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 Rn R n . The formulation of MBD is presented in the continuous setting, where D  is a simply connected region in Rn 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. We present several important properties of MBD, including the theorems: on the equivalence between the MBD ?A and its alternative definition ?A; and on the convergence of their digital versions, ? A ^ and ? A ^ , to the continuous MBD ?A=?A as we increase a precision of sampling. This last result provides an estimation of the discrepancy between the value of ? A ^ and of its approximation ? A ^ . An efficient computational solution for the approximation ? A ^ of ? 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.
distance function image processing distance transform fuzzy subset path strength