Abstract
In this work a marker-controlled and regularized watershed segmentation is proposed. 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 fixed set of challenging images, the comparison of these two methods shows that the present level set formulation performs better than a standard watershed segmentation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Caselles, V., et al.: A geometric model for active contours in image processing. Numer. Math. 66(1), 1–31 (1993)
Nielsen, L.K., et al.: Reservoir Description using a Binary Level Set Model. In: Tai, X.-C., et al. (eds.) Image Processing Based on Partial Differential Equations, pp. 403–426. Springer, Heidelberg (2006)
Christiansen, O., Tai, X.C.: Fast Implementation of Piecewise Constant Level Set Methods. In: Tai, X.-C., et al. (eds.) Image Processing Based on Partial Differential Equations, pp. 289–308. Springer, Heidelberg (2006)
Lie, J., Lysaker, M., Tai, X.-C.: A binary level set model and some applications to Mumford-Shah image segmentation. IEEE Transactions on Image Processing 15(5), 1171–1181 (2006)
Vese, L.A., Chan, T.F.: A multiphase level set framework for image segmentation using the Mumford and Shah model. International Journal of Computer Vision 50(3), 271–293 (2002)
Cremers, D., et al.: Diffusion snakes: introducing statistical shape knowledge into the Mumford–Shah functional. International Journal of Computer Vision 50, 295–313 (2002), citeseer.ist.psu.edu/cremers02diffusion.html
Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. International Journal of Computer Vision 1(4), 321–331 (1988), http://dx.doi.org/10.1007/BF00133570
Najman, L., Schmitt, M.: Watershed of a continuous function. Signal Process. 38(1), 99–112 (1994)
Meyer, F.: Topographic distance and watershed lines. Signal Processing 38(1), 113–125 (1994)
Vincent, L., Soille, P.: Watersheds in digital spaces: An efficient algorithm based on immersion simulations. IEEE Trans. Pattern Anal. Mach. Intell. 13(6), 583–598 (1991), doi:10.1109/34.87344
Nguyen, H., Worring, M., van den Boomgaard, R.: Watersnakes: Energy-driven watershed segmentation. IEEE Trans. on PAMI 25(3), 330–342 (2003), citeseer.ist.psu.edu/article/.html
Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Communications on Pure Applied Mathematics 42, 577–685 (1989)
Chan, T., Vese, L.: Active contours without edges. IEEE Trans. on Image Processing 10, 266–277 (2001)
Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics 79, 12–49 (1988), citeseer.ist.psu.edu/osher88fronts.html
Lie, J., Lysaker, M., Tai, X.-C.: A variant of the level set method and applications to image segmentation. Mathematics of Computation 75(255), 1155–1174 (2006)
Felkel, P., Bruckschwaiger, M., Wegenkittl, R.: Implementation and complexity of the watershed-from-markers algorithm computed as a minimal cost forest. Computer Graphics Forum 20(3) (2001)
Chang, S.G., Yu, B., Vetterli, M.: Spatially adaptive wavelet thresholding with context modeling for image denoising. IEEE Transactions on Image Processing 9(9), 1522–1531 (2000)
Hodneland, E., et al.: Automated detection of tunneling nanotubes in 3d images. Cytometry Part A 69(9), 961–972 (2006)
Arbeléz, P.A., Cohen, L.D.: Energy partitions and image segmentation. J. Math. Imaging Vis. 20(1-2), 43–57 (2004)
Appel, K.I., Haken, W.: Every planar map is four colorable. Illinois J. Math. 21, 429–567 (1977)
Robertson, N., et al.: A new proof of the four colour theorem. Electronic Research Announcements of the American Mathematical Society 2(1) (1996), citeseer.ist.psu.edu/robertson96new.html
Dervieux, A., Thomasset, F.: A finite element method for the simulation of Rayleigh–Taylor instability. In: Rautman, R. (ed.) Approximation Methods for Navier–Stokes Problems. Lecture Notes in Mathematics, vol. 771, pp. 145–158. Springer, Heidelberg (1979)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Tai, XC., Hodneland, E., Weickert, J., Bukoreshtliev, N.V., Lundervold, A., Gerdes, HH. (2007). Level Set Methods for Watershed Image Segmentation. In: Sgallari, F., Murli, A., Paragios, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2007. Lecture Notes in Computer Science, vol 4485. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72823-8_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-72823-8_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72822-1
Online ISBN: 978-3-540-72823-8
eBook Packages: Computer ScienceComputer Science (R0)