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International Conference on Scale Space and Variational Methods in Computer Vision

SSVM 2013: Scale Space and Variational Methods in Computer Vision pp 392–403Cite as

Relations between Amoeba Median Algorithms and Curvature-Based PDEs

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Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 7893))

Abstract

This paper is concerned with the theoretical analysis of structure-adaptive median filter algorithms that approximate curvature-based PDEs for image filtering and segmentation. These so-called morphological amoeba filters, introduced by Lerallut et al. and further developped by Welk et al., achieve similar results as the well-known geodesic active contour and self-snakes PDEs. In the present work, the PDE approximated by amoeba active contours is derived in the general case. This PDE is structurally similar but not identical to the geodesic active contour equation. Implications for the qualitative behaviour of amoeba active contours as well as for the approximation of the pre-smoothed self-snakes equation are investigated.

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References

  1. Alvarez, L., Lions, P.L., Morel, J.M.: Image selective smoothing and edge detection by nonlinear diffusion. II. SIAM Journal on Numerical Analysis 29, 845–866 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Borgefors, G.: Distance transformations in digital images. Computer Vision, Graphics and Image Processing 34, 344–371 (1986)

    Article  Google Scholar 

  3. Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. In: Proc. Fifth International Conference on Computer Vision, pp. 694–699. IEEE Computer Society Press, Cambridge (1995)

    Chapter  Google Scholar 

  4. Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. International Journal of Computer Vision 22, 61–79 (1997)

    Article  MATH  Google Scholar 

  5. Feddern, C., Weickert, J., Burgeth, B., Welk, M.: Curvature-driven PDE methods for matrix-valued images. International Journal of Computer Vision 69(1), 91–103 (2006)

    Article  Google Scholar 

  6. Guichard, F., Morel, J.M.: Partial differential equations and image iterative filtering. In: Duff, I.S., Watson, G.A. (eds.) The State of the Art in Numerical Analysis. IMA Conference Series (New Series), vol. 63, pp. 525–562. Clarendon Press, Oxford (1997)

    Google Scholar 

  7. Ikonen, L., Toivanen, P.: Shortest routes on varying height surfaces using gray-level distance transforms. Image and Vision Computing 23(2), 133–141 (2005)

    Article  Google Scholar 

  8. Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Gradient flows and geometric active contour models. In: Proc. Fifth International Conference on Computer Vision, pp. 810–815. IEEE Computer Society Press, Cambridge (1995)

    Chapter  Google Scholar 

  9. Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Conformal curvature flows: from phase transitions to active vision. Archives for Rational Mechanics and Analysis 134, 275–301 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kimmel, R.: Fast edge integration. In: Osher, S., Paragios, N. (eds.) Geometric Level Set Methods in Imaging, Vision and Graphics, pp. 59–77. Springer, New York (2003)

    Chapter  Google Scholar 

  11. Lerallut, R., Decencière, E., Meyer, F.: Image processing using morphological amoebas. In: Ronse, C., Najman, L., Decencière, E. (eds.) Mathematical Morphology: 40 Years On. Computational Imaging and Vision, vol. 30. Springer, Dordrecht (2005)

    Google Scholar 

  12. Lerallut, R., Decencière, E., Meyer, F.: Image filtering using morphological amoebas. Image and Vision Computing 25(4), 395–404 (2007)

    Article  Google Scholar 

  13. 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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Osher, S., Rudin, L.I.: Feature-oriented image enhancement using shock filters. SIAM Journal on Numerical Analysis 27, 919–940 (1990)

    Article  MATH  Google Scholar 

  15. Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 629–639 (1990)

    Article  Google Scholar 

  16. Sapiro, G.: Vector (self) snakes: a geometric framework for color, texture and multiscale image segmentation. In: Proc. 1996 IEEE International Conference on Image Processing, Lausanne, Switzerland, vol. 1, pp. 817–820 (September 1996)

    Google Scholar 

  17. Tukey, J.W.: Exploratory Data Analysis. Addison–Wesley, Menlo Park (1971)

    Google Scholar 

  18. Welk, M.: Amoeba active contours. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds.) SSVM 2011. LNCS, vol. 6667, pp. 374–385. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  19. Welk, M., Breuß, M., Vogel, O.: Morphological amoebas are self-snakes. Journal of Mathematical Imaging and Vision 39, 87–99 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. You, Y.L., Kaveh, M., Xu, W., Tannenbaum, A.: Analysis and design of anisotropic diffusion for image processing. In: Proc. 1994 IEEE International Conference on Image Processing, Austin, Texas, USA, vol. 2, pp. 497–501 (November 1994)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Medical Informatics and Technology (UMIT), University for Health Sciences, Eduard-Wallnöfer-Zentrum 1, 6060, Hall/Tyrol, Austria

    Martin Welk

Authors
  1. Martin Welk
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Editors and Affiliations

  1. Fraunhofer Gesellschaft, Institut für Graphische Datenverarbeitung, Fraunhoferstraße 5, 64283, Darmstadt, Germany

    Arjan Kuijper

  2. Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010, Graz, Austria

    Kristian Bredies

  3. Institute for Computer Graphics and Vision, Graz University of Technology, Inffeldgasse 16, 8010, Graz, Austria

    Thomas Pock  & Horst Bischof  & 

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Welk, M. (2013). Relations between Amoeba Median Algorithms and Curvature-Based PDEs. In: Kuijper, A., Bredies, K., Pock, T., Bischof, H. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2013. Lecture Notes in Computer Science, vol 7893. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38267-3_33

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  • DOI: https://doi.org/10.1007/978-3-642-38267-3_33

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38266-6

  • Online ISBN: 978-3-642-38267-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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