author = {Berkels, Benjamin and Effland, Alexander and Rumpf, Martin},
  title = {A posteriori error control for the binary {M}umford-{S}hah model},
  journal = {Math. Comp.},
  year = {2017},
  volume = {86},
  pages = {1769--1791},
  number = {306},
  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.},
  doi = {10.1090/mcom/3138},
  eprint = {1505.05284},
  pdf = { 1},
  fjournal = {Mathematics of Computation}