Computational geometry for polygonal reconfiguration, pattern analysis and recognition and music information retrieval

建议的研究旨在开发有效的算法来解决以下领域中出现的几何问题。(1)不同运动类型约束下，不同类型连杆机构在2维和3维空间中重构算法的探索。另一个目标是探索在什么条件下连杆被“卡住”，即它们不能被重新配置为平坦的凸形构形。这些结果不仅与机器人学有关，而且与分子生物学中的蛋白质折叠有关。(2)最有前途的模式识别方法之一是最近邻决策规则(也称为基于实例的学习)。提出了通过结合邻近图来提高最新最近邻规则算法的空间和时间效率。第二个目标是为模式识别问题开发新的字符串相似性和多边形链相似性度量。(3)探索计算几何工具在音乐信息检索和音乐理论中的应用。这些问题从衡量节奏和旋律的相似性，到对节奏家族进行聚类和系统发育分析，目的是加深对音乐理论和音乐信息检索实际应用的理解。(4)传统上，对纺织品的科学分析是在织物的几个物理结构层次上进行的。然而，对纺织品上出现的图案的几何结构的科学分析在很大程度上被忽视了。我们提出了一种打破这一传统的纺织品科学研究的新方法：对装饰纺织品的几何图案进行系统发育分析。我们方法的核心是设计一种衡量两种纺织图案之间差异的方法，它模仿生物学家测量两种DNA分子序列之间的差异的方式，即将一种纺织品图案转换为另一种纺织品图案所需的最小突变数量(简单的局部和全局转换)。这项研究将对互联网上基于图像的搜索以及博物馆和图书馆使用的纺织品检索系统产生影响。

The proposed research aims to develop efficient algorithms for solving geometric problems that arise in the following areas. (1) The exploration of algorithms for the reconfiguration of different types of linkages in 2 and 3 dimensional space, under various restrictions on the types of motions allowed. Another goal is to explore under what conditions linkages are "stuck" in the sense that they cannot be reconfigured to a flat convex configuration. These results are relevant, not only to robotics, but to protein folding in molecular biology. (2) One of the most promising approaches to pattern recognition is the nearest-neighbor decision rule (also referred to as instance-based learning). It is proposed to improve the space and time efficiency of state-of-the-art nearest neighbor rule algorithms, by incorporating proximity graphs. A second objective is to develop new measures of string similarity and polygonal chain similarity for pattern recognition problems. (3) It is proposed to explore the application of computational geometric tools to problems that arise in music information retrieval and music theory. These problems range from measuring the similarity of rhythms and melodies, to performing cluster and phylogenetic analyses of families of rhythms, with the goal of obtaining a deeper understanding of both, music theory and practical applications to music information retrieval. (4) Traditionally the scientific analysis of textiles has been carried out on several physical structural levels of the fabrics. However, a scientific analysis of the geometric structure of the patterns that appear on the textiles has been largely ignored. We propose a new approach to the scientific study of textiles that breaks with this tradition: the phylogenetic analysis of geometric patterns that decorate the textiles. At the heart of our approach is the design of a measure of dissimilarity between two textile patterns, that mimics the way in which biologists measure the dissimilarity between two DNA molecular sequences, i.e., the minimum number of mutations (simple local and global transformations) required to transform one textile pattern into the other. This research will have an impact of image-based search on the internet, as well as textile retrieval systems for use in museums and libraries.