Inverse problems in science are the process of estimating the causal factors that produced a set of observations. Many image processing tasks can be cast as inverse problems: image restoration : noise reduction, deconvolution; segmentation, tomography, demosaicing, inpaiting, and many others, are examples of such tasks. Typically, inverse problems are ill-posed, and solving these problems efficiently and effectively is a major, ongoing topic of research. While imaging is often thought of as occurring on regular grids, it is also useful to be able to solve these problems on arbitrary graphs. The combined frameworks of discrete calculus and modern optimisation allow us to formulate and provide solutions to many of these problems in an elegant way. This tutorial article summarizes and illustrates some of the research results of the last decade from this point of view. We provide illustrations and major references.