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

In many applications of solving ill-posed problems there are a number of regularization methods to choose from as well as a free regularization parameter. The choice of the method and regularization can add bias if deblurred images are based on what the researcher expects. The presented project develops a tool for method choice and parameter selection by using a set of three statistical diagnostics to validate solutions. We include three regularization methods; Tikhonov, Truncated SVD, and Total Variation. The diagnostics are motivated by the idea that the residual after deblurring should be normally distributed if we assume that the added noise is normally distributed. The project develops a interface where users can freely choose a method, then find a range of regularization parameters that produce plausible solutions based on the diagnostics. 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 the method and 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 methods and 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 tool that generates a plausible range of regularization parameters is needed to create a cost effective methodology and to control for bias when determining optimal solutions.