Confidence and Misplaced Confidence in Image Reconstruction

这里考虑的基本问题是第一类积分方程解的离散化。即使有完整的信息，这个问题也是不适定的，从这个意义上说，数据的微小变化可能会对解决方案产生任意大的变化。不幸的是，在医学成像(CAT、MRI)、天文成像、光谱学或结构中裂纹的无损检测等应用中，没有完整的信息可用，并且该问题变成了病态问题的离散化版本。解的算法将问题正规化，将不适定的问题替换为适定的问题以计算解。所有正则化方法背后的基本思想是对模型施加额外的约束，以使问题变得适定。有两种非常不同的约束，通常没有很好的区分：保证100%确定性的数据约束，以及源于观察者预期看到的偏差约束。如果观察者对偏差约束的判断是错误的，那么解算法可能会产生一个非常可信但非常具有误导性的解。这项工作集中在不适定问题的解决方案中的三个主要悬而未决的问题：发展诊断学来验证候选解和识别偏差；发展改进的算法，通过发现新的过滤方法，可靠地选择参数，更好地理解Krylov方法，以及统一数据最小二乘、最小二乘和总体最小二乘问题的算法；以及利用数据约束(例如，非负性)来计算解的置信限。这项工作的更广泛的影响源于其为医疗应用(CAT、MRI等)、天文学、光谱学、定位油层、测试隐藏裂缝的结构以及其他应用产生更可靠图像的潜力。这些技术包括有效地使用关于图像的额外信息(例如，每个像素值都是非负的)，以便约束解图像。重点是改进方法和通过建立统计可信区间更准确地了解解决办法。这项工作在教育方面也有很大的价值。将开设高级数值线性代数的研究生课程，其中包括离散不适定问题的部分。这项工作将在马里兰州为历史黑人学院和大学的本科生举办的螺旋暑期项目上展示，因为它提供了一个视觉上吸引人的、易于解释的不适定问题的介绍。

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.