Functional and Shape Data Analysis
Functional and Shape Data Analysis
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DOI：
10.1007/978-1-4939-4020-2
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发表时间：
2016-01-01
期刊：
影响因子：
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通讯作者：
Klassen, E. P.
中科院分区：
文献类型：
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作者：
Srivastava, A.;Klassen, E. P.
函数和形状数据分析是统计学中的老话题，在过去的几十年里断断续续地研究。然而，在新千年的头几年，这些领域的重点和精力重新得到重视。这种关注既令人兴奋，因为它寻求新的方向和资源，也因为它是面向应用程序和数据驱动的，所以具有生产力。虽然新的兴趣是由许多因素推动的，其中最突出的是涉及函数和曲线数据的大型数据集的可用性增加，特别是在计算机视觉和医学成像领域。计算能力和存储能力的提高，人们对黎曼方法的兴趣日益浓厚，以及几何和统计学思想汇聚的良好氛围，也推动了这一进程。尽管在功能和形状分析方面的研究和方法有很长的历史，但在这一时期，一些研究人员对形状分析有了新的看法。因此，他们提出了新的方法，基于对这个社区来说是新的数学工具，并使它们使用优雅的计算解决方案变得实用。这本书最初是一本研究专著，阐述了曲线形状分析中的这些当代思想。这在很大程度上是因为我们希望为曲线形状分析中的这一新一代方法提供一个独立的处理方法，重点是统计建模和推理。然而，像许多其他图书项目一样，这个项目也超出了最初的计划。它不仅成为了一本有附录、背景材料和练习的教科书，而且它的范围也扩大到包括关于函数数据分析的详细处理。不出所料，完成这份手稿花了比预期长得多的时间。这一延迟使我们能够在这本教科书的基础上开设几门统计学的研究生课程。这本书的主要主题领域是函数和曲线的形状分析-一维、二维和更高维度-既有封闭的也有开放的。这种材料与以往方法的不同之处在于，它将配准问题整合到形状分析中。配准涉及在比较和量化对象的形状时跨对象的点的匹配。过去的方法大多将配准作为一个预处理步骤，使用任意的现成技术进行处理，然后使用不相关的度量进行形状比较。相反，这本教科书寻求一个统一的、全面的解决方案。它开发了优雅的黎曼框架，同时提供形状差异的量化和曲线的配准。此外，这些方法还用于对给定曲线数据进行统计汇总、执行降维以及
Function and shape data analysis are old topics in statistics, studied off and on over the last several decades. However, the early years of the new millennium saw a renewed focus and energy in these areas. This focus was both exciting, because it sought new directions and resources, and productive, because it was application oriented and data driven. While the new interest was fueled by many factors, the most prominent amongst them was increasing availability of large datasets involving function and curve data, especially in the fields of computer vision and medical imaging. It was also propelled by increases in computational power and storage, a growing interest in Riemannian methods, and a favorable atmosphere for the confluence of ideas from geometry and statistics. Despite a long history of research and methods in function and shape analysis, several researchers took a fresh look at shape analysis during this period. As a result, they came up with novel approaches, based on mathematical tools that were new to this community, and made them practical using elegant computational solutions. This book started as a research monograph, as an exposition of these contemporary ideas in shape analysis of curves. It was heavily motivated by our desire to provide a self-contained treatment of this new generation of methods in shape analysis of curves, with a focus on statistical modeling and inference. However, like many other book projects, this project also grew beyond its original plan. Not only did it become a textbook with appendices, background material, and exercises, but also its scope grew to include a detailed treatment on function data analysis. Not surprisingly, it took much longer than expected to finish this manuscript. This delay allowed us to offer several graduate courses in statistics on the basis of this textbook.The main topic area of this book is shape analysis of functions and curves—in one, two, and higher dimensions—both closed and open. What differentiates this material from past approaches is that it integrates the registration problem into shape analysis. Registration is concerned with matching of points across objects when their shapes are being compared and quantified. The past methods mostly treated registration as a pre-processing step, handled using an arbitrary off-the-shelf technique, followed by an unrelated metric for shape comparison. Instead, this textbook seeks a unified, comprehensive solution. It develops elegant Riemannian frameworks that provide both quantification of shape differences and registration of curves at the same time. Additionally, these methods are used for statistically summarizing given curve data, performing dimension reduction, and