1 Mathematical models and practical solvers for uniform motion deblurring

Recovering an un-blurred image from a single motion-blurred picture has long been a fundamental research problem. If one assumes that the blur kernel – or point spread function (PSF) – is shift invariant, the problem reduces to that of image deconvolution. Image deconvolution can be further categorized as non-blind and blind. In non-blind deconvolution, the motion blur kernel is assumed to be known or computed elsewhere; the task is to estimate the un-blurred latent image. The general problems to address in non-blind deconvolution include reducing possible unpleasant ringing artifacts that appear near strong edges, suppressing noise, and saving computation. Traditional methods such as Wiener deconvolution (Wiener 1949) and the Richardson–Lucy (RL) method (Richardson 1972, Lucy 1974) were proposed decades ago and find many variants thanks to their simplicity and efficiency. Recent developments involve new models with sparse regularization and the proposal of effective linear and non-linear optimization to improve result quality and further reduce running time. Blind deconvolution is a much more challenging problem, since both the blur kernel and the latent image are unknown. One can regard non-blind deconvolution as an inevitable step in blind deconvolution during the course of PSF estimation or after the PSF has been computed. Both blind and non-blind deconvolution are practically very useful; they are studied and employed in a variety of disciplines including, but not limited to, image processing, computer vision, medical and astronomic imaging, and digital communication. This chapter discusses shift-invariant single image motion deblurring methods, which assume that the image is uniformly blurred with only one PSF, which may not be known a priori. This set of problems has a long history in theoretical and empirical research and has notably advanced in the last 5–10 years with a few remarkably effective models and solvers.