A Posteriori Error Control for the Binary Mumford–Shah Model

The binary Mumford-Shah model is a widespread tool for image segmentation and can be considered as a basic model in shape optimization with a broad range of applications in computer vision, ranging from basic segmentation and labeling to object reconstruction. This paper presents robust a posteriori error estimates for a natural error quantity, namely the area of the non-properly segmented region. To this end, a suitable uniformly convex and non-constrained relaxation of the originally non-convex functional is investigated and Repin's functional approach for a posteriori error estimation is used to control the numerical error for the relaxed problem in the L2-norm. In combination with a suitable cut out argument, fully practical estimates for the area mismatch are derived. This estimate is incorporated in an adaptive mesh refinement strategy. Two different adaptive primal-dual finite element schemes, a dual gradient descent scheme, and the most frequently used finite difference discretization are investigated and compared. Numerical experiments show qualitative and quantitative properties of the estimates and demonstrate their usefulness in practical applications.