A Posteriori Error Control for the Binary Mumford-Shah Model

TL;DR

This paper develops robust a posteriori error estimates for the binary Mumford-Shah model, enabling adaptive meshing and improved accuracy in image segmentation tasks.

Contribution

It introduces a new a posteriori error estimation method for the binary Mumford-Shah model using a convex relaxation and Repin's approach, enhancing error control in segmentation.

Findings

Effective error estimates for non properly segmented regions

Comparison of finite element schemes and finite difference discretization

Numerical experiments demonstrate practical usefulness of the estimates

Abstract

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.