Efficient algorithm for finding the exact minimum barrier distance

The minimum barrier distance, MBD, introduced recently in [1], is a pseudometric defined on a compact subset D of the Euclidean space R and whose values depend on a fixed map (an image) f from D into R. 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 along the path. In this paper we present a polynomial time algorithm, that provably calculates the exact values of MBD for the digital images. We compare this new algorithm, theoretically and experimentally, with the algorithm presented in [1], which computes the approximate values of the MBD. Moreover, we noEmail addresses: [email protected] (Krzysztof Chris Ciesielski), [email protected] (Robin Strand), [email protected] (Filip Malmberg), [email protected] (Punam K. Saha) Preprint submitted to Computer Vision and Image Understanding March 11, 2014 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 K.C. Ciesielski, et al.: Efficient algorithm for MBD 1/13/2014 2 tice that every generalized distance function can be naturally translated to an image segmentation algorithm. The algorithms that fall under such category include: Relative Fuzzy Connectedness, and those associated with the minimum barrier, fuzzy distance, and geodesic distance functions. In particular, we compare experimentally these four algorithms on the 2D and 3D natural and medical images with known ground truth and at varying level of noise, blur, and inhomogeneity.