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Median Filters

  • Chapter
Nonlinear Digital Filters

Part of the book series: The Springer International Series in Engineering and Computer Science ((SECS,volume 84))

Abstract

A major approach to nonlinear filtering is based on robust estimation and especially on local L-estimators, i.e., on order statistics. The main advantage of this approach is its computational simplicity and speed. Filters based on order statistics.usually have good behavior in the presence of additive white Gaussian noise and long-tailed additive noise. They have good edge preservation properties and they can become adaptive. Thus, they are suitable in a variety of applications where classical linear filters fail, notably in digital image filtering. The best known and most widely used filter based on order statistics is the median filter. Originally, the median was widely used in statistics. It was introduced by Tukey in time series analysis in 1970. Later on, the median filter and its modifications have found numerous applications in digital image processing [2,3,13], in digital image analysis [15,46], in digital TV applications [44,47], in speech processing and coding [20,23], in cepstral analysis [45], and in various other applications. The reason for its success is its good performance and computational simplicity. The theoretical analysis of its deterministic and statistical properties has started at the end of the seventies. A description of the early theoretical results can be found in three very good review chapters in edited books, namely in [4,13,21]. The material of this chapter is based on the recently published results, as well as in the classical work described in [4, 13,21].

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References

  1. M.D. Levine, Vision in man and machine, Mc Graw-Hill, 1985.

    Google Scholar 

  2. W.K. Pratt, Digital image processing, Wiley, 1978.

    Google Scholar 

  3. H.C. Andrews, B.R. Hunt, Digital image restoration, Prentice-Hall 1977.

    Google Scholar 

  4. G.R. Arce, N.C. Gallagher, T.A. Nodes, “Median filters: theory for one-and two-dimensional filters”, in Advances in Computer Vision and Image Processing, T.S. Huang editor, JAI Press, 1986.

    Google Scholar 

  5. H.A. David, Order statistics,,Wiley, 1981.

    Google Scholar 

  6. J.W. Tukey, Exploratory data analysis, Addison-Wesley 1977.

    Google Scholar 

  7. F. Kuhlmann, G.L. Wise, “On the second moment properties of median filtered sequences of independent data”, IEEE Transactions on Communications, vol. COM-29, no. 9, pp. 1374–1379, Sept. 1981.

    Google Scholar 

  8. G.Y. Liao, T.A. Nodes, N.C. Gallagher, “Output distributions of two-dimensional median filters”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-33, no. 5, pp. 1280–1295, Oct. 1985.

    Article  Google Scholar 

  9. T.A. Nodes, N.C. Gallagher, “The output distribution of the median type filters”, IEEE Transactions on Communications, vol. COM-32, pp. 532541, May 1984.

    Google Scholar 

  10. E.L. Lehmann, Theory of point estimation, J.Wiley, 1983.

    Google Scholar 

  11. J.W. Tukey, “A survey of sampling for contaminated distributions”, in Contributions to probability and statistics, O1kin editor, Stanford University Press, 1960.

    Google Scholar 

  12. F. Hampel, E. Ronchetti, P. Rousseevw, W. Stahel, Robust statistics: An approach based on influence functions, Wiley, 1986.

    Google Scholar 

  13. B.I. Justusson, “Median filtering: statistical properties” in Two-dimensional digital signal processing II, T.S. Huang editor, Springer Verlag, 1981.

    Google Scholar 

  14. P.J. Huber, “Robust estimation of a location parameter”, Ann. Math. Statist., vol. 35, pp. 73–101, 1964.

    Article  MathSciNet  MATH  Google Scholar 

  15. G.J. Yong, T.S. Huang, “The effect of median filtering in edge location estimation”, Computer Vision, Graphics and Image Processing, vol. 15, pp. 224–245, 1981.

    Article  Google Scholar 

  16. E. Ataman, V.K. Aatre, K.M. Wong, “Some statistical properties of median filters”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-29, no. 5, pp. 1073–1075, Oct. 1981.

    Article  Google Scholar 

  17. A.C. Bovik, T.S. Huang, D.C. Munson, “The effect of median filtering on edge estimation and detection”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PAMI-9, no. 2, pp. 181–194, March 1987.

    Article  Google Scholar 

  18. D.H. Ballard, C.M. Brown, Computer vision, Prentice-Hall, 1983.

    Google Scholar 

  19. A.C. Bovik, “Streaking in median filtered images”, IEEE Transactions on Acoustics, Speech and signal Processing, vol. ASSP-35, pp. 493–503, April 1987.

    Google Scholar 

  20. L.R. Rabiner, M.R. Sambur, C.E. Schmidt, “Applications of a nonlinear smoothing algorithm to speech processing”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-23, pp. 552–557, Dec. 1975.

    Google Scholar 

  21. S.G Tyan, “Median filtering: Deterministic properties”, Two-dimensional Signal Processing II, T.S. Huang editor, Springer Verlag, 1981.

    Google Scholar 

  22. N.C. Gallagher, G.L. Wise, “A theoretical analysis of the properties of the median filter”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-29, no. 6, pp. 1135–1141, Dec. 1981.

    Google Scholar 

  23. G. Arce, N.C. Gallagher, “State description for the root-signal set of median filters”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-30, no. 6, pp. 894–902, Dec. 1982.

    Article  Google Scholar 

  24. G.R. Arce, N.C. Gallagher, “Root-signal set analysis for median filters”, Proc. Allerton Conference Commun. Contr. Comp., Oct. 1980.

    Google Scholar 

  25. J.P. Fitch, E.J. Coyle, N.C. Gallagher, “Root properties and convergence rates for median filters”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-33, no. 1, pp. 230–240, Feb. 1985.

    Article  Google Scholar 

  26. P. D. Wendt, E.J. Coyle, N.C. Gallagher, “Some convergence properties of median filters”, IEEE Transactions on Circuits and Systems, vol. CAS-33, no. 3, pp. 276–286, March 1986.

    Article  Google Scholar 

  27. O.R. Mitchell, E. Delp, “Multilevel graphics representation using block truncation coding”, Proc. IEEE, vol. 68, July 1980.

    Google Scholar 

  28. J.P. Fitch, E.J. Coyle, N.C. Gallagher, “Median filtering by threshold decomposition”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-32, no. 6, pp. 1183–1188, Dec. 1984.

    Article  Google Scholar 

  29. G.R. Arce, R.L. Stevenson, “On the synthesis of median filter systems”, IEEE Transactions on Circuits and Systems, vol. CAS-34, no. 4, pp. 420–429, April 1987.

    Article  Google Scholar 

  30. G.R. Arce, “Statistical threshold decomposition for recursive and nonrecursive median filters”, IEEE Transactions on Information Theory, vol. IT-29, March 1986.

    Google Scholar 

  31. P.M. Narendra, “A separable median filter for image noise smoothing”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PAMI-3, no. 1, pp. 20–29, Jan. 1981.

    Google Scholar 

  32. T.A. Nodes, N.C. Gallagher, “Two-dimensional root structures and convergence properties of the separable median filter”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-31, no. 6, pp. 13501365, Dec. 1983.

    Google Scholar 

  33. M.P. Loughlin, G.R. Arce, “Deterministic properties of the recursive separable median filter”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-35, no. 1, pp. 98–106, Jan. 1987.

    Article  Google Scholar 

  34. T.A. Nodes, N.C. Gallagher, “Median filters: some modifications and their properties”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-30, no. 5, pp. 739–746, Oct. 1982.

    Article  Google Scholar 

  35. G.R. Arce, R.J. Crinon, “Median filters: analysis of two-dimensional recursively filtered signals” IEEE Int. Conf. on Acoustics, Speech and Signal Processing, 1984.

    Google Scholar 

  36. R.J. Crinon, G.R. Arce, “Median filters: analysis for signals with additive impulsive noise”, Proc. 21st Allerton Conference, Oct. 1983.

    Google Scholar 

  37. J. Astola, P. Heinonen, Y. Neuvo, “On root structure of median and median-type filters”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-35, no. 8, pp. 1199–1201, Aug. 1987.

    Google Scholar 

  38. A. Papoulis, Probability, random variables and stochastic processes, McGraw-Hill, 1984.

    Google Scholar 

  39. P.D. Wendt, E.J. Coyle, N.C. Gallagher Jr., “Stack filters”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-34, no. 4, pp. 898–911, Aug. 1986.

    Article  Google Scholar 

  40. E.J. Coyle, J.H. Lin, “Stack filters and the mean absolute error criterion”, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-36, no. 8, pp. 1244–1254, Aug. 1988.

    Google Scholar 

  41. E.N. Gilbert, “Lattice-theoretic properties of frontal switching functions”, Journal of Mathematical Physics, vol. 33, pp. 57–67, Apr. 1954.

    MATH  Google Scholar 

  42. W.W. Boles, M. Kanewski, M. Simaan, “Recursive two-dimensional median filtering algorithms for fast image root extraction”, IEEE Transactions on Circuits and Systems, vol. CAS-35, no. 10, pp. 1323–1326, Oct. 1988.

    Google Scholar 

  43. G. Arce, N.C. Gallagher, “Stochastic analysis for the recursive median filter process”, IEEE Transactions on Information Theory, vol. IT-34, no. 4, pp. 669–679, July 1988.

    Article  MathSciNet  Google Scholar 

  44. S.S. Perlman, S. Eisenhandler, P.W. Lyons, M.J. Shumila, “Adaptive median filtering for impulse noise elimination in real-time TV signals”, IEEE Transactions on Communications, vol. COM-35, no. 6, pp. 646652, June 1987.

    Google Scholar 

  45. K.J. Hahn, K.M. Wong, “Median filtering of cepstra”, Proc. 1983 International Electrical and Electronics Conference,pp. 352–355, Toronto, Canada, 1983.

    Google Scholar 

  46. P. Zamperoni, “Feature extraction by rank-order filtering for image segmentation”, International Journal of Pattern Analysis and Artificial Intelligence, vol. 2, no. 2, pp. 301–319, 1988.

    Article  Google Scholar 

  47. S.S.H. Naqvi, N.C. Gallagher, E.J. Coyle, “An application of median filter to digital TV”, Proc. 1986 IEEE Int. Conf. on Acoustics, Speech and Signal Processing,Tokyo, Japan, 1986.

    Google Scholar 

  48. V.J. Gebski, “Some properties of splicing when applied to nonlinear smoothers”, Comput. Statist. Data Analysis, no. 3, pp. 151–157, 1985.

    Article  Google Scholar 

  49. D.R.K. Brownrigg, “Weighted median filters”, Commun. Assoc. Comput. Machinery, vol. 27, pp. 807–818, Aug. 1984.

    Google Scholar 

  50. O. Yli-Harja, “Median filters: extensions, analysis and design”, Technical Report, Lappeenranta University of Technology, Finland, 1989.

    Google Scholar 

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Authors and Affiliations

  1. Aristotelian University of Thessaloniki, Greece

    I. Pitas

  2. University of Toronto, Canada

    A. N. Venetsanopoulos

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  1. I. Pitas
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© 1990 Springer Science+Business Media New York

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Pitas, I., Venetsanopoulos, A.N. (1990). Median Filters. In: Nonlinear Digital Filters. The Springer International Series in Engineering and Computer Science, vol 84. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6017-0_4

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  • DOI: https://doi.org/10.1007/978-1-4757-6017-0_4

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-5120-5

  • Online ISBN: 978-1-4757-6017-0

  • eBook Packages: Springer Book Archive

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