State estimation for random closed sets

State estimation entails the estimation of an unobserved random closed set from (partial) observation of an associated random set. Examples include edge effect correction, cluster detection, filtering and prediction. We focus on inference for random sets based on points sampled on its boundary. Such data are subject to mis-alignment and noise. First, we ignore mis-alignment and discuss maximum likelihood estimation of the model and noise parameters in the Fourier domain. We estimate the unknown curve by back-transformation and derive the expectation of the integrated squared error. Then, we model mis-alignment by means of a shifted parametric diffeomorphism and minimise a suitable objective function simultaneously over the unknown curve and the mis-alignment parameters.