A Posteriori Error Control for the Binary Mumford-Shah Model
This work provides a practical error control tool for practitioners using the binary Mumford-Shah model in computer vision, though the approach is incremental as it adapts existing error estimation techniques to a specific model.
The paper develops a posteriori error estimates for the binary Mumford-Shah model in image segmentation, controlling the area of mis-segmented regions via a convex relaxation and Repin's functional approach. The estimates are integrated into an adaptive meshing strategy, with numerical experiments demonstrating their effectiveness across multiple discretization schemes.
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 strictly 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 $L^2$-norm. In combination with a suitable cut out argument, a fully practical estimate for the area mismatch is derived. This estimate is incorporated in an adaptive meshing strategy. Two different adaptive primal-dual finite element schemes, 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.