Confidence and Misplaced Confidence in Image Reconstruction

这里考虑的基本问题是第一类积分方程离散化的解。即使有完整的信息，这个问题也是不适定的，也就是说，数据的微小变化会对解决方案造成任意大的变化。不幸的是，在医学成像（CAT、MRI）、天文成像、光谱学或结构裂缝无损检测等应用中无法获得完整的信息，因此问题变成了不适定问题的离散化版本。解算法使问题正则化，用一个适定的问题代替不适定的问题，从而计算出解。所有正则化方法背后的基本思想都是在模型上施加额外的约束，以使问题适定。有两种非常不同的约束，通常不能很好地区分：保证100%确定的数据约束和由观察者期望看到的产生的偏见约束。如果观察者对偏差约束是错误的，那么解决算法可能会产生一个非常可信但非常误导的解决方案。这项工作的重点是解决不适定问题的三个主要开放问题：诊断的发展，以验证候选解决方案和识别偏差；通过发现新的过滤方法、可靠的参数选择、更好地理解Krylov方法以及统一数据最小二乘、最小二乘和总最小二乘问题的算法，开发改进的算法，产生有效的解决方案；利用数据约束（如非负性）计算解的置信限。这项工作的更广泛影响来自于它为医学应用（CAT， MRI等），天文学，光谱学，定位油藏，测试隐藏裂缝结构和其他应用产生更可靠图像的潜力。这些技术包括有效地利用关于图像的已知额外信息（例如，每个像素值都是非负的）来约束解图像。重点是改进方法和通过构建统计置信区间更精确地了解解决方案。这项工作也有很大的教育价值。高等数值线性代数的研究生课程将会提供一个关于离散不适定问题的部分。这项工作将在马里兰州的螺旋暑期项目上展示，面向来自传统黑人学院和大学的本科生，因为它为病态问题提供了一个视觉上吸引人且易于解释的介绍。

The underlying problem considered here is the solution of a discretization of an integral equation of the first kind. Even with complete information, the problem is ill-posed, in the sense that small changes in the data can make arbitrarily large changes in the solution. Unfortunately, complete information is not available in applications such as medical imaging (CAT, MRI), astronomical imaging, spectroscopy, or non-destructive testing for cracks in a structure, and the problem becomes a discretized version of the ill-posed problem. Solution algorithms regularize the problem, replacing the ill-posed problem by one that is well-posed in order to compute a solution. The basic idea behind all regularization methods is to impose additional constraints on the model in order to make the problem well-posed. There are two very different kinds of constraints, which are typically not well differentiated: data constraints that are guaranteed to hold with 100% certainty, and bias constraints, arising from what the observer expects to see. If the observer is wrong about the bias constraints, then the solution algorithm might produce a solution that is is quite believable but very misleading. This work focuses on three major open questions in the solution to ill-posed problems: development of diagnostics to validate candidate solutions and identify bias; development of improved algorithms that produce validated solutions through discovery of new filtering methods, reliable choice of parameters, better understanding of Krylov methods, and unification of algorithms for data least squares, least squares, and total least squares problems; and computation of confidence bounds for the solutions, making use of data constraints (e.g., nonnegativity).The broader impact of the work arises from its potential to produce more reliable images for medical applications (CAT, MRI, etc.), astronomy, spectroscopy, locating oil reservoirs, testing structures for hidden cracks, and other applications. The techniques involve effective use of extra information known about the image (for example, that each pixel value is nonnegative) in order to constrain the solution image. The focus is on improved methods and on more precise knowledge of the solution through the construction of statistical confidence intervals. The work also has great value in education. A graduate course in advanced numerical linear algebra will be offered that will include a section on discrete ill-posed problems. This work will be presented at Maryland's SPIRAL summer program for undergraduate students from Historically Black Colleges and Universities, since it provides a visually-appealing and easily-explained introduction to ill-posed problems.