Segmentation has been widely studied and is one of the fundamental and classical tasks of image processing. Starting from Mumford–Shah functional, we review multiphase segmentation using piecewise constant level set approach and relate to phase transition models for multiphase segmentation. In particular, we focus on Euler's elastica-based variational functionals, in the context of fast algorithms for image segmentation problem. Its relation to illusory contour is covered following the phase transition approach. Minimizing the functionals with curvature term helps to capture the geometry of the recovered shape, but introduces complexity due to their nonconvexity and high order derivatives. We review some of fast algorithms developed to efficiently compute the minimum of curvature-based segmentation functionals, such as Euler's elastica segmentation.