This article considers the problem of identifying the irregular boundary of a magnetic domain in a thin multilayer film, using data in the form of an electron micrograph. Two approaches are illustrated, both of which are based on the concept of two smooth intensity surfaces, one corresponding to the domain and one corresponding to the background, so that the objective is to determine the boundary between the parts of the image where the two surfaces are present. In the first, a hierarchical Bayesian approach, priors are assumed for the set of domain/background states and for the two intensity surfaces, and restoration is carried out using a version of Besag's ICM procedure. It is shown that it is important to initialize the method efficiently. The second approach uses a template-like model for the boundary and also relies on a Bayesian approach. The results from the second approach can be used as an end in themselves or as a way of initializing the more flexible first approach. Several illustrations are provided.