Abstract

We introduce a new nonlinear filter for signal and image restoration, the hybrid order statistic (HOS) filter. Because it exploits both rank- and spatial-order information, the HOS realizes the advantages of nonlinear filters in edge preservation and reduction of impulsive noise components while retaining the ability of the linear filter to suppress Gaussian noise. We show that the HOS filter exhibits improved performance over both the linear Wiener and the nonlinear L filters in reducing mean-squared error in the presence of contaminated Gaussian noise. In many cases it also performs favorably compared with the Ll and rank-conditioned rank selection filters.

© 2001 Optical Society of America

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References

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  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, 1342–1349 (1983).
    [Crossref]
  2. F. Palmieri, C. G. Boncelet, “Ll-filters—a new class of order statistic filters,” IEEE Trans. Acoust. Speech Signal Process. 37, 691–701 (1989).
    [Crossref]
  3. R. C. Hardie, K. E. Barner, “Rank conditioned rank selection filters for signal restoration,” IEEE Trans. Image Process. 3, 192–206 (1994).
    [Crossref] [PubMed]
  4. J. W. Tukey, “Nonlinear (nonsuperimposable) methods for smoothing data,” Conference Records of Electronics and Aerospace Systems Convention (EASCOM) (Institute of Electrical and Electronics Engineers, New York, 1974), p. 673.
  5. S. J. Tukey, Exploratory Data Analysis (Addison-Wesley, Reading, Mass., 1977).
  6. Y. H. Lee, S. A. Kassam, “Generalized median filtering and related nonlinear filtering techniques,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 672–683 (1985).
  7. P. P. Gandhi, I. Song, S. A. Kassam, “Nonlinear smoothing filters based on rank estimates of location,” IEEE Trans. Acoust. Speech Signal Process. 37, 1359–1379 (1989).
    [Crossref]
  8. R. C. Hardie, C. G. Boncelet, “LUM filters: a class rank order based filters for smoothing and sharpening,” IEEE Trans. Signal Process. 41, 1061–1076 (1993).
    [Crossref]
  9. P. P. Gandhi, S. A. Kassam, “Design and performance of combination filters for signal restoration,” IEEE Trans. Signal Process. 39, 1524–1540 (1991).
    [Crossref]
  10. S.-J. Ko, Y. H. Lee, “Center weighted median filters and their applications to image enhancement,” IEEE Trans. Circuits Syst. 38, 984–993 (1991).
    [Crossref]
  11. The idea for the hybrid filter came from conversations between the first author and K. E. Barner, 1998.
  12. S. Haykin, Adaptive Filter Theory, 3rd ed. (Prentice-Hall, Upper Saddle River, N.J., 1996), p. 747.

1994 (1)

R. C. Hardie, K. E. Barner, “Rank conditioned rank selection filters for signal restoration,” IEEE Trans. Image Process. 3, 192–206 (1994).
[Crossref] [PubMed]

1993 (1)

R. C. Hardie, C. G. Boncelet, “LUM filters: a class rank order based filters for smoothing and sharpening,” IEEE Trans. Signal Process. 41, 1061–1076 (1993).
[Crossref]

1991 (2)

P. P. Gandhi, S. A. Kassam, “Design and performance of combination filters for signal restoration,” IEEE Trans. Signal Process. 39, 1524–1540 (1991).
[Crossref]

S.-J. Ko, Y. H. Lee, “Center weighted median filters and their applications to image enhancement,” IEEE Trans. Circuits Syst. 38, 984–993 (1991).
[Crossref]

1989 (2)

P. P. Gandhi, I. Song, S. A. Kassam, “Nonlinear smoothing filters based on rank estimates of location,” IEEE Trans. Acoust. Speech Signal Process. 37, 1359–1379 (1989).
[Crossref]

F. Palmieri, C. G. Boncelet, “Ll-filters—a new class of order statistic filters,” IEEE Trans. Acoust. Speech Signal Process. 37, 691–701 (1989).
[Crossref]

1985 (1)

Y. H. Lee, S. A. Kassam, “Generalized median filtering and related nonlinear filtering techniques,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 672–683 (1985).

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, 1342–1349 (1983).
[Crossref]

Barner, K. E.

R. C. Hardie, K. E. Barner, “Rank conditioned rank selection filters for signal restoration,” IEEE Trans. Image Process. 3, 192–206 (1994).
[Crossref] [PubMed]

The idea for the hybrid filter came from conversations between the first author and K. E. Barner, 1998.

Boncelet, C. G.

R. C. Hardie, C. G. Boncelet, “LUM filters: a class rank order based filters for smoothing and sharpening,” IEEE Trans. Signal Process. 41, 1061–1076 (1993).
[Crossref]

F. Palmieri, C. G. Boncelet, “Ll-filters—a new class of order statistic filters,” IEEE Trans. Acoust. Speech Signal Process. 37, 691–701 (1989).
[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, 1342–1349 (1983).
[Crossref]

Gandhi, P. P.

P. P. Gandhi, S. A. Kassam, “Design and performance of combination filters for signal restoration,” IEEE Trans. Signal Process. 39, 1524–1540 (1991).
[Crossref]

P. P. Gandhi, I. Song, S. A. Kassam, “Nonlinear smoothing filters based on rank estimates of location,” IEEE Trans. Acoust. Speech Signal Process. 37, 1359–1379 (1989).
[Crossref]

Hardie, R. C.

R. C. Hardie, K. E. Barner, “Rank conditioned rank selection filters for signal restoration,” IEEE Trans. Image Process. 3, 192–206 (1994).
[Crossref] [PubMed]

R. C. Hardie, C. G. Boncelet, “LUM filters: a class rank order based filters for smoothing and sharpening,” IEEE Trans. Signal Process. 41, 1061–1076 (1993).
[Crossref]

Haykin, S.

S. Haykin, Adaptive Filter Theory, 3rd ed. (Prentice-Hall, Upper Saddle River, N.J., 1996), p. 747.

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, 1342–1349 (1983).
[Crossref]

Kassam, S. A.

P. P. Gandhi, S. A. Kassam, “Design and performance of combination filters for signal restoration,” IEEE Trans. Signal Process. 39, 1524–1540 (1991).
[Crossref]

P. P. Gandhi, I. Song, S. A. Kassam, “Nonlinear smoothing filters based on rank estimates of location,” IEEE Trans. Acoust. Speech Signal Process. 37, 1359–1379 (1989).
[Crossref]

Y. H. Lee, S. A. Kassam, “Generalized median filtering and related nonlinear filtering techniques,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 672–683 (1985).

Ko, S.-J.

S.-J. Ko, Y. H. Lee, “Center weighted median filters and their applications to image enhancement,” IEEE Trans. Circuits Syst. 38, 984–993 (1991).
[Crossref]

Lee, Y. H.

S.-J. Ko, Y. H. Lee, “Center weighted median filters and their applications to image enhancement,” IEEE Trans. Circuits Syst. 38, 984–993 (1991).
[Crossref]

Y. H. Lee, S. A. Kassam, “Generalized median filtering and related nonlinear filtering techniques,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 672–683 (1985).

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, 1342–1349 (1983).
[Crossref]

Palmieri, F.

F. Palmieri, C. G. Boncelet, “Ll-filters—a new class of order statistic filters,” IEEE Trans. Acoust. Speech Signal Process. 37, 691–701 (1989).
[Crossref]

Song, I.

P. P. Gandhi, I. Song, S. A. Kassam, “Nonlinear smoothing filters based on rank estimates of location,” IEEE Trans. Acoust. Speech Signal Process. 37, 1359–1379 (1989).
[Crossref]

Tukey, J. W.

J. W. Tukey, “Nonlinear (nonsuperimposable) methods for smoothing data,” Conference Records of Electronics and Aerospace Systems Convention (EASCOM) (Institute of Electrical and Electronics Engineers, New York, 1974), p. 673.

Tukey, S. J.

S. J. Tukey, Exploratory Data Analysis (Addison-Wesley, Reading, Mass., 1977).

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

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, 1342–1349 (1983).
[Crossref]

F. Palmieri, C. G. Boncelet, “Ll-filters—a new class of order statistic filters,” IEEE Trans. Acoust. Speech Signal Process. 37, 691–701 (1989).
[Crossref]

Y. H. Lee, S. A. Kassam, “Generalized median filtering and related nonlinear filtering techniques,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 672–683 (1985).

P. P. Gandhi, I. Song, S. A. Kassam, “Nonlinear smoothing filters based on rank estimates of location,” IEEE Trans. Acoust. Speech Signal Process. 37, 1359–1379 (1989).
[Crossref]

IEEE Trans. Circuits Syst. (1)

S.-J. Ko, Y. H. Lee, “Center weighted median filters and their applications to image enhancement,” IEEE Trans. Circuits Syst. 38, 984–993 (1991).
[Crossref]

IEEE Trans. Image Process. (1)

R. C. Hardie, K. E. Barner, “Rank conditioned rank selection filters for signal restoration,” IEEE Trans. Image Process. 3, 192–206 (1994).
[Crossref] [PubMed]

IEEE Trans. Signal Process. (2)

R. C. Hardie, C. G. Boncelet, “LUM filters: a class rank order based filters for smoothing and sharpening,” IEEE Trans. Signal Process. 41, 1061–1076 (1993).
[Crossref]

P. P. Gandhi, S. A. Kassam, “Design and performance of combination filters for signal restoration,” IEEE Trans. Signal Process. 39, 1524–1540 (1991).
[Crossref]

Other (4)

J. W. Tukey, “Nonlinear (nonsuperimposable) methods for smoothing data,” Conference Records of Electronics and Aerospace Systems Convention (EASCOM) (Institute of Electrical and Electronics Engineers, New York, 1974), p. 673.

S. J. Tukey, Exploratory Data Analysis (Addison-Wesley, Reading, Mass., 1977).

The idea for the hybrid filter came from conversations between the first author and K. E. Barner, 1998.

S. Haykin, Adaptive Filter Theory, 3rd ed. (Prentice-Hall, Upper Saddle River, N.J., 1996), p. 747.

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Figures (12)

Fig. 1
Fig. 1

Block diagram showing the signals involved in optimization.

Fig. 2
Fig. 2

Training image.

Fig. 3
Fig. 3

Uncorrupted image.

Fig. 4
Fig. 4

Corrupted image with contaminated Gaussian noise with σ1 = 30, σ2 = 100, and ε = 0.3.

Fig. 5
Fig. 5

Wiener filtering with a MSE of 417.2722.

Fig. 6
Fig. 6

L filtering with a MSE of 420.2862.

Fig. 7
Fig. 7

HOS filtering with a MSE of 347.1582.

Fig. 8
Fig. 8

MSE versus ε in contaminated Gaussian noise with σ1 = 10 and σ2 = 100.

Fig. 9
Fig. 9

MSE versus ε in contaminated Gaussian noise with σ1 = 30 and σ2 = 100.

Fig. 10
Fig. 10

MSE versus ε in contaminated Gaussian noise with σ1 = 10 and σ2 = 100 for a nonideal case utilizing ε = 0.3 for the training image.

Fig. 11
Fig. 11

MSE versus ε in contaminated Gaussian noise with σ1 = 30 and σ2 = 100 for a nonideal case utilizing ε = 0.3 for the training image.

Fig. 12
Fig. 12

MSE versus ε in contaminated Gaussian noise with σ1 = 30 and σ2 = 100 for a HOS filter contrasting ideal versus nonideal training.

Equations (14)

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

xn=x1n, x2n,  xNnT.
y=wTx.
xos=x1, x2, x3,, xNT,
x1x2x3xN.
yL=wLTxos,
xext=x1, x2, x3,, xN, x1, x2,, xNT.
yHos=wextTxext,
en=dn-yn=dn-wextTnxextn.
J=Ee2n=Edn-wextTnxextn2.
wext=Rext-1pext,
Rext=ExextxextT,
pext=Exextd.
J=Ee2n+αwn2,
wext=Rext+αI-1pext.

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