Automated Parameter Selection Tool for Solution to Ill-Posed Problems Mid-Year Report

In many ill-posed problems it can be assumed that the error in the data is dominated by noise which is independent identically normally distributed. Given this assumption the residual should also be normally distributed with similar mean and variance. This idea has been used to develop three statistical diagnostic tests to constrain the region of plausible solutions. This project aims to develop software that automates the generation of a range of plausible regularization parameters based on diagnostic tests. 1 Background In medical images such as MRI or CT scans (figure 1 ), the images may be distorted and/or noisy due to the physics of the measurement and the structure of the material (human) being imaged. These images are expensive to produce and often are critical in making medical decisions. Image deblurring is an example of an ill-posed inverse problem. To find suitable approximate solutions to ill-posed inverse problems we use our knowledge about the particular problem to come up with constraints [4]. These constraints are used to determine parameters to regularize the problem, replacing the ill-posed problem by one that is well-posed, and thus has an acceptable solution. Finding and selecting good regularization parameters can be very expensive and subject to bias. Researchers often have invaluable information that is crucial in finding a good approximate solution, but without validation, there is risk of seeing what is expected and not the true solution or image (figure 2 ). An effective automated tool that generates a plausible range of regularization parameters is needed to create both a cost effective methodology and control for bias when determining optimal solutions. The goal of this project is to develop a software package with graphical user interface (GUI) for parameter selection in regularizing ill-posed problems, and apply it to debluring and denoising problems with Gaussian noise. The software will use generalized cross-validation (GCV) for initial parameter selection