In this paper we introduce a minimum barrier distance, MBD, defined for the (graphs of) real-valued bounded
functions f_{A}, whose domain D is a compact subsets of the Euclidean space **R ^{n}**.
The formulation of MBD
is presented in the continuous setting, where D is a
simply connected region in

We present several important properties of MBD, including the theorems: on the equivalence between the MBD
ρ_{A} and its alternative definition φ_{A};
and on the convergence of their digital versions, \hat{ρ_{A}} and \hat{φ_{A}}, to
the continuous MBD ρ_{A}=φ_{A} as we increase a precision of sampling.
This last result provides an estimation of the discrepancy between the value of \hat{ρ_{A}}
and of its approximation \hat{φ_{A}}. An efficient computational solution for the approximation
\hat{φ_{A}} of \hat{ρ_{A}} is presented. We experimentally
investigate the robustness of MBD to noise and blur, as well as its stability with respect to the change
of a position of points within the same object (or its background).
These experiments are used to compare MBD with other distance functions: fuzzy distance,
geodesic distance, and max-arc distance.
A favorable outcome for MBD of this comparison
suggests that the proposed minimum barrier distance is potentially useful in different imaging tasks,
such as image segmentation.

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**Last modified February 11, 2013.**