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. The formulation of MBD is presented in the continuous setting, where D is a simply connected region in R, 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, ρ̂A and φ̂A, 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 ρ̂A and of its Email addresses: [email protected] (Robin Strand), [email protected] (Krzysztof Chris Ciesielski), [email protected] (Filip Malmberg), [email protected] (Punam K. Saha) Preprint submitted to Computer Vision and Image Understanding September 6, 2012 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 approximation φ̂A. An efficient computational solution for the approximation φ̂A of ρ̂A 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.