Watershed of a continuous function

The notion of watershed, used in morphological segmentation, has only a digital definition. In this paper, we propose to extend this definition to the continuous plane. Using this continuous definition, we present the watershed differences with classical edge detectors. We then exhibit a metric in the plane for which the watershed is a skeleton by influence zones and show the lower semicontinuous behaviour of the associated skeleton. This theoretical approach suggests an algorithm for solving the eikonal equation: ‖∇ƒ‖ = g. Finally, we end with some new watershed algorithms, which present the advantage of allowing the use of markers and/or anchor points, thus opening the way towards grey-tone skeletons.

[1]  Rachid Deriche,et al.  On corner and vertex detection , 1991, Proceedings. 1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[2]  P. Lions,et al.  Shape-from-shading, viscosity solutions and edges , 1993 .

[3]  Y. J. Tejwani,et al.  Robot vision , 1989, IEEE International Symposium on Circuits and Systems,.

[4]  G. Matheron.  Random Sets and Integral Geometry , 1976 .

[5]  Luc Vincent,et al.  Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[6]  Serge Beucher.  Segmentation d'images et morphologie mathématique , 1990 .

[7]  Tomaso A. Poggio,et al.  On Edge Detection , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[8]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[9]  Francoise J. Preteux,et al.  New concepts in mathematical morphology: the topographical and differential distance functions , 1991, Optics & Photonics.

[10]  Warren M. Kruegger,et al.  The Geometry of Differential Operators with Application to Image Processing , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[11]  Piet W. Verbeek,et al.  Shading from shape, the eikonal equation solved by grey-weighted distance transform , 1990, Pattern Recognit. Lett..

[12]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[13]  John F. Canny,et al.  A Computational Approach to Edge Detection , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[14]  B.J.H. Verwer.  Distance transforms, metrics, algorithms and applications , 1991 .

[15]  M. Schmitt.  Géodésic arcs in non-euclidean mehics: application to the propagation function , 1989 .

[16]  D Marr,et al.  Theory of edge detection , 1979, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[17]  I. G. BONNER CLAPPISON.  Editor , 1960, The Electric Power Engineering Handbook - Five Volume Set.

[18]  Rachid Deriche,et al.  Accurate corner detection: an analytical study , 1990, [1990] Proceedings Third International Conference on Computer Vision.

[19]  Serge Beucher,et al.  Use of watersheds in contour detection , 1979 .

[20]  Michel Schmitt,et al.  Some examples of algorithm analysis in computational geometry by means of mathematical morphological techniques , 1988, Geometry and Robotics.

[21]  Emes Dynamiques.  Ecole Nationale Superieure des Mines de Paris , 1993 .