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

生物医学的最新进展在很大程度上依赖于计算机科学技术。随着生物医学领域的迅速扩大，需要更强大的计算技术来促进其持续发展。这个项目是为了开发有效的计算机算法，以解决在几种生物医学成像应用中出现的一组几何问题。特别是，它将为三个具有挑战性的问题（以及它们的相关问题）设计算法：(1)Voronoi图（这是几何中的基本结构）的概括，称为聚类诱导Voronoi图（CIVD），(2)基于地球移动距离（EMD）的几何设置中的一般匹配形式，以及(3)各种类型的约束聚类问题的统一框架。问题(1)是为了建立一个数学模型来理解免疫系统中树突状细胞和T细胞的相互作用。问题(2)的动机是医学成像应用将运动器官的解剖和功能图像结合起来，以获得更好的成像质量。问题(3)的目的是利用先验知识来提高图像分类的准确性。该项目将使用计算几何技术为提出的问题开发新的算法。它将介绍几种通用算法技术到计算几何领域，丰富和推动其进一步发展。这些算法技术也有可能应用于其他领域，如机器学习、计算机视觉、信息安全、数据挖掘和模式识别，并为这些领域带来新的思路。这个项目可能会产生几个长期影响。它可能有助于理解癌症中的免疫功能和失调，提高我们恢复快速器官运动（例如心脏和肺运动）的能力，并在各种应用中实现更准确的图像分类。

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.