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Root Images of Median Filters — Semi-topological Approach

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Geometry, Morphology, and Computational Imaging

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2616))

Abstract

Median filters are frequently used in signal analysis because they are robust edge-preserving smoothing filters. Since median filters are nonlinear filters, the tools of linear theory are not applicable to them. One approach to deal with nonlinear filters consists in investigating their root images (fixed elements or signals transparent to the filter). Whereas for one-dimensional median filters the set of all root signals can be completely characterized, this is not true for higher dimensional filters. Tyan (1981) and Döhler (1989) proposed a method for construction of small root images for two-dimensional median filters. Although the Tyan-Döhler construction is valid for a wide class of median filters, their arguments were not correct and their assertions do not hold universally. In this paper we give a rigorous treatment for the construction of Tyan and Döhler. Moreover, the approach is generalized to the d-dimensional case.

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References

  1. Alpers, A.: Digital topology: Regular sets and root images of the cross-median filter. J. Math. Imaging and Vision, to appear 186

    Google Scholar 

  2. Döhler, H.-U.: Generation of root signals of two dimensional median filters. Signal Processing 18 (1989) 269–276.

    Article  Google Scholar 

  3. Eckhardt, U.: Root Images of Median Filters. J. Math. Imaging and Vision, to appear 179, 194

    Google Scholar 

  4. Eckhardt, U., Hundt, E.: Topological approach to mathematical morphology. In: Solina, F., Kropatsch, W.G., Klette, R., Bajcsy, R. (eds.): Advances in Computer Vision. Springer-Verlag, Wien New York (1997) 11–20

    Google Scholar 

  5. Eckhardt, U., Latecki, L., Maderlechner, G.: Irreducible and thin binary sets. In: Arcelli, C., Cordella, L.P., Sanniti di Baja, G. (eds.): Aspects of Visual Form Processing. 2nd International Workshop on Visual Form, Capri, Italy, May 30-June 2, 1994. World Scientific Publishing Co. Pte. Ltd., Singapore New Jersey London Hong Kong (1994) 199–208

    Google Scholar 

  6. Eckhardt, U., Maderlechner, G.: Invariant thinning. Int. J. Pattern Recognition and Artificial Intelligence 7 (1993) 1115–1144

    Article  Google Scholar 

  7. Estrakh, D. D., Mitchell, H.B., Schaefer, P. A., Mann, Y., Peretz, Y.: “Soft” median adaptive predictor for lossless picture compression. Signal Processing 81 (2001) 1985–1989

    MATH  Google Scholar 

  8. Evans, L.C.: Convergence of an algorithm for mean curvature motion. Indiana Univ. Math. J. 42 (1993) 533–557

    Article  MATH  MathSciNet  Google Scholar 

  9. Gallagher, N.C., Wise, G. L.: A theoretical analysis of the properties of median filters IEEE Trans. ASSP-29 (1981) 1136–1141

    Google Scholar 

  10. Gan, Z. J., Mao, M.: Two convergence theorems on deterministic properties of median filters. IEEE Trans. Signal Processing 39 (1991) 1689–1691

    Article  Google Scholar 

  11. Gilbert, E. N.: Lattice theoretic properties of frontal switching functions. J. Math. Phys. 33 (1954) 57–67

    MATH  Google Scholar 

  12. Goles, E., Olivos, J.: Comportement periodique des fonctions à seuil binaires et applications. Discr. Appl. Math. 3 (1981) 93–105

    Article  MATH  MathSciNet  Google Scholar 

  13. Guichard, F., Morel, J.-M.: Partial differential equations and image iterative filtering. In: Du., I. S., Watson, G.A. (eds.): The State of the Art in Numerical Analysis, Based on the proceedings of a conference organized by the Institute of Mathematics and its Applications (IMA), University of York, York, GB, April 1-4, 1996. Inst. Math. Appl. Conf. Ser. New Ser. v. 63. Clarendon Press, Oxford (1997) pp525-562 177

    Google Scholar 

  14. Heijmans, H. J. A. M.: Morphological Image Operators. Advances in Electronics and Electron Physics. Academic Press, Inc., Harcourt, Brace & Company, Publishers, Boston San Diego New York London Sydney Tokyo Toronto (1994)

    Google Scholar 

  15. Krabs, W.: Mathematical Foundations of Signal Theory. Sigma Series in Applied Mathematics, Vol. 6. Heldermann Verlag, Berlin (1995)

    Google Scholar 

  16. Latecki, L. J.: Discrete Representation of Spatial Objects in Computer Vision. Computational Imaging and Vision. Kluwer Academic Publishers, Dordrecht Boston London (1998)

    Google Scholar 

  17. Latecki, L.: Digitale und Allgemeine Topologie in der bildhaften Wissensrepr äsentation. DISKI Dissertationen zur Künstlichen Intelligenz 9. infix, St. Augustin (1992)

    Google Scholar 

  18. Merriman, B., Bence, J., Osher, S.: Diffusion generated motions by mean curvature. UCLA Computational and Applied Mathematics Reports (1992) 92–18

    Google Scholar 

  19. Pitas, I., Venetsanopoulos, A.N.: Order statistics in digital image processing. Proc. IEEE 80 (1992) 1893–1921

    Google Scholar 

  20. Pitas, I., Venetsanopoulos, A.N.: Nonlinear Digital Filters: Principles and Applications. The Kluwer International Series in Engineering and Computer Science, 84. VLSI, Computer Archtecture and Digital Signal Processing. Kluwer Academic Publishers Group, Boston Dordrecht London (1990)

    Google Scholar 

  21. Rinow, W.: Lehrbuch der Topologie. Hochschulbücher für Mathematik, Bd. 79. Deutscher Verlag der Wissenschaften, Berlin (1975)

    Google Scholar 

  22. Ronse, C.: Lattice-theoretical fixpoint theorems in morphological image filtering. J. Math. Imaging and Vision 4 (1994) 19–41

    Article  MATH  MathSciNet  Google Scholar 

  23. Serra J., (ed.): Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances. Academic Press, Harcourt Brace Jovanovich Publishers, London San Diego New York Boston Sydney Tokyo Toronto 1988)

    Google Scholar 

  24. Stone, M.H.: Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 41 (1937) 375–481

    Article  MATH  MathSciNet  Google Scholar 

  25. Tuckey, J.W.: Exploratory Data Analysis. Addison-Wesley, Reading. Mass. (1977)

    Google Scholar 

  26. Tyan, S.G.: Median filtering: Deterministic properties. In: Huang, T. S. (ed.): Two-Dimensional Digital Signal Processing II. Transforms and Median Filters. Topics in Applied Physics, Vol. 43. Springer-Verlag, Berlin Heidelberg New York (1981) 197–217

    Google Scholar 

  27. Valentine, F.A.: Convex Sets. McGraw-Hill Series in Higher Mathematics. Mc-Graw-Hill Book Company, New York San Francisco Toronto London (1964)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Fachbereich Mathematik - Optimierung und Approximation, Universität Hamburg, Bundesstraße 55, D-20146, Hamburg

    Ulrich Eckhardt

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  1. Ulrich Eckhardt
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Editor information

Editors and Affiliations

  1. JAIST, School of Information Science, 1-1 Asahidai, Tatsunokuchi, 923-1202, Ishikawa, Japan

    Tetsuo Asano

  2. Computer Science Dept. and CITR, University of Auckland, Tamaki Campus, Glen Innes, 1005, Auckland, New Zealand

    Reinhard Klette

  3. LSIIT UMR 7005 CNRS-ULP, Parc d’Innovation, Boulevard Sebastien Brant, BP 10413, 67412, Illkirch, France

    Chrisitan Ronse

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Eckhardt, U. (2003). Root Images of Median Filters — Semi-topological Approach. In: Asano, T., Klette, R., Ronse, C. (eds) Geometry, Morphology, and Computational Imaging. Lecture Notes in Computer Science, vol 2616. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36586-9_12

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  • DOI: https://doi.org/10.1007/3-540-36586-9_12

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00916-0

  • Online ISBN: 978-3-540-36586-0

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