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The Viscous Watershed Transform

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Abstract

The watershed transform is the basic morphological tool for image segmentation. Watershed lines, also called divide lines, are a topographical concept: a drop of water falling on a topographical surface follows a steepest descent line until it stops when reaching a regional minimum. Falling on a divide line, the same drop of water may glide towards one or the other of both adjacent catchment basins. For segmenting an image, one takes as topographic surface the modulus of its gradient: the associated watershed lines will follow the contour lines in the initial image. The trajectory of a drop of water is disturbed if the relief is not smooth: it is undefined for instance on plateaus. On the other hand, each regional minimum of the gradient image is the attraction point of a catchment basin. As gradient images generally present many minima, the result is a strong oversegmentation. For these reasons a more robust scheme is used for the construction of the watershed based on flooding: a set of sources are defined, pouring water in such a way that the altitude of the water increases with constant speed. As the flooding proceeds, the boundaries of the lakes propagate in the direction of the steepest descent line of the gradient. The set of points where lakes created by two distinct sources meet are the contours. As the sources are far less numerous than the minima, there is no more oversegmentation. And on the plateaus the flooding also is well defined and propagates from the boundary towards the inside of the plateau. Used in conjunction with markers, the watershed is a powerful, fast and robust segmentation method. Powerful: it has been used with success in a variety of applications. Robust: it is insensitive to the precise placement or shape of the markers. Fast: efficient algorithms are able to mimic the progression of the flood. In some cases however the resulting segmentation will be poor: the contours always belong to the watershed lines of the gradient and these lines are poorly defined when the initial image is blurred or extremely noisy. In such cases, an additional regularization has to take place. Denoising and filtering the image before constructing the gradient is a widely used method. It is however not always sufficient. In some cases, one desires smoothing the contour, despite the chaotic fluctuations of the watershed lines. For this two options are possible. The first consists in using a viscous fluid for the flooding: a viscous fluid will not be able to follow all irregularities of the relief and produce lakes with smooth boundaries. Simulating a viscous fluid is however computationally intensive. For this reason we propose an alternative solution, in which the topographic surface is modified in such a way that flooding it with a non viscous fluid will produce the same lakes as flooding the original relief with a viscous fluid. On this new relief, the standard watershed algorithm can be used, which has been optimized for various architectures. Two types of viscous fluids will be presented, yielding two distinct regularization methods. We will illustrate the method on various examples.

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Authors and Affiliations

  1. Centre de Mathé matiques et Leurs Applications, Ecole Normale Supérieure de Cachan, 61, av. du Pt Wilson, 94235, Cachan, France

    Corinne Vachier

  2. Centre de Morphologie Mathématique, Ecole Nationale Supérieure des Mines de Paris, 35, rue Saint Honoré, 77305, Fontainebleau, France

    Fernand Meyer

Authors
  1. Corinne Vachier
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  2. Fernand Meyer
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Correspondence to Corinne Vachier or Fernand Meyer.

Additional information

Corinne Vachier received an engineer degree from the Ecole Supérieure d’Electricité, Paris and a Ph.D. in Mathematical Morphology from the Ecole des Mines de Paris, respectively in 1991 and 1995. From 1992 to 1995, she was research engineer in General Electric Medical Systems, Buc, France and phd student in the Centre de Morphologie Mathématique (CMM) of the Ecole des Mines de Paris. She became in 1996 an associate professor at the University Paris 12. She joined Jean-Michel Morel’s Team at the Centre de Mathématiques et Leurs Applications (CMLA) at the Ecole Nationale Supérieure de Cachan in 2001. Her research interests include mathematical morphology with emphasis on multiscale representations. Current applicative interests are focused on medical imaging.

Fernand Meyer got an engineer degree from the Ecole des Mines de Paris in 1975. He works since 1975 at the Centre de Morphologie Mathématique (CMM) of the Ecole des Mines de Paris, where he is currently director. His first research area was “Early and automatic detection of cervical cancer on cytological smears,” subject of his PhD thesis, obtained in 1979. He participated actively to the development of mathematical morphology: particle reconstruction, top-hat transform, the morphological segmentation paradigm based on the watershed transform and markers, the theory of digital skeleton, the introduction of hierarchical queues for high speed watershed computations, morphological interpolations, the theory of levelings, multiscale segmentation.

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Vachier, C., Meyer, F. The Viscous Watershed Transform. J Math Imaging Vis 22, 251–267 (2005). https://doi.org/10.1007/s10851-005-4893-3

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