Mathematical Sciences: Inverse Estimation Problems

数学科学：逆估计问题

• 批准号: 9504485
• 负责人: Frits Ruymgaart
• 金额: $ 10.5万
• 依托单位: Texas Tech University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504485&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Inverse; Estimation; Problems

Proposal: DMS 9504485 PI: Frits Ruymgaart Institution: Texas Tech Title: Inverse Estimation Problems Abstract: Inverse estimation is concerned with indirect curve estimation: the curve of interest is to be recovered from observations, subject to random blur, on a transformation of the curve. This class of problems is very broad and formally contains ordinary, direct, curve estimation as a special case. Many interesting examples come from physics, signal processing, statistics, and applied mathematics. The estimation problem can essentially be solved by inverting the transformation involved. Since this inverse is not in general continuous, the problem is typically ill-posed and regularization of the inverse is required. Two important questions are how restriction, due to limitations in time or space, of the transformation relate to the unrestricted transformation, and how recovery of irregular curves can be modified so as to avoid the Gibbs phenomenon. Along with a comprehensive study of deconvolution on locally compact Abelian groups, this research addresses these questions. Also, the study of some examples of practical interest are explored. Inverse estimation arises when an object, e.g., part of the human brain, can only be indirectly observed, and recovery of information about the object is required from the indirect measurements. Indirect measurement techniques, such as those used in medical imagery, are attractive because they are noninvasive. Perfect reconstruction of the image would, in principle, be possible if an unlimited number of uncorrupted measurements were available. In practice, however, one can only obtain finitely many data that, moreover, are corrupted by measurement errors. Here statistical considerations and techniques play a role in the image reconstruction process. Aspects addressed in this research include how the reconstruction depends on the specific way in which the indirect data are collected, and al so how it should be tuned to prior information about the object to be recovered.

建议：DMS 9504485 PI：Frits Ruymgaart Institution：德克萨斯理工大学标题：逆估计问题摘要：逆估计涉及间接曲线估计：感兴趣的曲线将从观测中恢复，受到随机模糊的影响，对曲线进行变换。这类问题非常广泛，形式上包含了普通的、直接的、曲线估计作为特例。许多有趣的例子来自物理学、信号处理、统计学和应用数学。估计问题基本上可以通过倒置所涉及的变换来解决。由于这个逆在一般情况下不是连续的，所以问题通常是不适定的，并且需要逆的正则化。两个重要的问题是，由于时间或空间的限制，变换的限制如何与无限制的变换相联系，以及如何修改不规则曲线的恢复以避免Gibbs现象。随着对局部紧Abel群上反卷积的全面研究，本研究解决了这些问题。并对一些具有实用价值的实例进行了研究。当只能间接地观察到对象(例如，人脑的一部分)，并且需要从间接测量中恢复关于该对象的信息时，出现逆估计。间接测量技术，例如那些用于医学成像的技术，因为它们是非侵入性的，所以很有吸引力。原则上，如果有无限数量的未受破坏的测量数据可用，图像的完美重建将是可能的。然而，在实践中，人们只能获得有限数量的数据，而且这些数据受到测量误差的破坏。在这里，统计考虑和技术在图像重建过程中发挥作用。在这项研究中涉及的方面包括重建如何依赖于收集间接数据的具体方式，以及如何调整它以适应关于要恢复的对象的先前信息。