Confidence Regions for Means of Random Sets using Oriented Distance Functions

Image analysis frequently deals with shape estimation and image reconstruction. The ob jects of interest in these problems may be thought of as random sets, and one is interested in finding a representative, or expected, set. We consider a definition of set expectation using oriented distance functions and study the properties of the associated empirical set. Conditions are given such that the empirical average is consistent, and a method to calculate a confidence region for the expected set is introduced. The proposed method is applied to both real and simulated data examples.

💡 Research Summary

The paper addresses a fundamental problem in image analysis and shape reconstruction: how to define and estimate a representative “average” set when the objects of interest are naturally modeled as random sets. Traditional definitions of set expectation—based on Hausdorff distance, Minkowski sums, or indicator‑function averages—suffer from non‑linearity, boundary discontinuities, and a lack of a tractable probabilistic structure, which makes statistical inference difficult. To overcome these obstacles, the authors adopt the oriented distance function (ODF) as a functional representation of a set. For a set (A\subset\mathbb{R}^d), the ODF is defined as
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