Introduction to a simple yet effective Two-Dimensional Fuzzy Smoothing Filter

bibitem{14}
A. A. Alshennawy and A. A. Aly, emph{Edge detection in digital images using fuzzy logic technique}, World Academy of Science, Engineering and Technology, textbf{51} (2009), 178-187.

bibitem{33}
D. Androutsos, K. N. Plataniotis and A. N. Venetsanopoulos, emph{Novel vector-based approach to color image retrieval using a vector angular-based distance measure}, Computer Vision and Image Understanding, textbf{{75}(1,2)} (1999), 46–58.

bibitem{8}
F. Beritelli, S. Casale and A. Cavallaro, emph{Multi-channel speech/silence detector based on time delay estimation and fuzzy classification}, IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP’99), (1999), 93-96.

bibitem{19}
R. Caponetto, L. LoBello and O. Mirabella, emph{Fuzzy traffic smoothing: another step towards statistical real-time communication over ethernet networks}, 1st Internet workshop on Real-Time LANs in the Internet Age (RTILA’02), 2002.

bibitem{21}
G. C. Cardarilli, M. Re, R. Lojacono and L. Carfagnini, emph{High speed fuzzy filter for non-linear channel equalization}, In Proceedings of the International Symposium on Fuzzy Logic (ISFL'97), 1997.

bibitem{1}
Central Bank of Iran: www.cbi.ir

bibitem{26}
K. M. T. Chen and L. Chen, emph{Tri-state median filter for image de-noising}, IEEE Transactions on Image Processing, textbf{{8}(12)} (1999), 1834-1838.

bibitem{22}
S. F. Chen and R. Rosenfeld, emph{A gaussian prior for smoothing maximum entropy models}, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213, (Available on-line at: http://reports-archive.adm.cs.cmu.edu/anon/1999/CMU-CS-99-108.pdf), 1999.

bibitem{9}
R. C. Chen and P. Yu, emph{Nonlinear filter design based on fuzzy inference rules for image processing}, IEEE Workshop on Nonlinear Signal and Image Processing (NSIP97), 1997.

bibitem{11}
F. Farbiz, M. B. Menhaj, S. A. Motamedi and M. T. Hagan, emph{A new fuzzy logic control based filter for image enhancement}, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, textbf{{30}(1)} (2000), 110-119.

bibitem{20}
Y. R. Hwang and M. Tomizuka, emph{Fuzzy smoothing algorithms for variable structure systems}, IEEE Transactions on Fuzzy Systems, textbf{{2}(4)} (1994), 277-284.

bibitem{12}
H. R. Ishii, A. Taguchi and M. Sone, emph{The edge detection from images corrupted by mixed noise using fuzzy rules}, IX European Signal Processing Conference, 1998.

bibitem{35}
K. Kal Vin Toh, H. Ibrahim and M. N. Mahyuddin, emph{Salt-and-Pepper noise detection and reduction using fuzzy switching median filter}, IEEE Transactions on Consumer Electronics, textbf{{54}(4)} (2008), 1956-1961.

bibitem{30}
Y. H. Kuo, C. Lee and C. Chen, emph{High-stability AWFM filter for signal restoration and its hardware design}, Fuzzy Sets and Systems, textbf{{114}(2)} (2000), 185-202.

bibitem{10}
Y. H. Kuo, C. S. Lee and C. C. Liu, emph{A new fuzzy edge detection method for image enhancement}, The Sixth IEEE International Conference on Fuzzy Systems, textbf{1} (1997), 369-372.

bibitem{4}
J. S. Lee, emph{Digital image smoothing and the sigma filter}, In Computer Vision, Graphics, and Image Processing, textbf{24} (1983), 255-269.

bibitem{27}
C. S. Lee, Y. H. Kuo and P. T. Yau, emph{Weighted fuzzy mean filter for image processing}, Fuzzy Sets and Systems, textbf{{89}(2)} (1997), 157-180.

bibitem{32}
R. Lukac, emph{Adaptive color image filtering based on center-weighted vector directional filters}, Multidimensional Systems and Signal Processing, textbf{{15}(2)} (2004), 169-196.

bibitem{23}
R. Lukac, K. N. Plataniotis, B. Smolka and A. N. Venetsanopoulos, emph{cDNA micro-array image processing using fuzzy vector filtering framework}, Fuzzy Sets and Systems, textbf{{152}(1)} (2005), 17-35.

bibitem{16}
M. Nachtegael, S. Schulte, D. Van der Weken, V. De Witte and E. E. Kerre, emph{Gaussian noise reduction in grayscale images}, International Journal of Intelligent Systems Technologies and Applications, textbf{{1}(3,4)} (2006), 211-233.

bibitem{25}
M. Nachtegael, D. Van der Weken, D. Van der Ville, E. Kerre, W. Philips and I. Lemahieu, emph{An overview of fuzzy filters for noise reduction}, Proc. IEEE, International Fuzzy System Conference, (2001), 7-10.

bibitem{6}
A. V. Openheim, A. S. Willsky and S. H. Nawab, emph{Signals and Systems}, Prentice Hall, 1997.

bibitem{5}
I. Pitas and A. N. Venetsanopoulos, emph{Nonlinear digital filters}, Kluwer Academic Publishers, 1990.

bibitem{15}
F. Russo and G. Ramponi, emph{A fuzzy filter for images corrupted by impulse noise}, IEEE Signal Processing Letters, textbf{{3}(6)} (1996), 168-170.

bibitem{18}
F. Sahba, H. R. Tizhoosh and M. M. Salama, emph{A coarse-to-fine approach to prostate boundary segmentation in ultrasound images}, Bio-Medical Engineering On-Line, (Available on-line at: http://www.biomedical-engineering-online.com/content/4/1/58), 2005.

bibitem{29}
F. Sahba, H. R. Tizhoosh and M. M. Salama, emph{A New technique for Adaptive fuzzy image enhancement for edge detection}, International workshop on multidisciplinary image, video and audio retrieval and mining, 2004.

bibitem{3}
A. Savitzky and M. J. E. Golay, emph{Smoothing and differentiation of data by simplified least squares procedures}, Analytical Chemistry, textbf{36} (1964), 1627-1639.

bibitem{36}
S. Schulte, V. De Witte, M. Nachtegael, D. Van der Weken and Etienne E. Kerre, emph{Fuzzy two-step filter for impulse noise reduction from color images}, IEEE Transactions on Image Processing, textbf{{15}(11)} (2006), 3568-3579.

bibitem{31}
H. Shakouri G. and M. B. Menhaj, emph{A single fuzzy rule to smooth the sharpness of mixed data: time & frequency domains analysis}, Fuzzy Sets and Systems, textbf{ {159}(18)} (2008), 2446-2465.

bibitem{24}
H. Shakouri G., M. Rastad and J. Nazarzadeh, emph{A hybrid nonlinear model for maximum simultaneous electric power demand}, IEEE Trans. on Power Systems, textbf{{21}(3)} (2006), 1069-1078.

bibitem{2}
Statistical Center of Iran: www.sci.ir.

bibitem{28}
A. Taguchi, emph{A design method of fuzzy weighted median filters}, IEEE International Conference on Image Processing, (1996), 423-426.

bibitem{13}
H. R. Tizhoosh, emph{Fast fuzzy edge detection}, Proceedings of Fuzzy Information Processing Society, (2002), 239-242.

bibitem{17}
H. R. Tizhoosh, emph{Image thresholding using type II fuzzy sets}, Pattern Recognition, textbf{38} (2005), 2363-2372.

bibitem{7}
D. Van der Ville, M. Nachtegael, D. Van der Weken, Etienne  E. Kerre, W. Philips and I. Lemahieu, emph{Noise reduction by fuzzy image filtering}, IEEE Transactions on Fuzzy Systems, textbf{{11}(4)} (2003), 429-436.

bibitem{34}
J. Wang and H. Chiu, emph{HAF: an adaptive fuzzy filter for restoring highly corrupted images by histogram estimation}, Proc. National Science Counc. ROC(A), textbf{{23}(5)} (1999), 630-643.

%bibitem{1}
%-, Central Bank of Iran: www.cbi.ir
%
%bibitem{2}
%-, Statistical Center of Iran: www.sci.ir
%
%bibitem{14}
%A. A. Alshennawy, and Ayman A. Aly, Edge Detection in Digital Images Using Fuzzy Logic Technique, World Academy of Science, Engineering and Technology, Vol. 51, 2009, pp. 178-187.
%
%bibitem{3}
%A. Savitzky and M. J. E. Golay, Smoothing and Differentiation of Data by Simplified Least Squares Procedures, Analytical Chemistry, Vol. 36, 1964, pp. 1627-1639.
%
%bibitem{28}
%A. Taguchi, A design method of fuzzy weighted median filters, IEEE International Conference on Image Processing, 1996, pp. 423- 426.
%
%bibitem{6}
%A.V. Openheim, A.S.Willsky, S.H. Nawab, Signals and Systems, Prentice Hall, 1997.
%
%bibitem{27}
%C.S. Lee, Y.H. Kuo, P.T. Yau, Weighted fuzzy mean filter for image processing, Fuzzy Sets and Systems, Vol. 89, No. 2, 1997, pp. 157-180.
%
%bibitem{33}
%D. Androutsos, K. N. Plataniotis and A. N. Venetsanopoulos, Novel Vector-Based Approach to Color Image Retrieval Using a Vector Angular-Based Distance Measure, Computer Vision and Image Understanding, Vol. 75, Nos. 1/2, July/August, 1999, pp. 46–58.
%
%bibitem{7}
%D. Van De Ville, Mike Nachtegael, Dietrich Van der Weken, Etienne  E. Kerre, W. Philips, and I. Lemahieu, Noise Reduction by Fuzzy Image Filtering, IEEE Transactions on Fuzzy Systems, Vol. 11, No. 4, August 2003, pp 429-436.
%
%bibitem{8}
%F. Beritelli, S. Casale, A. Cavallaro, Multi-channel Speech/Silence Detector Based on Time Delay Estimation and Fuzzy Classification, IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP’99), Arizona, USA, March 15-19, 1999, pp. 93 - 96.
%
%bibitem{11}
%F. Farbiz, M.B. Menhaj, S.A. Motamedi, M.T. Hagan, A New Fuzzy Logic Control Based Filter for Image Enhancement, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, Vol. 30(1), Feb. 2000, pp. 110-119.
%
%bibitem{15}
%F. Russo, G. Ramponi, A Fuzzy Filter for Images Corrupted by Impulse Noise, IEEE Signal Processing Letters, Vol. 3. No. 6, 1996, pp. 168-170.
%
%bibitem{18}
%F. Sahba, H.R. Tizhoosh, M.M. Salama, A coarse-to-fine approach to prostate boundary segmentation in ultrasound images, Bio-Medical Engineering On-Line, 2005, (Available on-line at: http://www.biomedical-engineering-online.com/content/4/1/58)
%
%bibitem{29}
%F. Sahba, H.R. Tizhoosh, M.M. Salama, A New technique for Adaptive fuzzy image enhancement for edge detection, International workshop on multidisciplinary image, video and audio retrieval and mining, Sherbrooke, Canada, 2004.
%
%bibitem{21}
%G.C. Cardarilli, M. Re, R. Lojacono, L. Carfagnini, High Speed Fuzzy Filter for Non-Linear Channel Equalization, Proceedings of the International Symposium on Fuzzy Logic (ISFL'97), 1997.
%
%bibitem{13}
%H.R.Tizhoosh, Fast fuzzy edge detection, Proceedings of Fuzzy Information Processing Society, 2002, pp. 239-242.
%
%bibitem{17}
%H.R. Tizhoosh, Image Thresholding Using Type II Fuzzy Sets, Pattern Recognition, No. 38, 2005, pp. 2363-2372.
%
%bibitem{12}
%H. Roaki Ishii, Akira Taguchi, Mototaka Sone, The edge detection from images corrupted by mixed noise using fuzzy rules, IX European Signal Processing Conference, September 8 - 11, 1998.
%
%bibitem{31}
%H. Shakouri G. and M.B. Menhaj, A Single Fuzzy Rule to Smooth the Sharpness of Mixed Data: Time & Frequency Domains Analysis, Fuzzy Sets and Systems, No. 159, 2008, pp. 2446-2465.
%
%bibitem{24}
%H. Shakouri G., M. Rastad, J. Nazarzadeh, A Hybrid Nonlinear Model for Maximum Simultaneous Electric Power Demand, IEEE Trans. on Power Systems, Vol.21, No.3, Aug.  2006, pp. 1069-1078.
%
%bibitem{5}
%I. Pitas, A. N. Venetsanopoulos, Nonlinear Digital Filters,Kluwer Academic Publishers, Boston, 1990.
%
%bibitem{4}
%J.S. Lee, Digital image smoothing and the sigma filter, in Computer Vision, Graphics, and Image Processing 24, 1983, pp. 255-269.
%
%bibitem{34}
%Jung-Hua Wang & Hsien-Chu Chiu, HAF: an Adaptive Fuzzy Filter for Restoring Highly Corrupted Images by Histogram Estimation, Proc. National Science Counc. ROC(A), Vol. 23, No. 5, 1999, pp. 630-643.
%
%bibitem{26}
%K. M. T.Chen and L. Chen, Tri-state median filter for image de-noising, IEEE Transactions on Image Processing, 8(12):1834–1838, December 1999.
%
%bibitem{35}
%Kenny Kal Vin Toh, Haidi Ibrahim, Muhammad Nasiruddin Mahyuddin, Salt-and-Pepper Noise Detection and Reduction Using Fuzzy Switching Median Filter, IEEE Transactions on Consumer Electronics, Vol. 54, No. 4, Nov. 2008, pp. 1956-1961.
%
%bibitem{25}
%M. Nachtegael, D. Van der Weken, D. Van de Ville, E. Kerre, W. Philips, I. Lemahieu, An Overview of Fuzzy Filters for Noise Reduction, Proc. IEEE, International Fuzzy System Conference, 2001, pp 7-10.
%
%bibitem{16}
%M. Nachtegael, S. Schulte, D. Van Der Weken, V. De Witte, E. E. Kerre, Gaussian noise reduction in grayscale images,International Journal of Intelligent Systems Technologies and Applications, Vol. 1 ,  Issue 3/4, June 2006, pp. 211-233.
%
%bibitem{19}
%R. Caponetto, L. LoBello, O. Mirabella, Fuzzy Traffic Smoothing: Another Step towards Statistical Real-Time Communication over Ethernet Networks, 1stInternet workshop on Real-Time LANs in the Internet Age (RTILA’02), Tech. Univ. of Vienna, Austria, 2002.
%
%bibitem{9}
%R. Chung Chen, Pao-Ta Yu, Nonlinear Filter Design Based on Fuzzy Inference Rules for Image Processing,IEEE Workshop on Nonlinear Signal and Image Processing (NSIP97), Mackinac Island, Michigan, USA, Sept. 1997.
%
%bibitem{23}
%R. Lukac, K.N. Plataniotis, B. Smolka, A.N. Venetsanopoulos, cDNA micro-array image processing using fuzzy vector filtering framework, Fuzzy Sets and Systems, Vol. 152, No. 1, May 2005, pp. 17-35.
%
%bibitem{32}
%Rastislav Lukac, Adaptive Color Image Filtering Based on Center-Weighted Vector Directional Filters, Multidimensional Systems and Signal Processing, No. 15, 2004, pp. 169–196.
%
%bibitem{22}
%S.F. Chen, Ronald Rosenfeld, A Gaussian Prior for Smoothing Maximum Entropy Models, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213, February 1999. (Available on-line at: http://reports-archive.adm.cs.cmu.edu/anon/1999/CMU-CS-99-108.pdf)
%
%bibitem{36}
%Stefan Schulte, Valérie De Witte, Mike Nachtegael, Dietrich Van der Weken, and Etienne E. Kerre, Fuzzy Two-Step Filter for Impulse Noise Reduction From Color Images, IEEE Transactions on Image Processing, Vol. 15, No. 11, Nov 2006, pp. 3568-3579.
%
%bibitem{10}
%Y. H. Kuo, C. S. Lee, C. C. Liu, A New Fuzzy Edge Detection Method for Image Enhancement, The Sixth IEEE International Conference on Fuzzy Systems, Barcelona, Spain, July 1-5, 1997, Vol.1, pp. 369-372.
%
%bibitem{30}
%Y. H. Kuo, Chang-Shing Lee, Chao-Lieh Chen, High-stability AWFM filter for signal restoration and its hardware design, Fuzzy Sets and Systems, No. 114, 2000, pp. 185-202.
%
%bibitem{20}
%Y. R. Hwang, M. Tomizuka, Fuzzy Smoothing Algorithms for Variable Structure Systems, IEEE Transactions on Fuzzy Systems, Vol. 2, No. 4, Nov. 1994, pp. 277-284.