AF: Small: Algorithmic Techniques for Several Geometric Problems Arising in Biomedical Imaging Applications

Recent progress in biomedicine has heavily relied on computer science technology. As the territory of biomedicine is rapidly enlarging, more powerful computational techniques are needed to foster its continuous growth. This project is for developing efficient computer algorithms for a set of geometric problems arising in several biomedical imaging applications. Particularly, it will design algorithms for three challenging problems (as well as their related problems): (1) generalizations of Voronoi diagram (which is a fundamental structure in geometry) called Clustering Induced Voronoi Diagram (CIVD), (2) general forms of matching in geometric settings based on Earth Mover's Distance (EMD), and (3) uniform framework for various types of constrained clustering problems. Problem (1) is for developing a mathematical model for understanding the interaction of dendritic cells and T cells in the immune system. Problem (2) is motivated by a medical imaging application of combining anatomic and functional images of moving organs for better imaging quality. Problem (3) aims to achieve better accuracy in image classifications by using a priori knowledge. This project will use computational geometry techniques to develop novel algorithms for the proposed problems. It will introduce several general algorithmic techniques to the area of computational geometry, enriching and prodding its further development. These algorithmic techniques are also likely to be used in other areas, such as machine learning, computer vision, information security, data mining, and pattern recognition, and bring new ideas to these areas. This project could lead to several long term impacts. It could potentially help understanding the immune function and dysregulation in cancer, improving our ability to recovering fast organ motion (e.g., cardiac and lung motion), and achieving more accurate image classification in various applications.