Linear Time Algorithms for Exact Distance Transform

Authors:
Krzysztof Chris Ciesielski;Xinjian Chen;Jayaram K. Udupa;George J. Grevera
Affiliations:
Department of Mathematics, West Virginia University, Morgantown, USA 26506-6310 and Dept. of Radiology, MIPG, Univ. of Pennsylvania, Philadelphia, USA 19104-6021;Dept. of Radiology, MIPG, Univ. of Pennsylvania, Philadelphia, USA 19104-6021;Dept. of Radiology, MIPG, Univ. of Pennsylvania, Philadelphia, USA 19104-6021;Dept. of Radiology, MIPG, Univ. of Pennsylvania, Philadelphia, USA 19104-6021 and Mathematics and Computer Science Department, Saint Joseph's University, Philadelphia, USA 19131
Venue:
Journal of Mathematical Imaging and Vision
Year:
2011

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Abstract

In 2003, Maurer et al. (IEEE Trans. Pattern Anal. Mach. Intell. 25:265---270, 2003) published a paper describing an algorithm that computes the exact distance transform in linear time (with respect to image size) for the rectangular binary images in the k-dimensional space 驴 k and distance measured with respect to L p -metric for 1驴p驴驴, which includes Euclidean distance L 2. In this paper we discuss this algorithm from theoretical and practical points of view. On the practical side, we concentrate on its Euclidean distance version, discuss the possible ways of implementing it as signed distance transform, and experimentally compare implemented algorithms. We also describe the parallelization of these algorithms and discuss the computational time savings associated with them. All these implementations will be made available as a part of the CAVASS software system developed and maintained in our group (Grevera et al. in J. Digit. Imaging 20:101---118, 2007). On the theoretical side, we prove that our version of the signed distance transform algorithm, GBDT, returns the exact value of the distance from the geometrically defined object boundary. We provide a complete proof (which was not given of Maurer et al. (IEEE Trans. Pattern Anal. Mach. Intell. 25:265---270, 2003) that all these algorithms work correctly for L p -metric with 1pL 1 and L 驴 metrics. In addition, we show that the algorithm can be used to find, in linear time, the exact value of the diameter of an object, that is, the largest possible distance between any two of its elements.