Chapter 32: Total variation regularization

We now turn to mathematical image processing, where the unknown to be reconstructed from data is a digital image. The most basic mathematical image processing task is denoising, i.e., removing the noise in an image (for example, a photograph taken in low light conditions), which corresponds to taking the forward operator as the identity. More advanced image processing tasks include inpainting, deblurring, and superresolution. These correspond to filling in missing parts of an image, reducing blur caused by defocused lenses or motion, and recovering additional detail, and they involve more complicated linear forward operators. For an introduction to mathematical image processing, we refer to [215, 37]. In true inverse imaging problems, the given data is not itself an image but related to it via some mathematical model describing the physical measurements; examples are magnetic resonance imaging (MRI), involving the Fourier transform[183], or positron emission tomography (PET) and computed X-ray tomography (CT), both involving the Radon transform [178]. More challenging imaging modalities such as electrical impedance tomography (EIT) and more advanced MRI techniques require the forward operator \(A\) to be nonlinear. We do not treat such operators here but point toward the primal-dual method of Chapter 15 as one possible solution technique. Alternative Gauss–Newton type methods are introduced by [135].