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

提出的研究旨在开发有效的算法来解决以下领域出现的几何问题。(1)在允许的运动类型的各种限制下，探索在二维和三维空间中不同类型的连杆机构的重构算法。另一个目标是探索在什么条件下连杆会“卡住”，即它们不能被重新配置为平凸构型。这些结果不仅与机器人有关，而且与分子生物学中的蛋白质折叠有关。(2)模式识别最有前途的方法之一是最近邻决策规则（也称为基于实例的学习）。为了提高最近邻规则算法的空间效率和时间效率，提出了结合接近图的方法。第二个目标是为模式识别问题开发新的字符串相似度和多边形链相似度度量。(3)提出探索计算几何工具在音乐信息检索和音乐理论中出现的问题中的应用。这些问题的范围从测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，目的是更深入地了解音乐理论和音乐信息检索的实际应用。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，对出现在纺织品上的图案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测量两种纺织品图案之间差异的方法，它模仿了生物学家测量两种DNA分子序列之间差异的方式，即将一种纺织品图案转化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.