedicine to solve critical applied and theoretical problems in computational geometry and biomedical applications, and help improve the quality of life in today's society.