The minimum barrier distance

  • Authors:

  • Robin Strand;Krzysztof Chris Ciesielski;Filip Malmberg;Punam K. Saha

  • Affiliations:

  • Centre for Image Analysis, Uppsala University, Sweden;Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA and Department of Radiology, MIPG, University of Pennsylvania, Blockley Hall - 4th Floor, 423 Guardian Drive, Ph ...;Centre for Image Analysis, Uppsala University, Sweden;Department of Electrical and Computer Engineering, The University of Iowa, Iowa City, IA 52242, USA and Department of Radiology, The University of Iowa, Iowa City, IA 52242, USA

  • Venue:
  • Computer Vision and Image Understanding
  • Year:
  • 2013

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Abstract

In this paper we introduce a minimum barrier distance, MBD, defined for the (graphs of) real-valued bounded functions f"A, 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 f"A along the path. We present several important properties of MBD, including the theorems: on the equivalence between the MBD @r"A and its alternative definition @f"A; and on the convergence of their digital versions, @r"A@^ and @f"A@^, to the continuous MBD @r"A=@f"A as we increase a precision of sampling. This last result provides an estimation of the discrepancy between the value of @r"A@^ and of its approximation @f"A@^. An efficient computational solution for the approximation @f"A@^ of @r"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.