Averaging of Random Sets Based on Their Distance Functions

Authors:
Adrian Baddeley;Ilya Molchanov
Affiliations:
Department of Mathematics, University of Western Australia, Nedlands WA 6907, Australia;Department of Statistics, University of Glasgow, G12 8QW Glasgow, Scotland, U.K.
Venue:
Journal of Mathematical Imaging and Vision
Year:
1998

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Abstract

A new notion of expectation for random sets (or average of binaryimages) is introduced using the representation of sets by distancefunctions. The distance function may be the familiar Euclidean distancetransform, or some generalisation. The expectation of a random setX is defined as the set whose distance function is closest tothe expected distance function of X. This distance average canbe applied to obtain the average of non-convex and non-connected randomsets. We establish some basic properties, compute examples, and prove limittheorems for the empirical distance average of independent identicallydistributed random sets.