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.
If a photo taken on your vacation is blurry, it is disappointing. If a medical or scientific image is blurry, it can have serious consequences. And since the raw data collected from medical and scientific imaging devices is rarely the final product that a physician or scientist needs to see, it is important to provide accurate deblurred images along with an estimate of the uncertainty in the final image. We addressed exactly this issue in this project. Here are some of the accomplishments. We developed new algorithms for detecting edges in video sequences. Our software implementation has been used by the US Navy in analyzing aerial photographs. We developed new algorithms for near-optimal reconstruction of blurred or convolved images, using either training data or assumptions on the noise, and provided an estimate of the uncertainty in the final image. We developed and made available a graphical user interface (GUI) for training students or aiding scientists in reconstructing blurred images. We demonstrated its usefulness in an undergraduate course with students in a variety of scientific fields. We developed an algorithm for resolving rotational blur and used it in photoacoustic tomography. We guided the careers of three members of groups underrepresented in mathematics: two female PhD students and one female postdoctoral fellow. We made three software packages available: edge detection in images; a graphical user interface (GUI) for training students or aiding scientists in reconstructing blurred images; and an implementation of a variable projection algorithm for solving nonlinear least squares problems.
