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 Rⁿ. The formulation of MBD is presented in the continuous setting, where D is a simply connected region in Rⁿ, as well as in the case where D is a digital scene. The MBD is defined as the minimal value of the barrier strength of a path between the points, which constitutes the length of the smallest interval containing all values of f_A along the path.

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.