Expectations of Random Sets and Their Boundaries Using Oriented Distance Functions

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.

💡 Research Summary

The paper tackles the fundamental problem of shape estimation and object reconstruction in image analysis by treating objects as random sets. Traditional approaches to defining an “expected set” have relied on various criteria—such as convex averaging, Hausdorff averaging, or Minkowski sums—but none simultaneously satisfy three desiderata that are crucial for practical applications: (i) preservation of inclusion relations (if one random set is almost surely contained in another, the same relation should hold for their expectations), (ii) convexity preservation (the expectation of a convex random set should remain convex), and (iii) equivariance under rigid motions (the expectation should transform in the same way as the underlying sets).

To overcome these shortcomings, the authors introduce a new framework based on oriented distance functions (ODFs). An ODF for a deterministic closed set (A\subset\mathbb{R}^d) is defined as

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