Optimal Level Curves and Global Minimizers of Cost Functionals in Image Segmentation

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 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 characterized. None of the proposed segmentation models captures all the important scene variables but may be useful to get an insight into objects, surfaces or parts of objects in the scene. In this paper, we prove that the set of curves that minimizes the cost functionals is a subset of level lines, i.e. the boundaries of level sets of the image. For the completeness of the paper, we present a fast algorithm for computing partitions with connected components. It leads to a sound initialization-free algorithm without any hidden parameter to be tuned. We illustrate the performance of our algorithm with several examples on both 2D biomedical and aerial images, and synthetic images.