Variational models and algorithms for blind image deconvolution with applications

This thesis deals with numerical solutions to partial differential equations (PDEs) and their application in image processing, particularly image deblurring. The PDEs we deal with arise from the minimisation of variational models for techniques for image restoration from a single image (such as denoising [29, 56, 169, 180], deblurring [149, 9, 44, 94, 119, 178, 54, 176, 212] or inpainting [6, 16, 25, 52]) or reconstruction from several images (such as focus fusion [80, 128, 159] and many techniques aimed at super-resolution [107, 130, 146]) as well as the identification of objects (such as global and selective segmentation [53, 89, 162, 163]) and other tasks such as the restoration of an image from a limited selection of points to facilitate image compression [101, 158, 175]. The aim of image denoising is to remove noise corruption from an image and restore the true image, while the aim of image segmentation is to distinguish the foreground from the background of an image or to select a particular feature in an image automatically. Image deblurring, or deconvolution, aims to restore an image which has been corrupted by blur and noise which remains a particular problem in many areas including remote sensing, medical imaging, and consumer photography. Deblurring tasks can be categorised into 3 types, all of which remain challenging subjects. Non-blind deblurring [15, 18, 152, 207] assumes that we know and can model the cause or degradation of the image precisely. The aim is then to recover the hidden true image using the knowledge of the blur function which is not a trivial task, particularly in the presence of strong blur, limited boundary information and noise. In contrast, blind image deconvolution (BID) [54, 31, 117, 61, 209] is the technique of recovering an image from blur degradation with no assumptions about the blur function. Such techniques commonly involve attempting to minimise a BID functional in order to recover both the true image and blur function simultaneously. Others involve attempting to use multiple images or image statistics in order to gain some knowledge of the blur function before deblurring the image. Semi-blind deblurring [1233, 11, 2, 211] involves recovering an image from blur degradation making some assumptions about the blur function. For example, we might be able to make the assumption from observation that an image has been corrupted by motion blur. The task then would be to estimate the orientation and strength of the blur while recovering the image. Such techniques are often regarded as blind since crucial information is still not known. For example, the above problem might be called blind motion deblurring. While there has been much research in the restoration of images, the performance of such methods remains poor particularly when the level of noise or blur is high. Many techniques also suffer from slow implementation. Identification, whether automatic or visual, of the blurring function can also prove a challenging task. While it is sometimes but not always possible to identify the type of blur function (for example Gaussian or motion blur) there still remains the challenge of identifying the level or amount of blur. It is also often the case that several types of blur are present and that the image cannot be recovered by the assumption that the true image has been globally corrupted by a single blur function. The aim of this thesis is to develop fast image restoration methods which provide better quality deblurring and give fast and robust results in the blind, non-blind and semi-blind cases. We develop new models to achieve this aim and present experimental results demonstrating their effectiveness. We begin with a review of some preliminary mathematics in Chapter 2 which may be useful during the reading of this thesis. We then present some existing work in Chapter 3 which is relevant to the work presented in later chapters. We next present the application of some of the ideas introduced in Chapter 3 with some refinements before moving onto the main work of this thesis in Chapters 5-8 which deals with the implicit application of optimisation constraints to improve non-blind and blind deconvolution. We also consider convex relaxation, obtaining improved solution speeds, improving deblurring approximations by separating noise and formulating parametric approximations of piece-wise constant functions. Finally, we present an application of this work to the segmentation of blurred images.