Edge detection and image restoration of blurred noisy images using jump regression analysis

University of Minnesota Ph.D. dissertation. August 2013. Major: Statistics. Advisor: Peihua Qiu. 1 computer file (PDF); vii, 124 pages.We consider the problem of edge-preserving image restoration when images are degraded by spatial blur and pointwise noise. When the spatial blur described by a point spread function (psf) is not completely specified beforehand, this is a challenging ill-posed problem, because (i) theoretically, the true image can not be uniquely determined by the observed image when the psf is unknown, even in cases when the observed image contains no noise, and (ii) practically, besides blurring, observed images often contain noise, which can cause numerical instability in many existing image deblurring procedures. In the literature, most existing deblurring procedures are developed under the assumption that the psf is completely specified, or that the psf follows a parametric form with one or more unknown parameters. In this dissertation, we propose blind image deblurring (BID) methodologies that do not require such restrictive conditions on the psf. They even allow the psf to change over location. This dissertation has three chapters. Chapter 1 introduces some motivating applications for image processing along with presenting the overall scope of the dissertation. In Chapter 2, the problem of step edge detection in blurred noisy images is studied. In Chapter 3, a BID procedure based on edge detection is proposed. In Chapter 4, an efficient BID procedure without explicitly detecting edges is presented. Both theoretical justifications and numerical studies show that our proposed procedures work well in applications