Advanced methods in variational learning: segmentation with intensity inhomogeneity

Mathematical imaging aims to develop new mathematical tools for the emerging field of image and data sciences. Automatic segmentation in the variational framework is a challenging as well as a demanding task for a number of imaging tasks. Achieving robustness and reliability is a major problem. The twophase piecewise-constant model of the Mumford-Shah (2PCMS) energy is most suitable for images with simple and homogeneous features where the intensity variation is limited. However, it has been applied to many different types of synthetic and real images after some adjustments to the formulation. This chapter addresses the task of generalising this widely used model to account for intensity inhomogeneity which is common for real life images. We consider two ways of altering the fitting terms by ‘correcting’ the intensity. This allows us to process the type of images that are beyond 2PCMS. We first review the state of the art of such methods for treating inhomogeneity and then demonstrate inconsistencies in existing methods. Next we propose two modified variational models by introducing additional constraints and extend the concept to selective segmentation, where user input can aid the intensity correction. These models are minimised with convex relaxation methods, where the global minimiser can be found for a fixed fitting term. Finally, we present numerical results that demonstrate an improvement to existing methods in terms of reliability.