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

该项目旨在研究医学、生物学、生物医学成像和数据挖掘等应用领域中的一系列重要计算几何问题，并为生物医学应用中的这些问题开发新的算法解决方案。此外，该项目还寻求研究一系列基本的理论几何问题和打造新的几何计算技术。新兴的生物医学成像技术和模式正在给疾病诊断和预后领域带来革命性的变化，推动诊断和预后研究和实践的范式从定性转向定量，从组织/结构水平转向分子/细胞水平。分子/细胞成像具有改变现代诊断和预后的前景，并提供了许多优于传统实践的优势。该项目将应用几何计算技术和数据挖掘方法，为显微图像中的生命细胞识别和分析问题开发新的算法，例如计算和分析肿瘤引流淋巴结多光谱显微图像中树突状细胞和其他类型细胞的结构结构，用于乳腺癌预后的评估；以及关节组织图像中的细胞检测和分类，用于类风湿性关节炎和其他自身免疫性疾病的诊断。放射治疗/手术是现代癌症治疗的一种主要方式。这个项目将为一类有趣的几何运动规划问题设计新的算法，该问题寻求一组在特殊约束和标准下覆盖目标肿瘤区域的路径。这些问题出现在动态伽玛刀放射外科治疗中，是乳腺癌治疗的一种新的放射外科方法的核心。此外，该项目还将开发新的算法技术来解决一些理论问题，这些问题是计算几何中最基本的任务之一，如计算最优路径、可见性、Voronoi图、测地线直径和中心、几何聚类和形状近似。该项目的研究计划包括算法实现、实验、评估、软件开发和实际应用的关键部分。这项研究将整合和增强计算机算法和现代生物医学的力量，以解决计算几何和生物医学应用中的关键应用和理论问题，并有助于提高当今社会的生活质量。

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.