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 R n . The formulation of MBD is presented in the continuous setting, where D is a simply connected region in R n , 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. sent 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, ρ A ^ and φ A ^ , 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 ρ A ^ and of its approximation φ A ^ . An efficient computational solution for the approximation φ A ^ of 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.