Total variation and level set methods in image science

Abstract

We review level set methods and the related techniques that are common in many PDE-based image models. Many of these techniques involve minimizing the total variation of the solution and admit regularizations on the curvature of its level sets. We examine the scope of these techniques in image science, in particular in image segmentation, interpolation, and decomposition, and introduce some relevant level set techniques that are useful for this class of applications. Many of the standard problems are formulated as variational models. We observe increasing synergistic progression of new tools and ideas between the inverse problem community and the ‘imagers’. We show that image science demands multi-disciplinary knowledge and flexible, but still robust methods. That is why the level set method and total variation methods have become thriving techniques in this field.Our goal is to survey recently developed techniques in various fields of research that are relevant to diverse objectives in image science. We begin by reviewing some typical PDE-based applications in image processing. In typical PDE methods, images are assumed to be continuous functions sampled on a grid. We will show that these methods all share a common feature, which is the emphasis on processing the level lines of the underlying image. The importance of level lines has been known for some time. See, e.g., Alvarez, Guichard, Morel and Lions (1993). This feature places our slightly general definition of the level set method for image science in context. In Section 2 we describe the building blocks of a typical level set method in the continuum setting. Each important task that we need to do is formulated as the solution to certain PDEs. Then, in Section 3, we briefly describe the finite difference methods developed to construct approximate solutions to these PDEs. Some approaches to interpolation into small subdomains of an image are reviewed in Section 4. In Section 5 we describe the Chan–Vese segmentation algorithm and two new fast implementation methods. Finally, in Section 6, we describe some new techniques developed in the level set community.