The minimum barrier distance

In this paper we introduce a minimum barrier distance, MBD, defined for the (graphs of) real-valued bounded functions fA, whose domain D   is a compact subsets of the Euclidean space RnRn. The formulation of MBD is presented in the continuous setting, where D   is a simply connected region in RnRn, 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 fA along the path.We present several important properties of MBD, including the theorems: on the equivalence between the MBD ρA and its alternative definition φA; and on the convergence of their digital versions, ρA^ and φA^, to the continuous MBD ρA = φA as we increase a precision of sampling. This last result provides an estimation of the discrepancy between the value of ρA^ and of its approximation φA^. An efficient computational solution for the approximation φA^ of ρA^ 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.

► We introduce a new distance function on fuzzy subsets.
► Properties of the continuous and digital versions are proved.
► We give an approximation that can be computed efficiently.
► The method is robust to noise, blur and seed point position.