  Shape estimation and object reconstruction are common problems in image
analysis. Mathematically, viewing objects in the image plane as random sets
reduces the problem of shape estimation to inference about sets. Currently
existing definitions of the expected set rely on different criteria to
construct the expectation. This paper introduces new definitions of the
expected set and the expected boundary, based on oriented distance functions.
The proposed expectations have a number of attractive properties, including
inclusion relations, convexity preservation and equivariance with respect to
rigid motions. The paper introduces a special class of separable oriented
distance functions for parametric sets and gives the definition and properties
of separable random closed sets. Further, the definitions of the empirical mean
set and the empirical mean boundary are proposed and empirical evidence of the
consistency of the boundary estimator is presented. In addition, the paper
gives loss functions for set inference in frequentist framework and shows how
some of the existing expectations arise naturally as optimal estimators. The
proposed definitions of the set and boundary expectations are illustrated on
theoretical examples and real data.
