ABSTRACT

We want to address in this chapter one of the most important problems in image processing: the image restoration one. Given a noisy and/or blurry image f (the observation), we wish to restore its clean sharp version u. Noise and blur are inherent degradations during the acquisition process. Noise can be of random nature (described as a random variable following a probability distribution) or of deterministic nature (as a periodic pattern due to physical interference within the imaging device). In simpler cases, the noise term is additive or multiplicative. In more complicated cases, there is a more nonlinear, intricate relation between noise, the observed image and the underlying clean image. Noise is usually removed by averaging pixel values (in other words, smoothing). In contrast, blur is produced by averaging or smoothing. Therefore, in order to remove noise (process called denoising), we want to smooth the observed noisy image f , in such a way that edges, texture and other features are preserved. In order to remove blur (process called deblurring), we want to sharpen the observed image f (deblurring). These two opposite processes (denoising and deblurring) render the restoration problem very difficult. Due to these and other difficulties, the problem of restoring u from the observed image f (even knowing something about the noise and blur) is an ill-posed inverse problem. Inspired from Tikhonov regularization, described in Chapter 2 in a general setting, this can be addressed via regularization. This method introduces a priori assumptions on the unknown image u and the problem becomes well-posed. Moreover, interesting mathematical and numerical problems of variational nature arise.