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

摘要：P Lamm，密歇根州立大学反问题及其应用：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可靠的方法来实际选择变量正则化参数，作为反问题局部正则化方法的一部分。PI首先提出了一种确定性方法，通过局部差异原则的协调，并假设一些关于数据误差水平的局部信息是可用的，来解决变量参数选择问题。这项工作的一部分将需要结合局部正则化思想使用基于小波的近似。此外，该计划还计划研究一种统计参数估计的想法，该想法目前正在研究大气中臭氧水平检测问题的研究人员的测试中，并为这些想法提供目前缺乏的理论基础。有希望从这项工作中融合出强大的新方法来选择可变的局部正则化参数，这些方法不需要数据中的局部噪声级信息。作为该项目的一部分，PI计划将这类新方法应用于臭氧检测问题。逆问题在许多应用中广泛出现，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）、大气中臭氧水平的探测和地球物理勘探。虽然经典方法适用于解决此类问题，但经典方法通常效率低下，并且导致过于昂贵的解决方案技术。经典解方法的第二个缺点可以在成像应用中看到，重建图像可能有模糊的边缘和不够详细的特征。PI一直致力于通过基于局部正则化思想的新解决方法的开发来解决这些困难。使用这些更新颖的方法可以显著降低求解大量实际反问题的成本，并提高解的详细特征的分辨率。

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.