Mathematical Modeling and Essential Regularization for Imaging Applications: From Variational Models to Deep Learning Algorithms

To deal with an increasingly large and sophisticated class of real life problems, image processing methods range from the traditional filtering and thresholding techniques to advanced variational models and deep learning algorithms. Regularization is a key concept in developing a variational model to ensure that a model has at least one solution and hence efforts in devising efficient algorithms worthwhile. High order and nonlocal regularization is particularly important, especially when the underlying problem (i.e. input image) requires one to minimize intensity differences within a large neighbourhood (e.g. beyond immediate voxels) for smoothness consideration. This Element aims to survey, review and discuss the state of the art techniques towards the latter kind of methods, emphasizing foundations, algorithms (and codes) and open challenges of high order and nonlocal regularizers for imaging tasks in commonly practised application scenarios.