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Four-Color Theorem and Level Set Methods for Watershed Segmentation

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Abstract

A marker-controlled and regularized watershed segmentation is proposed for cell segmentation. Only a few previous studies address the task of regularizing the obtained watershed lines from the traditional marker-controlled watershed segmentation. In the present formulation, the topographical distance function is applied in a level set formulation to perform the segmentation, and the regularization is easily accomplished by regularizing the level set functions. Based on the well-known Four-Color theorem, a mathematical model is developed for the proposed ideas. With this model, it is possible to segment any 2D image with arbitrary number of phases with as few as one or two level set functions. The algorithm has been tested on real 2D fluorescence microscopy images displaying rat cancer cells, and the algorithm has also been compared to a standard watershed segmentation as it is implemented in MATLAB. For a fixed set of markers and a set of challenging images, the comparison of these two methods shows that the present level set formulation performs better than a standard watershed segmentation.

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Authors and Affiliations

  1. Department of Biomedicine, University of Bergen, Jonas Lies vei 91, 5009, Bergen, Norway

    Erlend Hodneland & Hans-Hermann Gerdes

  2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637616, Singapore

    Xue-Cheng Tai

  3. Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5007, Bergen, Norway

    Xue-Cheng Tai

Authors
  1. Erlend Hodneland
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  2. Xue-Cheng Tai
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  3. Hans-Hermann Gerdes
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Correspondence to Xue-Cheng Tai.

Additional information

The authors wish to thank Steffen Gurke and Nickolay Bukhoresthliev for providing the majority of pictures in this work. Erlend Hodneland was supported by the Norwegian Cancer Society, project number A05103/004.

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Hodneland, E., Tai, XC. & Gerdes, HH. Four-Color Theorem and Level Set Methods for Watershed Segmentation. Int J Comput Vis 82, 264–283 (2009). https://doi.org/10.1007/s11263-008-0199-4

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  • DOI: https://doi.org/10.1007/s11263-008-0199-4

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