Generalized Correlativity of Median Filtering Operator on Signals

ABSTRACT

The generalized correlativity of input signal and output signal of a stack filtering operator is defined and used for numerously measuring these filtering operators's behavior in removing noise in signals. We show that under the criterion of the generalized correlativity, of stack filtering operators the median filtering operator is optimal, which implies that this filtering operator possesses better filtering behavior than the others.

The generalized correlativity of input signal and output signal of a stack filtering operator is defined and used for numerously measuring these filtering operators's behavior in removing noise in signals. We show that under the criterion of the generalized correlativity, of stack filtering operators the median filtering operator is optimal, which implies that this filtering operator possesses better filtering behavior than the others.

Cite this paper

W. Ye and Z. Liao, "Generalized Correlativity of Median Filtering Operator on Signals,"*Open Journal of Discrete Mathematics*, Vol. 2 No. 3, 2012, pp. 83-87. doi: 10.4236/ojdm.2012.23015.

W. Ye and Z. Liao, "Generalized Correlativity of Median Filtering Operator on Signals,"

References

[1] J. W. Tukey, “Nonlinear (Nonsuperposable) Methods for Smoothing Data,” Proceedings of Congress Record EASCON, Washington DC, 7-9 October 1974, p. 673.

[2] H. X. Chen, R. K. Yang and M. Gabbouj, “On root Structures and Convergence Properties of Weighted Median Filters,” Circuits and System Signal Processing, Vol. 14, No. 6, 1995, pp. 735-747. doi:10.1007/BF01204682

[3] U. Eckhardt, “Root Images of Median Filters,” Journal of Mathematical Imeging and Vision, Vol. 19, No. 1, 2003, pp. 63-70. doi:10.1023/A:1024489020930

[4] D. Eberly, H. Longbotham and J. Aragon, “Complete Classification of Roots to One-Demensional Median and Rank-Order Filters,” IEEE Transactions on Signal Processing, Vol. 39, No. 1, 1991, pp. 197-199. doi:10.1109/78.80781

[5] S. G. Tyan, “Median Filters: Dete-rinistic Properties,” In: Two-Dimension Digital Signal Processing II: Transforms and Median Filters, Springer, Berlin, 1981. doi:10.1007/BFb0057598

[6] P.-T. Yu and W.-L. Wang, “Root Properties of Median Filters under There Appending Strategies,” IEEE Transactions on Signal Processing, Vol. 41, No. 2, 1993, pp. 965-970. doi:10.1109/78.193236

[7] J. Brandt, “Cycles of Medians,” Utilitas Mathematica, Vol. 54, 1998, pp. 111-126.

[8] W. Z. Ye, L. Wang and L. G. Xu, “Properties of Locally Convergent Sequences with Respect to Median Filter,” Discrete Mathematics, Vol. 309, No. 9, 2009, pp. 27752781. doi:10.1016/j.disc.2008.07.002

[9] W. Z. Ye, M. T. Zhang and Y. L. Ma, “Structure of Recurrent Sequences of Median Filters,” Discrete Mathematics, Vol. 310, No. 6-7, 2010, pp. 1253-1258. doi:10.1016/j.disc.2009.12.005

[10] Z.-J. Gan and M Mao, “Two Convergence Theorems on Deterministic Properties of Median Filters,” IEEE Transactions on Signal Processing, Vol. 39, No. 7, 1991, pp. 1689-1690. doi:10.1109/78.134410

[11] P. D. Wendt, E. J. Coyle and N. C. Gallagher Jr., “Stack Filter,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 34, No. 4, 1986, pp. 898-911. doi:10.1109/TASSP.1986.1164871

[12] W. Z. Ye and X. W. Zhou, “Criteria of Convergence of Median Filters,” IEEE Transactions on Signal Processing, Vol. 49, No. 2, 2001, pp. 360-363. doi:10.1109/78.902118

[13] T. A. Nodes and N. C. Gallagher, “The Output Distribution of Median-Type Filters,” IEEE Trans-actions on Communication, Vol. 32. No. 5, 1984, pp. 532-541. doi:10.1109/TCOM.1984.1096099

[14] A. C.Bovik, T. S. Huang and D. C. Munson, “A generalization of Median Filtering Using Linear Combinations of Order Statistics,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 31, No. 6, 1983, pp. 13421349. doi:10.1109/TASSP.1983.1164247

[15] R. Oten and R. J. P. de Figueiredo, “An Efficient Method for L-Filter Design,” IEEE Transactions on Signal Processing, Vol. 51, No. 1, 2003, pp. 193-203. doi:10.1109/TSP.2002.806573

[16] J. P. Fitch, E. J. Coyle and N. C. Gallagher Jr., “Median Filtering by Threshold Decompo-sition,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 32, No. 6, 1984, pp. 1183-1188.

[1] J. W. Tukey, “Nonlinear (Nonsuperposable) Methods for Smoothing Data,” Proceedings of Congress Record EASCON, Washington DC, 7-9 October 1974, p. 673.

[2] H. X. Chen, R. K. Yang and M. Gabbouj, “On root Structures and Convergence Properties of Weighted Median Filters,” Circuits and System Signal Processing, Vol. 14, No. 6, 1995, pp. 735-747. doi:10.1007/BF01204682

[3] U. Eckhardt, “Root Images of Median Filters,” Journal of Mathematical Imeging and Vision, Vol. 19, No. 1, 2003, pp. 63-70. doi:10.1023/A:1024489020930

[4] D. Eberly, H. Longbotham and J. Aragon, “Complete Classification of Roots to One-Demensional Median and Rank-Order Filters,” IEEE Transactions on Signal Processing, Vol. 39, No. 1, 1991, pp. 197-199. doi:10.1109/78.80781

[5] S. G. Tyan, “Median Filters: Dete-rinistic Properties,” In: Two-Dimension Digital Signal Processing II: Transforms and Median Filters, Springer, Berlin, 1981. doi:10.1007/BFb0057598

[6] P.-T. Yu and W.-L. Wang, “Root Properties of Median Filters under There Appending Strategies,” IEEE Transactions on Signal Processing, Vol. 41, No. 2, 1993, pp. 965-970. doi:10.1109/78.193236

[7] J. Brandt, “Cycles of Medians,” Utilitas Mathematica, Vol. 54, 1998, pp. 111-126.

[8] W. Z. Ye, L. Wang and L. G. Xu, “Properties of Locally Convergent Sequences with Respect to Median Filter,” Discrete Mathematics, Vol. 309, No. 9, 2009, pp. 27752781. doi:10.1016/j.disc.2008.07.002

[9] W. Z. Ye, M. T. Zhang and Y. L. Ma, “Structure of Recurrent Sequences of Median Filters,” Discrete Mathematics, Vol. 310, No. 6-7, 2010, pp. 1253-1258. doi:10.1016/j.disc.2009.12.005

[10] Z.-J. Gan and M Mao, “Two Convergence Theorems on Deterministic Properties of Median Filters,” IEEE Transactions on Signal Processing, Vol. 39, No. 7, 1991, pp. 1689-1690. doi:10.1109/78.134410

[11] P. D. Wendt, E. J. Coyle and N. C. Gallagher Jr., “Stack Filter,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 34, No. 4, 1986, pp. 898-911. doi:10.1109/TASSP.1986.1164871

[12] W. Z. Ye and X. W. Zhou, “Criteria of Convergence of Median Filters,” IEEE Transactions on Signal Processing, Vol. 49, No. 2, 2001, pp. 360-363. doi:10.1109/78.902118

[13] T. A. Nodes and N. C. Gallagher, “The Output Distribution of Median-Type Filters,” IEEE Trans-actions on Communication, Vol. 32. No. 5, 1984, pp. 532-541. doi:10.1109/TCOM.1984.1096099

[14] A. C.Bovik, T. S. Huang and D. C. Munson, “A generalization of Median Filtering Using Linear Combinations of Order Statistics,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 31, No. 6, 1983, pp. 13421349. doi:10.1109/TASSP.1983.1164247

[15] R. Oten and R. J. P. de Figueiredo, “An Efficient Method for L-Filter Design,” IEEE Transactions on Signal Processing, Vol. 51, No. 1, 2003, pp. 193-203. doi:10.1109/TSP.2002.806573

[16] J. P. Fitch, E. J. Coyle and N. C. Gallagher Jr., “Median Filtering by Threshold Decompo-sition,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 32, No. 6, 1984, pp. 1183-1188.