Level lines as global minimizers of energy functionals in image segmentation

Abstract

We propose a variational framework for determining global minimizers of rough energy functionals used in image segmentation. Segmentation is achieved by minimizing an energy model, which is comprised of two parts: the first part is the interaction between the observed data and the model, the second is a regularity term. The optimal boundaries are the set of curves that globally minimize the energy functional. Our motivation comes from the observation that energy functionals are traditionally complex, for which it is usually difficult to precise global minimizers corresponding to “best” segmentations. Therefore, we focus on basic energy models, which global minimizers can be explicitly determined. In this paper, we prove that the set of curves that minimizes the image moment-based energy functionals is a family of level lines, i.e. the boundaries of level sets (connected components) of the image. For the completeness of the paper, we present a non-iterative algorithm for computing partitions with connected components. It leads to a sound initialization-free algorithm without any hidden parameter to be tuned.