Abstract

Vector operators based on robust order statistics have proved successful in digital multichannel imaging applications, particularly color image filtering and enhancement, in dealing with impulsive noise while preserving edges and fine image details. These operators often have very high computational requirements, which limits their use in time-critical applications. This paper introduces techniques to speed up vector filters using the minimax approximation theory. Extensive experiments on a large and diverse set of color images show that proposed approximations achieve an excellent balance among ease of implementation, accuracy, and computational speed.

© 2009 Optical Society of America

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  1. K. N. Plataniotis and A. N. Venetsanopoulos, Color Image Processing and Applications (Springer, 2000).
  2. R. Lukac and K. N. Plataniotis, “A taxonomy of color image filtering and enhancement solutions,” in Advances in Imaging & Electron Physics, P.W.Hawkes, ed. (Academic, 2006), (Vol. 140), pp. 187-264.
    [CrossRef]
  3. M. E. Celebi, H. A. Kingravi, and Y. A. Aslandogan, “Nonlinear vector filtering for impulsive noise removal from color images,” J. Electron. Imaging 16, 033008 (2007).
    [CrossRef]
  4. R. Lukac, B. Smolka, K. Martin, K. N. Plataniotis, and A. N. Venetsanopoulos, “Vector filtering for color imaging,” IEEE Signal Process. Mag. 22, 74-86 (2005).
    [CrossRef]
  5. M. E. Celebi and Y. A. Aslandogan, “Robust switching vector median filter for impulsive noise removal,” J. Electron. Imaging 17, 043006 (2008).
    [CrossRef]
  6. M. Barni, “A fast algorithm for 1-norm vector median filtering,” IEEE Trans. Image Process. 6, 1452-1455 (1997).
    [CrossRef] [PubMed]
  7. M. Barni, F. Buti, F. Bartolini, and V. Cappellini, “A quasi-Euclidean norm to speed up vector median filtering,” IEEE Trans. Image Process. 9, 1704-1709 (2000).
    [CrossRef]
  8. J. Astola, P. Haavisto, and Y. Neuvo, “Vector median filters,” Proc. IEEE 78, 678-689 (1990).
    [CrossRef]
  9. M. E. Celebi, H. A. Kingravi, B. Uddin, and Y. A. Aslandogan, “Fast switching filter for impulsive noise removal from color images,” J. Imaging Sci. Technol. 51, 155-165 (2007).
    [CrossRef]
  10. P. E. Trahanias and A. N. Venetsanopoulos, “Vector directional filters: A new class of multichannel image processing filters,” IEEE Trans. Image Process. 2, 528-534 (1993).
    [CrossRef] [PubMed]
  11. K. N. Plataniotis, D. Androutsos, S. Vinayagamoorthy, and A. N. Venetsanopoulos, “Color image processing using adaptive multichannel filters,” IEEE Trans. Image Process. 6, 933-949 (1997).
    [CrossRef] [PubMed]
  12. R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Entropy vector median filter,” in Proceedings of the IbPRIA Conference, Lecture Notes in Computer Science 2652 (Springer, 2003), pp. 1117-1125.
  13. E. W. Cheney, Introduction to Approximation Theory (AMS, 2000).
  14. J.-M. Muller, Elementary Functions: Algorithms and Implementation (Birkhäuser, 2006).
  15. W. Fraser, “A survey of methods of computing minimax and near-minimax polynomial approximations for functions of a single independent variable,” J. ACM 12, 295-314 (1965).
    [CrossRef]
  16. N. Nikolaidis and I. Pitas, “Nonlinear processing and analysis of angular signals,” IEEE Trans. Signal Process. 46, 3181-3194 (1998).
    [CrossRef]
  17. P. E. Trahanias, D. Karakos, and A. N. Venetsanopoulos, “Directional processing of color images: Theory and experimental results,” IEEE Trans. Image Process. 5, 868-880 (1996).
    [CrossRef] [PubMed]
  18. D. G. Karakos and P. E. Trahanias, “Generalized multichannel image filtering structures,” IEEE Trans. Image Process. 6, 1038-1045 (1997).
    [CrossRef] [PubMed]
  19. L. Khriji and M. Gabbouj, “Adaptive fuzzy order statistics-rational hybrid filters for color image processing,” Fuzzy Sets Syst. 128, 35-46 (2002).
    [CrossRef]
  20. R. Lukac, “Adaptive color image filtering based on center-weighted vector directional filters,” Multidimens. Syst. Signal Process. 15, 169-196 (2004).
    [CrossRef]
  21. K. N. Plataniotis, D. Androutsos, and A. N. Venetsanopoulos, “Color image processing using adaptive vector directional filters,” IEEE Trans. on Circuits and Systems-II 45, 1414-1419 (1998).
    [CrossRef]
  22. R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Vector sigma filters for noise detection and removal in color images,” J. Visual Commun. Image Represent 17, 1-26 (2006).
    [CrossRef]
  23. K. N. Plataniotis, D. Androutsos, and A. N. Venetsanopoulos, “Adaptive fuzzy systems for multichannel signal processing,” Proc. IEEE 87, 1601-1622 (1999).
    [CrossRef]
  24. B. Smolka, K. N. Plataniotis, R. Lukac, and A. N. Venetsanopoulos, “Kernel density estimation based impulsive noise reduction filter,” in Proceedings of the IEEE International Conference on Image Processing (ICIP'03) (IEEE, 2003), Vol. 2, pp. 137-140.
  25. B. Smolka, R. Lukac, A. Chydzinski, K. N. Plataniotis, and K. Wojciechowski, “Fast adaptive similarity based impulsive noise reduction filter,” Real-Time Imag. 9, 261-276 (2003).
    [CrossRef]
  26. B. Smolka, K. N. Plataniotis, A. Chydzinski, M. Szczepanski, A. N. Venetsanopoulos, and K. Wojciechowski, “Self-adaptive algorithm of impulsive noise reduction in color images,” Pattern Recogn. 35, 1771-1784 (2002).
    [CrossRef]
  27. R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Generalized entropy vector filters,” in Proceedings of the 4th EURASIP EC-VIP-MC, Video, Image Processing and Multimedia Communications Conference (IEEE, 2003), pp. 239-244.
  28. T. Viero, K. Oistamo, and Y. Neuvo, “Three-dimensional median-related filters for color image sequence filtering,” IEEE Trans. Circuits Syst. Video Technol. 4, 129-142 (1994).
    [CrossRef]

2008 (1)

M. E. Celebi and Y. A. Aslandogan, “Robust switching vector median filter for impulsive noise removal,” J. Electron. Imaging 17, 043006 (2008).
[CrossRef]

2007 (2)

M. E. Celebi, H. A. Kingravi, and Y. A. Aslandogan, “Nonlinear vector filtering for impulsive noise removal from color images,” J. Electron. Imaging 16, 033008 (2007).
[CrossRef]

M. E. Celebi, H. A. Kingravi, B. Uddin, and Y. A. Aslandogan, “Fast switching filter for impulsive noise removal from color images,” J. Imaging Sci. Technol. 51, 155-165 (2007).
[CrossRef]

2006 (1)

R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Vector sigma filters for noise detection and removal in color images,” J. Visual Commun. Image Represent 17, 1-26 (2006).
[CrossRef]

2005 (1)

R. Lukac, B. Smolka, K. Martin, K. N. Plataniotis, and A. N. Venetsanopoulos, “Vector filtering for color imaging,” IEEE Signal Process. Mag. 22, 74-86 (2005).
[CrossRef]

2004 (1)

R. Lukac, “Adaptive color image filtering based on center-weighted vector directional filters,” Multidimens. Syst. Signal Process. 15, 169-196 (2004).
[CrossRef]

2003 (1)

B. Smolka, R. Lukac, A. Chydzinski, K. N. Plataniotis, and K. Wojciechowski, “Fast adaptive similarity based impulsive noise reduction filter,” Real-Time Imag. 9, 261-276 (2003).
[CrossRef]

2002 (2)

B. Smolka, K. N. Plataniotis, A. Chydzinski, M. Szczepanski, A. N. Venetsanopoulos, and K. Wojciechowski, “Self-adaptive algorithm of impulsive noise reduction in color images,” Pattern Recogn. 35, 1771-1784 (2002).
[CrossRef]

L. Khriji and M. Gabbouj, “Adaptive fuzzy order statistics-rational hybrid filters for color image processing,” Fuzzy Sets Syst. 128, 35-46 (2002).
[CrossRef]

2000 (1)

M. Barni, F. Buti, F. Bartolini, and V. Cappellini, “A quasi-Euclidean norm to speed up vector median filtering,” IEEE Trans. Image Process. 9, 1704-1709 (2000).
[CrossRef]

1999 (1)

K. N. Plataniotis, D. Androutsos, and A. N. Venetsanopoulos, “Adaptive fuzzy systems for multichannel signal processing,” Proc. IEEE 87, 1601-1622 (1999).
[CrossRef]

1998 (2)

K. N. Plataniotis, D. Androutsos, and A. N. Venetsanopoulos, “Color image processing using adaptive vector directional filters,” IEEE Trans. on Circuits and Systems-II 45, 1414-1419 (1998).
[CrossRef]

N. Nikolaidis and I. Pitas, “Nonlinear processing and analysis of angular signals,” IEEE Trans. Signal Process. 46, 3181-3194 (1998).
[CrossRef]

1997 (3)

K. N. Plataniotis, D. Androutsos, S. Vinayagamoorthy, and A. N. Venetsanopoulos, “Color image processing using adaptive multichannel filters,” IEEE Trans. Image Process. 6, 933-949 (1997).
[CrossRef] [PubMed]

D. G. Karakos and P. E. Trahanias, “Generalized multichannel image filtering structures,” IEEE Trans. Image Process. 6, 1038-1045 (1997).
[CrossRef] [PubMed]

M. Barni, “A fast algorithm for 1-norm vector median filtering,” IEEE Trans. Image Process. 6, 1452-1455 (1997).
[CrossRef] [PubMed]

1996 (1)

P. E. Trahanias, D. Karakos, and A. N. Venetsanopoulos, “Directional processing of color images: Theory and experimental results,” IEEE Trans. Image Process. 5, 868-880 (1996).
[CrossRef] [PubMed]

1994 (1)

T. Viero, K. Oistamo, and Y. Neuvo, “Three-dimensional median-related filters for color image sequence filtering,” IEEE Trans. Circuits Syst. Video Technol. 4, 129-142 (1994).
[CrossRef]

1993 (1)

P. E. Trahanias and A. N. Venetsanopoulos, “Vector directional filters: A new class of multichannel image processing filters,” IEEE Trans. Image Process. 2, 528-534 (1993).
[CrossRef] [PubMed]

1990 (1)

J. Astola, P. Haavisto, and Y. Neuvo, “Vector median filters,” Proc. IEEE 78, 678-689 (1990).
[CrossRef]

1965 (1)

W. Fraser, “A survey of methods of computing minimax and near-minimax polynomial approximations for functions of a single independent variable,” J. ACM 12, 295-314 (1965).
[CrossRef]

Androutsos, D.

K. N. Plataniotis, D. Androutsos, and A. N. Venetsanopoulos, “Adaptive fuzzy systems for multichannel signal processing,” Proc. IEEE 87, 1601-1622 (1999).
[CrossRef]

K. N. Plataniotis, D. Androutsos, and A. N. Venetsanopoulos, “Color image processing using adaptive vector directional filters,” IEEE Trans. on Circuits and Systems-II 45, 1414-1419 (1998).
[CrossRef]

K. N. Plataniotis, D. Androutsos, S. Vinayagamoorthy, and A. N. Venetsanopoulos, “Color image processing using adaptive multichannel filters,” IEEE Trans. Image Process. 6, 933-949 (1997).
[CrossRef] [PubMed]

Aslandogan, Y. A.

M. E. Celebi and Y. A. Aslandogan, “Robust switching vector median filter for impulsive noise removal,” J. Electron. Imaging 17, 043006 (2008).
[CrossRef]

M. E. Celebi, H. A. Kingravi, and Y. A. Aslandogan, “Nonlinear vector filtering for impulsive noise removal from color images,” J. Electron. Imaging 16, 033008 (2007).
[CrossRef]

M. E. Celebi, H. A. Kingravi, B. Uddin, and Y. A. Aslandogan, “Fast switching filter for impulsive noise removal from color images,” J. Imaging Sci. Technol. 51, 155-165 (2007).
[CrossRef]

Astola, J.

J. Astola, P. Haavisto, and Y. Neuvo, “Vector median filters,” Proc. IEEE 78, 678-689 (1990).
[CrossRef]

Barni, M.

M. Barni, F. Buti, F. Bartolini, and V. Cappellini, “A quasi-Euclidean norm to speed up vector median filtering,” IEEE Trans. Image Process. 9, 1704-1709 (2000).
[CrossRef]

M. Barni, “A fast algorithm for 1-norm vector median filtering,” IEEE Trans. Image Process. 6, 1452-1455 (1997).
[CrossRef] [PubMed]

Bartolini, F.

M. Barni, F. Buti, F. Bartolini, and V. Cappellini, “A quasi-Euclidean norm to speed up vector median filtering,” IEEE Trans. Image Process. 9, 1704-1709 (2000).
[CrossRef]

Buti, F.

M. Barni, F. Buti, F. Bartolini, and V. Cappellini, “A quasi-Euclidean norm to speed up vector median filtering,” IEEE Trans. Image Process. 9, 1704-1709 (2000).
[CrossRef]

Cappellini, V.

M. Barni, F. Buti, F. Bartolini, and V. Cappellini, “A quasi-Euclidean norm to speed up vector median filtering,” IEEE Trans. Image Process. 9, 1704-1709 (2000).
[CrossRef]

Celebi, M. E.

M. E. Celebi and Y. A. Aslandogan, “Robust switching vector median filter for impulsive noise removal,” J. Electron. Imaging 17, 043006 (2008).
[CrossRef]

M. E. Celebi, H. A. Kingravi, B. Uddin, and Y. A. Aslandogan, “Fast switching filter for impulsive noise removal from color images,” J. Imaging Sci. Technol. 51, 155-165 (2007).
[CrossRef]

M. E. Celebi, H. A. Kingravi, and Y. A. Aslandogan, “Nonlinear vector filtering for impulsive noise removal from color images,” J. Electron. Imaging 16, 033008 (2007).
[CrossRef]

Cheney, E. W.

E. W. Cheney, Introduction to Approximation Theory (AMS, 2000).

Chydzinski, A.

B. Smolka, R. Lukac, A. Chydzinski, K. N. Plataniotis, and K. Wojciechowski, “Fast adaptive similarity based impulsive noise reduction filter,” Real-Time Imag. 9, 261-276 (2003).
[CrossRef]

B. Smolka, K. N. Plataniotis, A. Chydzinski, M. Szczepanski, A. N. Venetsanopoulos, and K. Wojciechowski, “Self-adaptive algorithm of impulsive noise reduction in color images,” Pattern Recogn. 35, 1771-1784 (2002).
[CrossRef]

Fraser, W.

W. Fraser, “A survey of methods of computing minimax and near-minimax polynomial approximations for functions of a single independent variable,” J. ACM 12, 295-314 (1965).
[CrossRef]

Gabbouj, M.

L. Khriji and M. Gabbouj, “Adaptive fuzzy order statistics-rational hybrid filters for color image processing,” Fuzzy Sets Syst. 128, 35-46 (2002).
[CrossRef]

Haavisto, P.

J. Astola, P. Haavisto, and Y. Neuvo, “Vector median filters,” Proc. IEEE 78, 678-689 (1990).
[CrossRef]

Karakos, D.

P. E. Trahanias, D. Karakos, and A. N. Venetsanopoulos, “Directional processing of color images: Theory and experimental results,” IEEE Trans. Image Process. 5, 868-880 (1996).
[CrossRef] [PubMed]

Karakos, D. G.

D. G. Karakos and P. E. Trahanias, “Generalized multichannel image filtering structures,” IEEE Trans. Image Process. 6, 1038-1045 (1997).
[CrossRef] [PubMed]

Khriji, L.

L. Khriji and M. Gabbouj, “Adaptive fuzzy order statistics-rational hybrid filters for color image processing,” Fuzzy Sets Syst. 128, 35-46 (2002).
[CrossRef]

Kingravi, H. A.

M. E. Celebi, H. A. Kingravi, B. Uddin, and Y. A. Aslandogan, “Fast switching filter for impulsive noise removal from color images,” J. Imaging Sci. Technol. 51, 155-165 (2007).
[CrossRef]

M. E. Celebi, H. A. Kingravi, and Y. A. Aslandogan, “Nonlinear vector filtering for impulsive noise removal from color images,” J. Electron. Imaging 16, 033008 (2007).
[CrossRef]

Lukac, R.

R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Vector sigma filters for noise detection and removal in color images,” J. Visual Commun. Image Represent 17, 1-26 (2006).
[CrossRef]

R. Lukac, B. Smolka, K. Martin, K. N. Plataniotis, and A. N. Venetsanopoulos, “Vector filtering for color imaging,” IEEE Signal Process. Mag. 22, 74-86 (2005).
[CrossRef]

R. Lukac, “Adaptive color image filtering based on center-weighted vector directional filters,” Multidimens. Syst. Signal Process. 15, 169-196 (2004).
[CrossRef]

B. Smolka, R. Lukac, A. Chydzinski, K. N. Plataniotis, and K. Wojciechowski, “Fast adaptive similarity based impulsive noise reduction filter,” Real-Time Imag. 9, 261-276 (2003).
[CrossRef]

R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Generalized entropy vector filters,” in Proceedings of the 4th EURASIP EC-VIP-MC, Video, Image Processing and Multimedia Communications Conference (IEEE, 2003), pp. 239-244.

B. Smolka, K. N. Plataniotis, R. Lukac, and A. N. Venetsanopoulos, “Kernel density estimation based impulsive noise reduction filter,” in Proceedings of the IEEE International Conference on Image Processing (ICIP'03) (IEEE, 2003), Vol. 2, pp. 137-140.

R. Lukac and K. N. Plataniotis, “A taxonomy of color image filtering and enhancement solutions,” in Advances in Imaging & Electron Physics, P.W.Hawkes, ed. (Academic, 2006), (Vol. 140), pp. 187-264.
[CrossRef]

R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Entropy vector median filter,” in Proceedings of the IbPRIA Conference, Lecture Notes in Computer Science 2652 (Springer, 2003), pp. 1117-1125.

Martin, K.

R. Lukac, B. Smolka, K. Martin, K. N. Plataniotis, and A. N. Venetsanopoulos, “Vector filtering for color imaging,” IEEE Signal Process. Mag. 22, 74-86 (2005).
[CrossRef]

Muller, J.-M.

J.-M. Muller, Elementary Functions: Algorithms and Implementation (Birkhäuser, 2006).

Neuvo, Y.

T. Viero, K. Oistamo, and Y. Neuvo, “Three-dimensional median-related filters for color image sequence filtering,” IEEE Trans. Circuits Syst. Video Technol. 4, 129-142 (1994).
[CrossRef]

J. Astola, P. Haavisto, and Y. Neuvo, “Vector median filters,” Proc. IEEE 78, 678-689 (1990).
[CrossRef]

Nikolaidis, N.

N. Nikolaidis and I. Pitas, “Nonlinear processing and analysis of angular signals,” IEEE Trans. Signal Process. 46, 3181-3194 (1998).
[CrossRef]

Oistamo, K.

T. Viero, K. Oistamo, and Y. Neuvo, “Three-dimensional median-related filters for color image sequence filtering,” IEEE Trans. Circuits Syst. Video Technol. 4, 129-142 (1994).
[CrossRef]

Pitas, I.

N. Nikolaidis and I. Pitas, “Nonlinear processing and analysis of angular signals,” IEEE Trans. Signal Process. 46, 3181-3194 (1998).
[CrossRef]

Plataniotis, K. N.

R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Vector sigma filters for noise detection and removal in color images,” J. Visual Commun. Image Represent 17, 1-26 (2006).
[CrossRef]

R. Lukac, B. Smolka, K. Martin, K. N. Plataniotis, and A. N. Venetsanopoulos, “Vector filtering for color imaging,” IEEE Signal Process. Mag. 22, 74-86 (2005).
[CrossRef]

B. Smolka, R. Lukac, A. Chydzinski, K. N. Plataniotis, and K. Wojciechowski, “Fast adaptive similarity based impulsive noise reduction filter,” Real-Time Imag. 9, 261-276 (2003).
[CrossRef]

B. Smolka, K. N. Plataniotis, A. Chydzinski, M. Szczepanski, A. N. Venetsanopoulos, and K. Wojciechowski, “Self-adaptive algorithm of impulsive noise reduction in color images,” Pattern Recogn. 35, 1771-1784 (2002).
[CrossRef]

K. N. Plataniotis, D. Androutsos, and A. N. Venetsanopoulos, “Adaptive fuzzy systems for multichannel signal processing,” Proc. IEEE 87, 1601-1622 (1999).
[CrossRef]

K. N. Plataniotis, D. Androutsos, and A. N. Venetsanopoulos, “Color image processing using adaptive vector directional filters,” IEEE Trans. on Circuits and Systems-II 45, 1414-1419 (1998).
[CrossRef]

K. N. Plataniotis, D. Androutsos, S. Vinayagamoorthy, and A. N. Venetsanopoulos, “Color image processing using adaptive multichannel filters,” IEEE Trans. Image Process. 6, 933-949 (1997).
[CrossRef] [PubMed]

R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Entropy vector median filter,” in Proceedings of the IbPRIA Conference, Lecture Notes in Computer Science 2652 (Springer, 2003), pp. 1117-1125.

R. Lukac and K. N. Plataniotis, “A taxonomy of color image filtering and enhancement solutions,” in Advances in Imaging & Electron Physics, P.W.Hawkes, ed. (Academic, 2006), (Vol. 140), pp. 187-264.
[CrossRef]

K. N. Plataniotis and A. N. Venetsanopoulos, Color Image Processing and Applications (Springer, 2000).

B. Smolka, K. N. Plataniotis, R. Lukac, and A. N. Venetsanopoulos, “Kernel density estimation based impulsive noise reduction filter,” in Proceedings of the IEEE International Conference on Image Processing (ICIP'03) (IEEE, 2003), Vol. 2, pp. 137-140.

R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Generalized entropy vector filters,” in Proceedings of the 4th EURASIP EC-VIP-MC, Video, Image Processing and Multimedia Communications Conference (IEEE, 2003), pp. 239-244.

Smolka, B.

R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Vector sigma filters for noise detection and removal in color images,” J. Visual Commun. Image Represent 17, 1-26 (2006).
[CrossRef]

R. Lukac, B. Smolka, K. Martin, K. N. Plataniotis, and A. N. Venetsanopoulos, “Vector filtering for color imaging,” IEEE Signal Process. Mag. 22, 74-86 (2005).
[CrossRef]

B. Smolka, R. Lukac, A. Chydzinski, K. N. Plataniotis, and K. Wojciechowski, “Fast adaptive similarity based impulsive noise reduction filter,” Real-Time Imag. 9, 261-276 (2003).
[CrossRef]

B. Smolka, K. N. Plataniotis, A. Chydzinski, M. Szczepanski, A. N. Venetsanopoulos, and K. Wojciechowski, “Self-adaptive algorithm of impulsive noise reduction in color images,” Pattern Recogn. 35, 1771-1784 (2002).
[CrossRef]

R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Generalized entropy vector filters,” in Proceedings of the 4th EURASIP EC-VIP-MC, Video, Image Processing and Multimedia Communications Conference (IEEE, 2003), pp. 239-244.

B. Smolka, K. N. Plataniotis, R. Lukac, and A. N. Venetsanopoulos, “Kernel density estimation based impulsive noise reduction filter,” in Proceedings of the IEEE International Conference on Image Processing (ICIP'03) (IEEE, 2003), Vol. 2, pp. 137-140.

R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Entropy vector median filter,” in Proceedings of the IbPRIA Conference, Lecture Notes in Computer Science 2652 (Springer, 2003), pp. 1117-1125.

Szczepanski, M.

B. Smolka, K. N. Plataniotis, A. Chydzinski, M. Szczepanski, A. N. Venetsanopoulos, and K. Wojciechowski, “Self-adaptive algorithm of impulsive noise reduction in color images,” Pattern Recogn. 35, 1771-1784 (2002).
[CrossRef]

Trahanias, P. E.

D. G. Karakos and P. E. Trahanias, “Generalized multichannel image filtering structures,” IEEE Trans. Image Process. 6, 1038-1045 (1997).
[CrossRef] [PubMed]

P. E. Trahanias, D. Karakos, and A. N. Venetsanopoulos, “Directional processing of color images: Theory and experimental results,” IEEE Trans. Image Process. 5, 868-880 (1996).
[CrossRef] [PubMed]

P. E. Trahanias and A. N. Venetsanopoulos, “Vector directional filters: A new class of multichannel image processing filters,” IEEE Trans. Image Process. 2, 528-534 (1993).
[CrossRef] [PubMed]

Uddin, B.

M. E. Celebi, H. A. Kingravi, B. Uddin, and Y. A. Aslandogan, “Fast switching filter for impulsive noise removal from color images,” J. Imaging Sci. Technol. 51, 155-165 (2007).
[CrossRef]

Venetsanopoulos, A. N.

R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Vector sigma filters for noise detection and removal in color images,” J. Visual Commun. Image Represent 17, 1-26 (2006).
[CrossRef]

R. Lukac, B. Smolka, K. Martin, K. N. Plataniotis, and A. N. Venetsanopoulos, “Vector filtering for color imaging,” IEEE Signal Process. Mag. 22, 74-86 (2005).
[CrossRef]

B. Smolka, K. N. Plataniotis, A. Chydzinski, M. Szczepanski, A. N. Venetsanopoulos, and K. Wojciechowski, “Self-adaptive algorithm of impulsive noise reduction in color images,” Pattern Recogn. 35, 1771-1784 (2002).
[CrossRef]

K. N. Plataniotis, D. Androutsos, and A. N. Venetsanopoulos, “Adaptive fuzzy systems for multichannel signal processing,” Proc. IEEE 87, 1601-1622 (1999).
[CrossRef]

K. N. Plataniotis, D. Androutsos, and A. N. Venetsanopoulos, “Color image processing using adaptive vector directional filters,” IEEE Trans. on Circuits and Systems-II 45, 1414-1419 (1998).
[CrossRef]

K. N. Plataniotis, D. Androutsos, S. Vinayagamoorthy, and A. N. Venetsanopoulos, “Color image processing using adaptive multichannel filters,” IEEE Trans. Image Process. 6, 933-949 (1997).
[CrossRef] [PubMed]

P. E. Trahanias, D. Karakos, and A. N. Venetsanopoulos, “Directional processing of color images: Theory and experimental results,” IEEE Trans. Image Process. 5, 868-880 (1996).
[CrossRef] [PubMed]

P. E. Trahanias and A. N. Venetsanopoulos, “Vector directional filters: A new class of multichannel image processing filters,” IEEE Trans. Image Process. 2, 528-534 (1993).
[CrossRef] [PubMed]

R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Entropy vector median filter,” in Proceedings of the IbPRIA Conference, Lecture Notes in Computer Science 2652 (Springer, 2003), pp. 1117-1125.

K. N. Plataniotis and A. N. Venetsanopoulos, Color Image Processing and Applications (Springer, 2000).

B. Smolka, K. N. Plataniotis, R. Lukac, and A. N. Venetsanopoulos, “Kernel density estimation based impulsive noise reduction filter,” in Proceedings of the IEEE International Conference on Image Processing (ICIP'03) (IEEE, 2003), Vol. 2, pp. 137-140.

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B. Smolka, K. N. Plataniotis, A. Chydzinski, M. Szczepanski, A. N. Venetsanopoulos, and K. Wojciechowski, “Self-adaptive algorithm of impulsive noise reduction in color images,” Pattern Recogn. 35, 1771-1784 (2002).
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R. Lukac, B. Smolka, K. N. Plataniotis, and A. N. Venetsanopoulos, “Generalized entropy vector filters,” in Proceedings of the 4th EURASIP EC-VIP-MC, Video, Image Processing and Multimedia Communications Conference (IEEE, 2003), pp. 239-244.

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Figures (6)

Fig. 1
Fig. 1

Indexing convention inside a 3 × 3 window.

Fig. 2
Fig. 2

Function arccos(z) in the interval [0, 1].

Fig. 3
Fig. 3

Function arcsin(z) in the interval [0, 0.5].

Fig. 4
Fig. 4

Function exp(–z) in the interval [0, 10].

Fig. 5
Fig. 5

Function z log(z) in the interval [0.05, 1].

Fig. 6
Fig. 6

Comparison of the exact and approximate filters on the Lena image: (a) Original ( 512 × 512 ) , (b) 10% noisy (MAE 6.373, MSE 987.418, time 0), (c) BVDF exact (MAE 3.936, MSE 43.571, time 10.360), (d) BVDF approx. (MAE 3.936, MSE 43.558, time 0.688), (e) AMNFE exact (MAE 3.471, MSE 30.303, time 0.594), (f) AMNFE approx. (MAE 3.471, MSE 30.303, time 0.422), (g) EVMF exact (MAE 1.139, MSE 25.898, time 0.594), (h) EVMF approx. (MAE 1.138, MSE 25.819, time 0.250).

Tables (6)

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Table 1 Fourth-Degree Minimax Polynomials for the ARCSIN and ARCOS Functions

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Table 2 Minimax Polynomials for the EXP Function

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Table 3 Minimax Rational for the EXP Function

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Table 4 Minimax Rational for the ENT Function

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Table 5 Performance Statistics at 10% Noise Level

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Table 6 Performance Statistics at Higher Noise Levels

Equations (18)

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

ε = max z [ a , b ] f ( z ) P ( z ) .
P n ( [ a , b ] ) = { a 0 + a 1 z + + a n z n : z [ a , b ] , a i R , i = 0 , 1 , , n }
R m n ( [ a , b ] ) = { p ( z ) q ( z ) : p ( z ) P n ( [ a , b ] ) , q ( z ) P m ( [ a , b ] ) } ,
max z [ a , b ] f ( z ) r * ( z ) = dist ( f , R m n ) ,
y ( r , c ) = arg min x i W ( r , c ) ( j = 1 n A ( x i , x j ) ) ,
A ( x i , x j ) = arccos ( x i 1 x j 1 + x i 2 x j 2 + x i 3 x j 3 x i 2 x j 2 ) ,
arccos ( z ) = 2 arcsin ( 0.5 ( 1 z ) ) ,
τ = 1 z
arccos ( z ) = 2 arcsin ( τ 2 ) ,
y ( r , c ) = i = 1 n x i ( h i 3 K ( ( x C x i ) h i ) j = 1 n h j 3 K ( ( x C x j ) h j ) ) ,
h i = n κ 3 j = 1 n x i x j 1 ,
y ( r , c ) = { arg min x i W ( r , c ) ( j = 1 n x i x j 2 ) P C > β C x ( r , c ) otherwise , }
P i = x i x ¯ 2 j = 1 n x j x ¯ 2 ; β i = P i log P i j = 1 n P j log P j ,
Uncorrelated Impulsive Noise
x = { x 1 , x 2 , x 3 }
x k = { o k with probability 1 ϕ k , r k with probability ϕ k , }
Correlated Impulsive Noise
x = { o with probability 1 ϕ , { r 1 , o 2 , o 3 } with probability ϕ 1 ϕ , { o 1 , r 2 , o 3 } with probability ϕ 2 ϕ , { o 1 , o 2 , r 3 } with probability ϕ 3 ϕ , { r 1 , r 2 , r 3 } with probability ( 1 ( ϕ 1 + ϕ 2 + ϕ 3 ) ) ϕ }

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