Curvature regularity for region-based image segmentation and inpainting: A linear programming relaxation

We consider a class of region-based energies for image segmentation and inpainting which combine region integrals with curvature regularity of the region boundary. To minimize such energies, we formulate an integer linear program which jointly estimates regions and their boundaries. Curvature regularity is imposed by respective costs on pairs of adjacent boundary segments. By solving the associated linear programming relaxation and thresholding the solution one obtains an approximate solution to the original integer problem. To our knowledge this is the first approach to impose curvature regularity in region-based formulations in a manner that is independent of initialization and allows to compute a bound on the optimal energy. In a variety of experiments on segmentation and inpainting, we demonstrate the advantages of higher-order regularity. Moreover, we demonstrate that for most experiments the optimality gap is smaller than 2% of the global optimum. For many instances we are even able to compute the global optimum.