Mathematical Sciences: Inverse Estimation Problems

提案：DMS 9504485 PI：Frits Ruymgaart机构：Texas Tech标题：逆估计问题 摘要： 逆估计与间接曲线估计有关：感兴趣的曲线将从曲线变换上的随机模糊的观察中恢复。 这类问题非常广泛，形式上包含普通的，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将所涉及的变换求逆来解决。 由于这个逆通常不是连续的，所以这个问题通常是不适定的，需要对逆进行正则化。 两个重要的问题是，如何限制，由于在时间或空间上的限制，转换与无限制的转换，以及如何恢复的不规则曲线可以修改，以避免吉布斯现象。 沿着对局部紧阿贝尔群上的反卷积进行了全面的研究，本研究解决了这些问题。 同时，对一些具有实际意义的实例进行了探讨。 当对象，例如， 人脑的一部分，只能被间接观察到，并且需要从间接测量中恢复关于对象的信息。 间接测量技术，如那些用于医学影像，是有吸引力的，因为它们是非侵入性的。 如果有无限数量的未损坏的测量值，则原则上可以完美地重建图像。 然而，在实践中，人们只能获得非常多的数据，而且这些数据被测量误差所破坏。 这里，统计考虑和技术在图像重建过程中起作用。 本研究涉及的方面包括重建如何取决于特定的方式， 收集间接数据，以及如何将其调整到关于要恢复的对象的先验信息。

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.