Confidence and Misplaced Confidence in Image Reconstruction

这里考虑的根本问题是第一类积分方程的离散化的解决方案。 即使有完整的信息，这个问题也是不适定的，在这个意义上，数据中的小变化可以使解决方案发生任意大的变化。 不幸的是，完整的信息是不可用的应用程序，如医学成像（CAT，MRI），天文成像，光谱学，或非破坏性测试的结构中的裂缝，和问题成为一个离散化版本的不适定问题。 求解算法将问题正则化，用适定问题代替不适定问题以计算解。 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，以使问题良定。 有两种非常不同的约束，它们通常没有很好的区分：保证100%确定性的数据约束，以及由观察者期望看到的内容引起的偏差约束。 如果观察者在偏差约束上是错误的，那么求解算法可能会产生一个相当可信但非常误导的解。 这项工作的重点是在解决不适定问题的三个主要的开放性问题：诊断的发展，以验证候选解决方案和识别偏差;改进的算法，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，并统一算法的数据最小二乘，最小二乘和总最小二乘问题，产生有效的解决方案的发展;以及利用数据约束（例如，这项工作更广泛的影响来自于它为医学应用（CAT，MRI等）产生更可靠图像的潜力，天文学、光谱学、定位油藏、测试结构是否存在隐藏裂缝以及其他应用。 该技术涉及有效使用关于图像已知的额外信息（例如，每个像素值是非负的），以约束解图像。 重点是改进的方法和更精确的知识的解决方案，通过建设统计置信区间。 这项工作在教育方面也有很大的价值。 高级数值线性代数的研究生课程将提供，其中将包括一节离散不适定问题。 这项工作将在马里兰州的螺旋夏季计划的本科生从历史上的黑人学院和大学，因为它提供了一个视觉吸引力和容易解释的介绍不适定的问题。

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.