Abstract

A new generalized version of the median filter is proposed. The new version preserves clear definitions of image details while keeping the ability of removing the impulsive noise; it is still simple in principle and easy to use. The input–output relationship and noise-removing power of the new version are compared with those of the standard median filter.

© 1988 Optical Society of America

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References

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  1. J. W. Tukey, Exploratory Data Analysis (Addison-Wesley, Reading, MA, 1977, Prelim. ed. 1971).
  2. B. R. Frieden, “New Restoring Algorithm for the Preferential Enhancement of Edge Gradients,” J. Opt. Soc. Am. 66, 280 (1976).
    [CrossRef]
  3. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), p. 330.
  4. T. S. Huang, Ed., Two-Dimensional Digital Signal Processing II; Transforms and Median Filters (Springer-Verlag, Berlin, 1981).
  5. E. Ataman, V. K. Aatre, K. M. Wong, “Some Statistical Properties of Median Filters,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1073 (1981).
    [CrossRef]
  6. T. Gotoh, T. Toriu, E. Yamamoto, “Analysis of Median Filter Characteristics and Its Application to Contour Extraction,” in Proceedings, Seventh International Conference on Pattern Recognition, Vol. 2 (IEEE Computer Press, Silver Spring, MD, 1984), p. 1098.
  7. J. P. Fitch, E. J. Coyle, N. C. Gallagher, “Median Filtering by Threshold Decomposition,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1183 (1984).
    [CrossRef]
  8. J. P. Fitch, E. J. Coyle, N. C. Gallagher, “The Analog Median Filter,” IEEE Trans. Circuits Syst. CAS-33, 94 (1986).
    [CrossRef]
  9. S. L. Hurt, A. Rosenfeld, “Noise Reduction in Three-Dimensional Digital Images,” Pattern Recognition 17, 407 (1981).
    [CrossRef]
  10. G. A. Mastin, “Adaptive Filters for Digital Image Noise Smoothing,” Comput. Vision Graphics Image Process. 31, 103 (1985).
    [CrossRef]
  11. E. Ochoa, J. P. Allebach, D. W. Sweeney, “Optical Median Filtering Using Threshold Decomposition,” Appl. Opt. 26, 252 (1987).
    [CrossRef] [PubMed]
  12. P. M. Narendra, “A Separable Median Filter for Image Noise Smoothing,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-3, 20 (1981).
    [CrossRef]
  13. T. A. Nodes, N. C. Gallagher, “Median Filters: Some Modifications and Their Properties,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 739 (1982).
    [CrossRef]
  14. A. C. Bovik, T. S. Huang, D. C. Munson, “A Generalization of Median Filtering Using Linear Combinations of Order Statistics,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1342 (1983).
    [CrossRef]
  15. D. R. K. Brownrigg, “The Weighted Median Filter,” Commun. ACM 27, 807 (1984).
    [CrossRef]
  16. E. R. Davies, “The Median Filters: An Appraisal and a New Truncated Version,” in Proceedings Seventh International Conference on Pattern Recognition, Vol. 1 (IEEE Computer Press, Silver Spring, MD, 1984), p. 590.
  17. A. Lev, S. W. Zucker, A. Rosenfeld, “Iterative Enhancement of Noisy Images,” IEEE Trans. Syst. Man Cybern. SMC-7, 435 (1977).
    [CrossRef]
  18. L. S. Davis, A. Rosenfeld, “Noise Cleaning by Iterated Local Averaging,” IEEE Trans. Syst. Man Cybern. SMC-8, 705 (1978).
  19. F. Tomita, S. Tsuji, “Extraction of Multiple Regions by Smoothing in Selected Neighborhoods” IEEE Trans. Syst. Man Cybern. SMC-7, 107 (1977).
  20. M. Nagao, T. Matsuyama, “Edge Preservation Smoothing,” Comput. Graphics Image Process. 9, 394 (1979).
    [CrossRef]
  21. N. Ford, How Machines Think (Wiley, New York, 1987).

1987 (1)

1986 (1)

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “The Analog Median Filter,” IEEE Trans. Circuits Syst. CAS-33, 94 (1986).
[CrossRef]

1985 (1)

G. A. Mastin, “Adaptive Filters for Digital Image Noise Smoothing,” Comput. Vision Graphics Image Process. 31, 103 (1985).
[CrossRef]

1984 (2)

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “Median Filtering by Threshold Decomposition,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1183 (1984).
[CrossRef]

D. R. K. Brownrigg, “The Weighted Median Filter,” Commun. ACM 27, 807 (1984).
[CrossRef]

1983 (1)

A. C. Bovik, T. S. Huang, D. C. Munson, “A Generalization of Median Filtering Using Linear Combinations of Order Statistics,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1342 (1983).
[CrossRef]

1982 (1)

T. A. Nodes, N. C. Gallagher, “Median Filters: Some Modifications and Their Properties,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 739 (1982).
[CrossRef]

1981 (3)

P. M. Narendra, “A Separable Median Filter for Image Noise Smoothing,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-3, 20 (1981).
[CrossRef]

S. L. Hurt, A. Rosenfeld, “Noise Reduction in Three-Dimensional Digital Images,” Pattern Recognition 17, 407 (1981).
[CrossRef]

E. Ataman, V. K. Aatre, K. M. Wong, “Some Statistical Properties of Median Filters,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1073 (1981).
[CrossRef]

1979 (1)

M. Nagao, T. Matsuyama, “Edge Preservation Smoothing,” Comput. Graphics Image Process. 9, 394 (1979).
[CrossRef]

1978 (1)

L. S. Davis, A. Rosenfeld, “Noise Cleaning by Iterated Local Averaging,” IEEE Trans. Syst. Man Cybern. SMC-8, 705 (1978).

1977 (2)

F. Tomita, S. Tsuji, “Extraction of Multiple Regions by Smoothing in Selected Neighborhoods” IEEE Trans. Syst. Man Cybern. SMC-7, 107 (1977).

A. Lev, S. W. Zucker, A. Rosenfeld, “Iterative Enhancement of Noisy Images,” IEEE Trans. Syst. Man Cybern. SMC-7, 435 (1977).
[CrossRef]

1976 (1)

Aatre, V. K.

E. Ataman, V. K. Aatre, K. M. Wong, “Some Statistical Properties of Median Filters,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1073 (1981).
[CrossRef]

Allebach, J. P.

Ataman, E.

E. Ataman, V. K. Aatre, K. M. Wong, “Some Statistical Properties of Median Filters,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1073 (1981).
[CrossRef]

Bovik, A. C.

A. C. Bovik, T. S. Huang, D. C. Munson, “A Generalization of Median Filtering Using Linear Combinations of Order Statistics,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1342 (1983).
[CrossRef]

Brownrigg, D. R. K.

D. R. K. Brownrigg, “The Weighted Median Filter,” Commun. ACM 27, 807 (1984).
[CrossRef]

Coyle, E. J.

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “The Analog Median Filter,” IEEE Trans. Circuits Syst. CAS-33, 94 (1986).
[CrossRef]

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “Median Filtering by Threshold Decomposition,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1183 (1984).
[CrossRef]

Davies, E. R.

E. R. Davies, “The Median Filters: An Appraisal and a New Truncated Version,” in Proceedings Seventh International Conference on Pattern Recognition, Vol. 1 (IEEE Computer Press, Silver Spring, MD, 1984), p. 590.

Davis, L. S.

L. S. Davis, A. Rosenfeld, “Noise Cleaning by Iterated Local Averaging,” IEEE Trans. Syst. Man Cybern. SMC-8, 705 (1978).

Fitch, J. P.

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “The Analog Median Filter,” IEEE Trans. Circuits Syst. CAS-33, 94 (1986).
[CrossRef]

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “Median Filtering by Threshold Decomposition,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1183 (1984).
[CrossRef]

Ford, N.

N. Ford, How Machines Think (Wiley, New York, 1987).

Frieden, B. R.

Gallagher, N. C.

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “The Analog Median Filter,” IEEE Trans. Circuits Syst. CAS-33, 94 (1986).
[CrossRef]

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “Median Filtering by Threshold Decomposition,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1183 (1984).
[CrossRef]

T. A. Nodes, N. C. Gallagher, “Median Filters: Some Modifications and Their Properties,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 739 (1982).
[CrossRef]

Gotoh, T.

T. Gotoh, T. Toriu, E. Yamamoto, “Analysis of Median Filter Characteristics and Its Application to Contour Extraction,” in Proceedings, Seventh International Conference on Pattern Recognition, Vol. 2 (IEEE Computer Press, Silver Spring, MD, 1984), p. 1098.

Huang, T. S.

A. C. Bovik, T. S. Huang, D. C. Munson, “A Generalization of Median Filtering Using Linear Combinations of Order Statistics,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1342 (1983).
[CrossRef]

Hurt, S. L.

S. L. Hurt, A. Rosenfeld, “Noise Reduction in Three-Dimensional Digital Images,” Pattern Recognition 17, 407 (1981).
[CrossRef]

Lev, A.

A. Lev, S. W. Zucker, A. Rosenfeld, “Iterative Enhancement of Noisy Images,” IEEE Trans. Syst. Man Cybern. SMC-7, 435 (1977).
[CrossRef]

Mastin, G. A.

G. A. Mastin, “Adaptive Filters for Digital Image Noise Smoothing,” Comput. Vision Graphics Image Process. 31, 103 (1985).
[CrossRef]

Matsuyama, T.

M. Nagao, T. Matsuyama, “Edge Preservation Smoothing,” Comput. Graphics Image Process. 9, 394 (1979).
[CrossRef]

Munson, D. C.

A. C. Bovik, T. S. Huang, D. C. Munson, “A Generalization of Median Filtering Using Linear Combinations of Order Statistics,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1342 (1983).
[CrossRef]

Nagao, M.

M. Nagao, T. Matsuyama, “Edge Preservation Smoothing,” Comput. Graphics Image Process. 9, 394 (1979).
[CrossRef]

Narendra, P. M.

P. M. Narendra, “A Separable Median Filter for Image Noise Smoothing,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-3, 20 (1981).
[CrossRef]

Nodes, T. A.

T. A. Nodes, N. C. Gallagher, “Median Filters: Some Modifications and Their Properties,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 739 (1982).
[CrossRef]

Ochoa, E.

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), p. 330.

Rosenfeld, A.

S. L. Hurt, A. Rosenfeld, “Noise Reduction in Three-Dimensional Digital Images,” Pattern Recognition 17, 407 (1981).
[CrossRef]

L. S. Davis, A. Rosenfeld, “Noise Cleaning by Iterated Local Averaging,” IEEE Trans. Syst. Man Cybern. SMC-8, 705 (1978).

A. Lev, S. W. Zucker, A. Rosenfeld, “Iterative Enhancement of Noisy Images,” IEEE Trans. Syst. Man Cybern. SMC-7, 435 (1977).
[CrossRef]

Sweeney, D. W.

Tomita, F.

F. Tomita, S. Tsuji, “Extraction of Multiple Regions by Smoothing in Selected Neighborhoods” IEEE Trans. Syst. Man Cybern. SMC-7, 107 (1977).

Toriu, T.

T. Gotoh, T. Toriu, E. Yamamoto, “Analysis of Median Filter Characteristics and Its Application to Contour Extraction,” in Proceedings, Seventh International Conference on Pattern Recognition, Vol. 2 (IEEE Computer Press, Silver Spring, MD, 1984), p. 1098.

Tsuji, S.

F. Tomita, S. Tsuji, “Extraction of Multiple Regions by Smoothing in Selected Neighborhoods” IEEE Trans. Syst. Man Cybern. SMC-7, 107 (1977).

Tukey, J. W.

J. W. Tukey, Exploratory Data Analysis (Addison-Wesley, Reading, MA, 1977, Prelim. ed. 1971).

Wong, K. M.

E. Ataman, V. K. Aatre, K. M. Wong, “Some Statistical Properties of Median Filters,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1073 (1981).
[CrossRef]

Yamamoto, E.

T. Gotoh, T. Toriu, E. Yamamoto, “Analysis of Median Filter Characteristics and Its Application to Contour Extraction,” in Proceedings, Seventh International Conference on Pattern Recognition, Vol. 2 (IEEE Computer Press, Silver Spring, MD, 1984), p. 1098.

Zucker, S. W.

A. Lev, S. W. Zucker, A. Rosenfeld, “Iterative Enhancement of Noisy Images,” IEEE Trans. Syst. Man Cybern. SMC-7, 435 (1977).
[CrossRef]

Appl. Opt. (1)

Commun. ACM (1)

D. R. K. Brownrigg, “The Weighted Median Filter,” Commun. ACM 27, 807 (1984).
[CrossRef]

Comput. Graphics Image Process. (1)

M. Nagao, T. Matsuyama, “Edge Preservation Smoothing,” Comput. Graphics Image Process. 9, 394 (1979).
[CrossRef]

Comput. Vision Graphics Image Process. (1)

G. A. Mastin, “Adaptive Filters for Digital Image Noise Smoothing,” Comput. Vision Graphics Image Process. 31, 103 (1985).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (4)

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “Median Filtering by Threshold Decomposition,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1183 (1984).
[CrossRef]

E. Ataman, V. K. Aatre, K. M. Wong, “Some Statistical Properties of Median Filters,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1073 (1981).
[CrossRef]

T. A. Nodes, N. C. Gallagher, “Median Filters: Some Modifications and Their Properties,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 739 (1982).
[CrossRef]

A. C. Bovik, T. S. Huang, D. C. Munson, “A Generalization of Median Filtering Using Linear Combinations of Order Statistics,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1342 (1983).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

J. P. Fitch, E. J. Coyle, N. C. Gallagher, “The Analog Median Filter,” IEEE Trans. Circuits Syst. CAS-33, 94 (1986).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell. (1)

P. M. Narendra, “A Separable Median Filter for Image Noise Smoothing,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-3, 20 (1981).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (3)

A. Lev, S. W. Zucker, A. Rosenfeld, “Iterative Enhancement of Noisy Images,” IEEE Trans. Syst. Man Cybern. SMC-7, 435 (1977).
[CrossRef]

L. S. Davis, A. Rosenfeld, “Noise Cleaning by Iterated Local Averaging,” IEEE Trans. Syst. Man Cybern. SMC-8, 705 (1978).

F. Tomita, S. Tsuji, “Extraction of Multiple Regions by Smoothing in Selected Neighborhoods” IEEE Trans. Syst. Man Cybern. SMC-7, 107 (1977).

J. Opt. Soc. Am. (1)

Pattern Recognition (1)

S. L. Hurt, A. Rosenfeld, “Noise Reduction in Three-Dimensional Digital Images,” Pattern Recognition 17, 407 (1981).
[CrossRef]

Other (6)

J. W. Tukey, Exploratory Data Analysis (Addison-Wesley, Reading, MA, 1977, Prelim. ed. 1971).

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), p. 330.

T. S. Huang, Ed., Two-Dimensional Digital Signal Processing II; Transforms and Median Filters (Springer-Verlag, Berlin, 1981).

T. Gotoh, T. Toriu, E. Yamamoto, “Analysis of Median Filter Characteristics and Its Application to Contour Extraction,” in Proceedings, Seventh International Conference on Pattern Recognition, Vol. 2 (IEEE Computer Press, Silver Spring, MD, 1984), p. 1098.

E. R. Davies, “The Median Filters: An Appraisal and a New Truncated Version,” in Proceedings Seventh International Conference on Pattern Recognition, Vol. 1 (IEEE Computer Press, Silver Spring, MD, 1984), p. 590.

N. Ford, How Machines Think (Wiley, New York, 1987).

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Figures (5)

Fig. 1
Fig. 1

Input–output relationship of the NNM filters. The ranked orders of the gray levels of the central pixels after filtering (outputs) are plotted as functions of those before filtering (inputs).

Fig. 2
Fig. 2

Noise removal by the generalized median filters: (A) original image; (B) original image suffering from salt-and-pepper noise (1% occurrence rate for each pixel); (C) image restored by the conventional median (same as the 9-NNM filter); and images restored by NNM filters of (D) k = 7, (E) k = 5, and (F) k = 3.

Fig. 3
Fig. 3

Noise removal by the generalized median filters: (A) original image; (B) original image suffering from salt-and-pepper noise (1% occurrence rate for each pixel); (C) image restored by the conventional median (same as the 9-NNM filter); and images restored by NNM filters of (D) k = 7, (E) k = 5, and (F) k = 3.

Fig. 4
Fig. 4

Noise removal by the generalized median filters: (A) original image suffering from salt-and-pepper noise (12% occurrence rate for each pixel); (B) image restored by the conventional median (identical with the 9-NNM filter); and images restored by NNM filters of (C) k = 7, (D) k = 5, and (E) k = 3.

Fig. 5
Fig. 5

Improvement of SNRs after filtering by the NNM filters vs SNRs of input pictures. Noise patterns with various occurrence rates ranging from 1 to 20% were superimposed on the original image shown in Fig. 3(A) and then filtered.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

p = p 0 + P ( 1 - p 0 ) ( 1 - p 0 ) - P ( q 0 - N ) ( q 0 - N ) ,
q = q 0 + P ( 1 - p 0 ) ( 1 - p 0 ) - P ( q 0 - N ) ( q 0 - N ) ,
p 0 = c - ( k - 1 ) / 2 ,
q 0 = c + ( k - 1 ) / 2 ,
P ( x ) = { 1 , x > 0 0 , x 0.
{ a i , j } = [ 1 1 1 1 m 1 1 1 1 ] .
S c = S 9 ( maximum ) ,
S c = S 1 ( minimum ) .
N i , j = D i , j - O i , j
ρ = 10 log [ ( O i , j - O ¯ ) 2 / ( N i , j - N ¯ ) 2 ] ,
O ¯ = O i , j , N ¯ = N i , j ,
η = [ SNR of restored image ] - [ SNR of input image ] .

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