Abstract

Two new noise-reduction algorithms, namely, the adaptive symmetric mean filter (ASMF) and the hybrid filter, are presented in this paper. The idea of the ASMF is to find the largest symmetric region on a slope facet by incorporation of the gradient similarity criterion and the symmetry constraint into region growing. The gradient similarity criterion allows more pixels to be included for a statistically better estimation, whereas the symmetry constraint promises an unbiased estimate if the noise is completely removed. The hybrid filter combines the advantages of the ASMF, the double-window modified-trimmed mean filter, and the adaptive mean filter to optimize noise reduction on the step and the ramp edges. The experimental results have shown the ASMF and the hybrid filter are superior to three conventional filters for the synthetic and the natural images in terms of the root-mean-squared error, the root-mean-squared difference of gradient, and the visual presentation.

© 2001 Optical Society of America

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References

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  1. R. M. Haralick, L. Watson, “A facet model for image data,” Comput. Graph. Image Process. 15, 113–129 (1981).
    [CrossRef]
  2. V. Koivunen, “A robust nonlinear filter for image restoration,” IEEE Trans. Image Process. 4, 569–578 (1995).
    [CrossRef] [PubMed]
  3. I. Pitas, A. N. Venetsanopoulos, “Order statistics in digital image processing,” Proc. IEEE 80, 1893–1921 (1992).
    [CrossRef]
  4. C. A. Pomalaza-Raez, C. D. McGillem, “An adaptive, nonlinear edge-preserving filter,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 571–576 (1984).
    [CrossRef]
  5. S. T. Acton, A. C. Bovik, “Nonlinear image estimation using piecewise and local image models,” IEEE Trans. Image Process. 7, 979–991 (1998).
    [CrossRef]
  6. D. Geman, C. Yang, “Nonlinear image recovery with half-quadratic regularization,” IEEE Trans. Image Process. 4, 932–946 (1995).
    [CrossRef] [PubMed]
  7. D. C. C. Wang, A. H. Vagucci, “Gradient inverse weighted smoothing scheme and the evaluation of its performance,” Comput. Vision Graph. Image Process. 15, 167–181 (1981).
  8. J.-S. Lee, “Digital image enhancement and noise filtering by use of local statistics,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 165–168 (1981).
    [CrossRef]
  9. Y. H. Lee, S. A. Kassam, “Generalized median filtering and related nonlinear filtering techniques,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 672–683 (1984).
  10. S. R. Peterson, S. A. Kassam, “Edge preserving signal enhancement using generalizations of ordered statistics filtering,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 672–675.
    [CrossRef]
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    [CrossRef] [PubMed]
  12. D. Jones, J. Malik, “A computational framework for determining stereo correspondence from a set of linear spatial filters,” in Second European Conference on Computer Vision, G. Sandini, ed. (Springer-Verlag, New York, 1992), pp. 395–410.
  13. J. Little, “Accurate early detection of discontinuities,” in Vision Interface’92 (Canadian Image Processing and Pattern Recognition Society, Toronto, Canada, 1992), pp. 97–102.
  14. Y.-S. Fong, C. A. Pomalaza-Raez, X.-H Wang, “Comparison study of nonlinear filters in image processing applications,” Opt. Eng. 28, 749–760 (1989).
  15. Y. Boykov, O. Veksler, R. Zabih, “A variable window approach to early vision,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1283–1294 (1998).
    [CrossRef]
  16. R. A. Kirsch, “Computer determination of the constitute structure of biological images,” Comput. Biomed. Res. 4, 315–328 (1971).
    [CrossRef] [PubMed]
  17. R. Ding, A. N. Venetsanopoulos, “Generalized homomorphic and adaptive order statistic filters for the removal of impulsive and signal-dependent noise,” IEEE Trans. Circuits Syst. CAS-34, 948–955 (1987).
    [CrossRef]

1998 (2)

S. T. Acton, A. C. Bovik, “Nonlinear image estimation using piecewise and local image models,” IEEE Trans. Image Process. 7, 979–991 (1998).
[CrossRef]

Y. Boykov, O. Veksler, R. Zabih, “A variable window approach to early vision,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1283–1294 (1998).
[CrossRef]

1995 (2)

D. Geman, C. Yang, “Nonlinear image recovery with half-quadratic regularization,” IEEE Trans. Image Process. 4, 932–946 (1995).
[CrossRef] [PubMed]

V. Koivunen, “A robust nonlinear filter for image restoration,” IEEE Trans. Image Process. 4, 569–578 (1995).
[CrossRef] [PubMed]

1994 (1)

1992 (1)

I. Pitas, A. N. Venetsanopoulos, “Order statistics in digital image processing,” Proc. IEEE 80, 1893–1921 (1992).
[CrossRef]

1989 (1)

Y.-S. Fong, C. A. Pomalaza-Raez, X.-H Wang, “Comparison study of nonlinear filters in image processing applications,” Opt. Eng. 28, 749–760 (1989).

1987 (1)

R. Ding, A. N. Venetsanopoulos, “Generalized homomorphic and adaptive order statistic filters for the removal of impulsive and signal-dependent noise,” IEEE Trans. Circuits Syst. CAS-34, 948–955 (1987).
[CrossRef]

1984 (2)

C. A. Pomalaza-Raez, C. D. McGillem, “An adaptive, nonlinear edge-preserving filter,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 571–576 (1984).
[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 (1984).

1981 (3)

R. M. Haralick, L. Watson, “A facet model for image data,” Comput. Graph. Image Process. 15, 113–129 (1981).
[CrossRef]

D. C. C. Wang, A. H. Vagucci, “Gradient inverse weighted smoothing scheme and the evaluation of its performance,” Comput. Vision Graph. Image Process. 15, 167–181 (1981).

J.-S. Lee, “Digital image enhancement and noise filtering by use of local statistics,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 165–168 (1981).
[CrossRef]

1971 (1)

R. A. Kirsch, “Computer determination of the constitute structure of biological images,” Comput. Biomed. Res. 4, 315–328 (1971).
[CrossRef] [PubMed]

Acton, S. T.

S. T. Acton, A. C. Bovik, “Nonlinear image estimation using piecewise and local image models,” IEEE Trans. Image Process. 7, 979–991 (1998).
[CrossRef]

Bovik, A. C.

S. T. Acton, A. C. Bovik, “Nonlinear image estimation using piecewise and local image models,” IEEE Trans. Image Process. 7, 979–991 (1998).
[CrossRef]

Boykov, Y.

Y. Boykov, O. Veksler, R. Zabih, “A variable window approach to early vision,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1283–1294 (1998).
[CrossRef]

Ding, R.

R. Ding, A. N. Venetsanopoulos, “Generalized homomorphic and adaptive order statistic filters for the removal of impulsive and signal-dependent noise,” IEEE Trans. Circuits Syst. CAS-34, 948–955 (1987).
[CrossRef]

Fong, Y.-S.

Y.-S. Fong, C. A. Pomalaza-Raez, X.-H Wang, “Comparison study of nonlinear filters in image processing applications,” Opt. Eng. 28, 749–760 (1989).

Geman, D.

D. Geman, C. Yang, “Nonlinear image recovery with half-quadratic regularization,” IEEE Trans. Image Process. 4, 932–946 (1995).
[CrossRef] [PubMed]

Haralick, R. M.

R. M. Haralick, L. Watson, “A facet model for image data,” Comput. Graph. Image Process. 15, 113–129 (1981).
[CrossRef]

Jones, D.

D. Jones, J. Malik, “A computational framework for determining stereo correspondence from a set of linear spatial filters,” in Second European Conference on Computer Vision, G. Sandini, ed. (Springer-Verlag, New York, 1992), pp. 395–410.

Kassam, S. A.

Y. H. Lee, S. A. Kassam, “Generalized median filtering and related nonlinear filtering techniques,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 672–683 (1984).

S. R. Peterson, S. A. Kassam, “Edge preserving signal enhancement using generalizations of ordered statistics filtering,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 672–675.
[CrossRef]

Kirsch, R. A.

R. A. Kirsch, “Computer determination of the constitute structure of biological images,” Comput. Biomed. Res. 4, 315–328 (1971).
[CrossRef] [PubMed]

Koivunen, V.

V. Koivunen, “A robust nonlinear filter for image restoration,” IEEE Trans. Image Process. 4, 569–578 (1995).
[CrossRef] [PubMed]

Lee, J.-S.

J.-S. Lee, “Digital image enhancement and noise filtering by use of local statistics,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 165–168 (1981).
[CrossRef]

Lee, Y. H.

Y. H. Lee, S. A. Kassam, “Generalized median filtering and related nonlinear filtering techniques,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 672–683 (1984).

Little, J.

J. Little, “Accurate early detection of discontinuities,” in Vision Interface’92 (Canadian Image Processing and Pattern Recognition Society, Toronto, Canada, 1992), pp. 97–102.

Malik, J.

D. Jones, J. Malik, “A computational framework for determining stereo correspondence from a set of linear spatial filters,” in Second European Conference on Computer Vision, G. Sandini, ed. (Springer-Verlag, New York, 1992), pp. 395–410.

McGillem, C. D.

C. A. Pomalaza-Raez, C. D. McGillem, “An adaptive, nonlinear edge-preserving filter,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 571–576 (1984).
[CrossRef]

Paranjape, R. B.

Peterson, S. R.

S. R. Peterson, S. A. Kassam, “Edge preserving signal enhancement using generalizations of ordered statistics filtering,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 672–675.
[CrossRef]

Pitas, I.

I. Pitas, A. N. Venetsanopoulos, “Order statistics in digital image processing,” Proc. IEEE 80, 1893–1921 (1992).
[CrossRef]

Pomalaza-Raez, C. A.

Y.-S. Fong, C. A. Pomalaza-Raez, X.-H Wang, “Comparison study of nonlinear filters in image processing applications,” Opt. Eng. 28, 749–760 (1989).

C. A. Pomalaza-Raez, C. D. McGillem, “An adaptive, nonlinear edge-preserving filter,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 571–576 (1984).
[CrossRef]

Rabie, T. F.

Rangayyan, R. M.

Vagucci, A. H.

D. C. C. Wang, A. H. Vagucci, “Gradient inverse weighted smoothing scheme and the evaluation of its performance,” Comput. Vision Graph. Image Process. 15, 167–181 (1981).

Veksler, O.

Y. Boykov, O. Veksler, R. Zabih, “A variable window approach to early vision,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1283–1294 (1998).
[CrossRef]

Venetsanopoulos, A. N.

I. Pitas, A. N. Venetsanopoulos, “Order statistics in digital image processing,” Proc. IEEE 80, 1893–1921 (1992).
[CrossRef]

R. Ding, A. N. Venetsanopoulos, “Generalized homomorphic and adaptive order statistic filters for the removal of impulsive and signal-dependent noise,” IEEE Trans. Circuits Syst. CAS-34, 948–955 (1987).
[CrossRef]

Wang, D. C. C.

D. C. C. Wang, A. H. Vagucci, “Gradient inverse weighted smoothing scheme and the evaluation of its performance,” Comput. Vision Graph. Image Process. 15, 167–181 (1981).

Wang, X.-H

Y.-S. Fong, C. A. Pomalaza-Raez, X.-H Wang, “Comparison study of nonlinear filters in image processing applications,” Opt. Eng. 28, 749–760 (1989).

Watson, L.

R. M. Haralick, L. Watson, “A facet model for image data,” Comput. Graph. Image Process. 15, 113–129 (1981).
[CrossRef]

Yang, C.

D. Geman, C. Yang, “Nonlinear image recovery with half-quadratic regularization,” IEEE Trans. Image Process. 4, 932–946 (1995).
[CrossRef] [PubMed]

Zabih, R.

Y. Boykov, O. Veksler, R. Zabih, “A variable window approach to early vision,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1283–1294 (1998).
[CrossRef]

Appl. Opt. (1)

Comput. Biomed. Res. (1)

R. A. Kirsch, “Computer determination of the constitute structure of biological images,” Comput. Biomed. Res. 4, 315–328 (1971).
[CrossRef] [PubMed]

Comput. Graph. Image Process. (1)

R. M. Haralick, L. Watson, “A facet model for image data,” Comput. Graph. Image Process. 15, 113–129 (1981).
[CrossRef]

Comput. Vision Graph. Image Process. (1)

D. C. C. Wang, A. H. Vagucci, “Gradient inverse weighted smoothing scheme and the evaluation of its performance,” Comput. Vision Graph. Image Process. 15, 167–181 (1981).

IEEE Trans. Acoust. Speech Signal Process. (2)

Y. H. Lee, S. A. Kassam, “Generalized median filtering and related nonlinear filtering techniques,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 672–683 (1984).

C. A. Pomalaza-Raez, C. D. McGillem, “An adaptive, nonlinear edge-preserving filter,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 571–576 (1984).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

R. Ding, A. N. Venetsanopoulos, “Generalized homomorphic and adaptive order statistic filters for the removal of impulsive and signal-dependent noise,” IEEE Trans. Circuits Syst. CAS-34, 948–955 (1987).
[CrossRef]

IEEE Trans. Image Process. (3)

S. T. Acton, A. C. Bovik, “Nonlinear image estimation using piecewise and local image models,” IEEE Trans. Image Process. 7, 979–991 (1998).
[CrossRef]

D. Geman, C. Yang, “Nonlinear image recovery with half-quadratic regularization,” IEEE Trans. Image Process. 4, 932–946 (1995).
[CrossRef] [PubMed]

V. Koivunen, “A robust nonlinear filter for image restoration,” IEEE Trans. Image Process. 4, 569–578 (1995).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

J.-S. Lee, “Digital image enhancement and noise filtering by use of local statistics,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 165–168 (1981).
[CrossRef]

Y. Boykov, O. Veksler, R. Zabih, “A variable window approach to early vision,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1283–1294 (1998).
[CrossRef]

Opt. Eng. (1)

Y.-S. Fong, C. A. Pomalaza-Raez, X.-H Wang, “Comparison study of nonlinear filters in image processing applications,” Opt. Eng. 28, 749–760 (1989).

Proc. IEEE (1)

I. Pitas, A. N. Venetsanopoulos, “Order statistics in digital image processing,” Proc. IEEE 80, 1893–1921 (1992).
[CrossRef]

Other (3)

S. R. Peterson, S. A. Kassam, “Edge preserving signal enhancement using generalizations of ordered statistics filtering,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 672–675.
[CrossRef]

D. Jones, J. Malik, “A computational framework for determining stereo correspondence from a set of linear spatial filters,” in Second European Conference on Computer Vision, G. Sandini, ed. (Springer-Verlag, New York, 1992), pp. 395–410.

J. Little, “Accurate early detection of discontinuities,” in Vision Interface’92 (Canadian Image Processing and Pattern Recognition Society, Toronto, Canada, 1992), pp. 97–102.

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Figures (18)

Fig. 1
Fig. 1

(a) Side view of an ideal ramp edge. (b) The front view of the denoising window, derived with the DWMTM filter or the AMF, for a target pixel on the ideal ramp edge. (c) The front view of the denoising window, derived with the ANNS filter, for a target pixel on the ideal ramp edge.

Fig. 2
Fig. 2

Symmetric denoising window derived with the ASMF, in which the central square is the target pixel of the denoising window and the other two squares represent a pair of pixels symmetric with respect to the target pixel.

Fig. 3
Fig. 3

Uncorrupted synthetic images with (a) σ b = 0 and (b) σ b = 2.

Fig. 4
Fig. 4

Original natural images (a) Lenna, (b) Cameraman, and (c) Peppers.

Fig. 5
Fig. 5

(a) RMSEs and (b) the RMSDGs achieved by the five filters with σ n = 15 and σ b = 0, 0.5, 1, 2, 3, and 4.

Fig. 6
Fig. 6

(a) RMSEs and (b) the RMSDGs achieved by the five filters with σ n = 25 and σ b = 0, 0.5, 1, 2, 3, and 4.

Fig. 7
Fig. 7

(a) RMSEs and (b) the RMSDGs achieved by the five filters with σ n = 35 and σ b = 0, 0.5, 1, 2, 3, and 4.

Fig. 8
Fig. 8

(a) Corrupted synthetic image with (σ b , σ n ) = (2, 25), and the denoised image derived with (b) the DWMTM filter, (c) the AMF, (d) the ANNS filter, (e) the ASMF, and (f) the hybrid filter.

Fig. 9
Fig. 9

(a) Gradient magnitude of the uncorrupted synthetic image with σ b = 2, and the gradient magnitude of the denoised image derived with (b) the DWMTM filter, (c) the AMF, (d) the ANNS filter, (e) the ASMF, and (f) the hybrid filter for the corrupted image (σ b , σ n ) = (2, 25).

Fig. 10
Fig. 10

(a) RMSEs and (b) the RMSDGs achieved by the five filters with σ n = 15 for the three natural images, i.e., Lenna, Camerman, and Peppers.

Fig. 11
Fig. 11

(a) RMSEs and (b) the RMSDGs achieved by the five filters with σ n = 25 for the three natural images, i.e., Lenna, Camerman, and Peppers.

Fig. 12
Fig. 12

(a) RMSEs and (b) the RMSDGs achieved by the five filters with σ n = 35 for the three natural images, i.e., Lenna, Camerman, and Peppers.

Fig. 13
Fig. 13

(a) Corrupted Lenna image with σ n = 25, and the denoised image derived with (b) the DWMTM filter, (c) the AMF, (d) the ANNS filter, (e) the ASMF, and (f) the hybrid filter. The white ellipse indicates the fine details that may be easily smeared out by the filters.

Fig. 14
Fig. 14

(a) Corrupted Cameraman image with σ n = 25, and the denoised image derived with (b) the DWMTM filter, (c) the AMF, (d) the ANNS filter, (e) the ASMF, and (f) the hybrid filter. The white ellipse indicates the fine details that may be easily smeared out by the filters.

Fig. 15
Fig. 15

(a) Corrupted Peppers image with σ n = 25, and the denoised image derived with (b) the DWMTM filter, (c) the AMF, (d) the ANNS filter, (e) the ASMF, and (f) the hybrid filter. The white ellipse indicates the fine details that may be easily smeared out by the filters.

Fig. 16
Fig. 16

(a) Gradient magnitude of the uncorrupted Lenna image, and the gradient magnitude of the denoised image derived with (b) the DWMTM filter, (c) the AMF, (d) the ANNS filter, (e) the ASMF, and (f) the hybrid filter for the corrupted Lenna image with σ n = 25. The white ellipse indicates the edges that may be easily smeared out by the filters.

Fig. 17
Fig. 17

(a) Gradient magnitude of the uncorrupted Cameraman image, and the gradient magnitude of the denoised image derived with (b) the DWMTM filter, (c) the AMF, (d) the ANNS filter, (e) the ASMF, and (f) the hybrid filter for the corrupted Cameraman image with σ n = 25. The white ellipse indicates the edges that may be easily smeared out by the filters.

Fig. 18
Fig. 18

(a) The gradient magnitude of the uncorrupted Peppers image, and the gradient magnitude of the denoised image derived with (b) the DWMTM filter, (c) the AMF, (d) the ANNS filter, (e) the ASMF, and (f) the hybrid filter for the corrupted Peppers image with σ n = 25. The white ellipse indicates the edges that may be easily smeared out by the filters.

Tables (1)

Tables Icon

Table 1 Parameter Settings Used in the Experiments for the DWMTM Filter, the AMF, the ANNS Filter, the ASMF, and the Hybrid Filter

Equations (14)

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Wk=piIpi-mk|<Tdwmtm, piDLWk,
Wk=piIpi-Ipk|<Tamf, piAWk.
yk=EIWk+1-σn2σd2+σn21/2×Ipk-EIWk,
σd2=VIWk-σn2for VIWk>σn20for VIWkσn2.
-1Wkirsch×Wkirsch-1/20Wkirsch×11Wkirsch×Wkirsch-1/2,
EIWk=ESWk+ENWk=ESWk=Spk.
εk=EIWk-Spk=0.
εk=EIWk-Spk=ENWk,
-Tdwmtm+δk<εk<Tdwmtm+δk,
-Tamf+Npk<εk<Tamf+Npk.
εk=1-σn2σd2+σn21/2Npk.
εk=1-11+σd2/σn21/2Npkσd22σn2 Npk.
RMSE=1Nf2i=1Nfj=1Nffi, j-f˜i, j21/2,
RMSDG=1Nf2i=1Nfj=1Nffi, j-f˜i, j21/2,

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