Inverse Problems and Their Applications: Deterministic and Statistical Methods for Variable Local Regularization

摘要：0405978 P Lamm，密歇根州立大学逆问题及其应用：变量局部正则化的确定性和统计方法在这个项目中，主要研究者计划开发理论上合理的方法，用于实际选择变量正则化参数，作为逆问题局部正则化方法的一部分。 PI首先提出通过局部差异原则的协调，并在假设有关数据误差水平的局部信息可用的情况下，对可变参数选择问题进行确定性处理，部分工作将涉及结合局部正则化思想使用的基于小波的近似。此外，PI计划研究一个统计参数估计的想法，目前正在测试的研究人员在探测大气中臭氧水平的问题，并给予这些想法的理论基础，他们目前缺乏。 有希望的是，强大的新方法选择可变的局部正则化参数，方法不需要信息的局部噪声水平的数据，将出现从这项工作。 作为该项目的一部分，PI计划将这类新的方法应用于臭氧检测问题。逆问题在许多应用中广泛存在，包括生物医学成像（CT扫描和X射线），图像重建（来自卫星或其他来源），大气中臭氧水平的检测和地球物理勘探。 虽然经典的方法存在解决这些问题，经典的方法往往是非常低效的，并导致过于昂贵的解决方案的技术。 第二个缺点的经典解决方案的方法可以看到在成像应用中，重建图像可能有模糊的边缘和不够详细的功能。 PI一直致力于解决这些困难，开发了基于局部正则化思想的新解决方案。这些新方法的使用可以显著降低求解一类实际反问题的成本，并提高解的细节特征的分辨率。

Abstract: 0405978 P Lamm, Michigan State UniversityInverse Problems and their Applications: Deterministic and Statistical Methods for Variable Local Regularization In this project the principal investigator plans to developtheoretically-sound methods for the practical selection of variableregularization parameters as part of local regularization methods forinverse problems. The PI first proposes to undertake a deterministicapproach to the problem of variable parameter selection via thecoordination of local discrepancy principles and under the assumptionthat some local information about data error-level is available.Parts of this work will entail wavelet-based approximations used inconjunction with the local regularization ideas. In addition, the PIplans to study a statistical parameter estimation idea currently beingtested by researchers working on the problem of detecting ozone levelsin the atmosphere, and to give these ideas the theoretical basis theyare currently lacking. There is hope that powerful new methods forthe selection of variable local regularization parameters, methods notrequiring information about local noise-levels in the data, willemerge from this work. As part of the project the PI plans to applythis new class of methods to the ozone detection problem.Inverse problems occur widely in many applications, including problemsof biomedical imaging (CT scans and X-rays), image reconstruction(from satellites or other sources), the detection of ozone levels inthe atmosphere, and geophysical exploration. While classical methodsexist for for solving such problems, classical methods are often veryinefficient and lead to overly expensive solution techniques. Asecond disadvantage of classical solution methods can be seen inimaging applications where reconstructed images may have blurred edgesand inadequately detailed features. The PI has been working toaddress these difficulties with the development of new solutionmethods based on the ideas of local regularization. The use of thesenewer methods can lead to a significant decrease in cost for thesolution of a wide class of practical inverse problems, with improvedresolution of detailed features of solutions.