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Shape estimation and object reconstruction are common problems in imageanalysis. Mathematically, viewing objects in the image plane as random setsreduces the problem of shape estimation to inference about sets. Currentlyexisting definitions of the expected set rely on different criteria toconstruct the expectation. This paper introduces new definitions of theexpected set and the expected boundary, based on oriented distance functions.The proposed expectations have a number of attractive properties, includinginclusion relations, convexity preservation and equivariance with respect torigid motions. The paper introduces a special class of separable orienteddistance functions for parametric sets and gives the definition and propertiesof separable random closed sets. Further, the definitions of the empirical meanset and the empirical mean boundary are proposed and empirical evidence of theconsistency of the boundary estimator is presented. In addition, the papergives loss functions for set inference in frequentist framework and shows howsome of the existing expectations arise naturally as optimal estimators. Theproposed definitions of the set and boundary expectations are illustrated ontheoretical examples and real data.

Expectations of Random Sets and Their Boundaries Using Oriented Distance Functions