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Impulsive noise detection by double noise detector and removal using adaptive neural-fuzzy inference system
Int. J. Electron. Commun. (AEÜ) 65 (2011) 429–434
Contents lists available at ScienceDirect
International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.de/aeue
Impulsive noise detection by double noise detector and removal using adaptive neural-fuzzy inference system Xuan Wang a,∗ , Xue-qing Zhao a , Fang-xia Guo a , Jian-feng Ma b a b
School of Physics and Information Technology, Shaanxi Normal University, Xi’An 710062, China The Key Laboratory of the Ministry of Education for Computer Networks and Information Security Xidian University, Xi’An 710071, China
a r t i c l e
i n f o
Article history: Received 21 May 2009 Accepted 5 June 2010 Keywords: Double noise detector Nonlinear filtering ANFIS Impulsive noise
a b s t r a c t A novel impulsive noise elimination filter combining a double noise detector with an adaptive neuralfuzzy inference system (ANFIS) is proposed. The proposed filter has a two-phase scheme for removing impulse noise: detecting noisy pixels and removing noise. In the detecting noisy pixels phase, noise candidates identified with the noise detection algorithm of the adaptive median filter are judged again by local fuzzy membership function to improve the accurate rate of noise detection. In the removing noise phase, the corrupted pixels are restored using adaptive neural-fuzzy inference system based on their neighbor uncorrupted pixels. Experimental results show that the proposed filter yields a better performance than the recent filters in terms of details preservation and noise suppression, and our scheme can remove impulsive noise with noise level as high as 90%. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction Efficient removal of noise from image date is of key importance because the performances of subsequent image processing tasks are strictly dependent on the results of the noise removal operation [1]. Two common types of image noise are the impulse noise and the Gaussian noise. There are many works on the restoration of images corrupted by the two types of noise [2–4]. Median filter is a nonlinear filtering technique widely used for removal of impulse noise [5]; it yields a reasonable noise removal performance with the cost of introducing undesirable blurring effects into image details [6,7]. In order to alleviate the undesirable blurring effects, switching median filters were introduced [8], e.g., the adaptive median filter [9], the multistate median filter [10], the median filter based on homogeneity information [11,12], and the fuzzy median filter [13]. Since these switching filters only replace the identified possible noisy pixels with the median values of their filtering windows, and leaving all other pixels unchanged, these filters present a better performance than that of the conventional median filters. Their main drawback is that the noisy pixels are replaced with the median values of their filtering windows without taking into account local features such as the presence of edges. Hence, the edges and details of images are not recovered satisfactorily.
∗ Corresponding author. Tel.: +86 29 85303867; fax: +86 29 85303867. E-mail address: [email protected] (X. Wang). 1434-8411/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2010.06.004
Recently, some variations of switching median filters were proposed to achieve good performances by improving impulse noise detection accuracy [14,15] or restoring the noisy pixels based on their local features such as the presence of edges and details [16–18]. Nonlinear approaches based on fuzzy systems and artificial neural networks [19,20] have emerged as attractive alternatives to classical noise suppression techniques due to their ability to handle information in the form of rules that emulate human thinking and decision making. There are two problems to be considered in constructing a switching median filter, one is how to detect noisy pixels accurately, and the other is how to restore the noisy pixels efficiently. Poor noise detection leads to some uncorrupted pixels are treated as the noisy pixels or some noisy pixels are not restored, and thus the performance of the constructed switching median filter is depressed. To detect noisy pixels accurately, and restore details of images satisfactorily, a new switching median filter combining a double noise detector with an adaptive neural-fuzzy inference system is proposed in this paper. Experimental results show that it yields a better performance than the recent filters including Progressive switching median filter (PSM) [16], two-state recursive signal-dependent rank order mean filter (SDROMR) [17], recursive adaptive-center weighted median filter (ACWMR) [19], Yüksel’s fuzzy filter [20], Russo’s if-then-else fuzzy reasoning filter (RUSSO) [3], etc. The remaining of the paper is organized as follows. An adaptive neural-fuzzy inference system is given in Section 2. In Section 3, we introduce our proposed filter. The experimental results and conclusions are presented in Sections 4 and 5, respectively.
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X. Wang et al. / Int. J. Electron. Commun. (AEÜ) 65 (2011) 429–434
Every node in this layer is a square node with a linear function [24] ij as ij = ω ¯ ij fij = ω ¯ ij (pij x + qij y + rij )
(5)
Layer 5: The single node in this layer is labeled with , which computes the overall output, g, as the summation of all incoming signals, i.e., g=
2 2
i,j
(6)
i=1 j=1
3. The proposed method Fig. 1. The ANFIS structure, which has been used in this paper.
2. The adaptive neural-fuzzy inference system (ANFIS) ANFIS is a fuzzy inference system implemented in the framework of adaptive networks [23–25], which serves as a basis for constructing a set of fuzzy if-then-else rules with appropriate membership functions to generate the stipulated input–output pairs. The ANFIS structure used in this method is illustrated in Fig. 1, which possesses two inputs (x, y) and one output (g). Here, (x, y) and (g) denote the spatial positions and the intensity gray values of the pixels, respectively. Each input of the ANFIS structure has two different triangular membership functions and the rule base contains a total of 4 rules, which are detailed as follows:
In the proposed method, a novel double noise detector was proposed to improve the accurate rate of noise detection; the corrupted pixels identified by the double noise detector were restored using an adaptive neural-fuzzy inference system based on their neighbor uncorrupted pixels. 3.1. The double noise detector The double noise detector consists of two steps. In the first step, noise candidates were identified with the noise detection algorithm ω of the adaptive median filter, which was detailed as follows: Let Di,j indicate a block of size ω × ω pixels, c(i, j) be the center pixel of the block, and ωmax × ωmax be the maximum block. The center pixel c(i, j) was handled by following steps: 1. Set the initial parameter value ω = 3. max,ω 2. Compute the minimum gray value Di,j , median gray value
Rule 1: if x is A1 and y is B1 , then f11 = p11 x + q11 y + r11 Rule 2: if x is A1 and y is B2 , then f12 = p12 x + q12 y + r12 Rule 3: if x is A2 and y is B1 , then f21 = p21 x + q21 y + r21 Rule 4: if x is A2 and y is B2 , then f22 = p22 x + q22 y + r22
med,ω min,ω Di,j and the maximum gray value Di,j of pixels in the block ω Di,j
where p, q and r denote the consequent parameters [24] .A combination of least-squares and back-propagation gradient descent method has been used at the training phase of the fuzzy structure. In order to compute the parameters of the membership functions, which were used to model given set of inputs (x, y) and singleoutput data (g), the details are given as follows. Let the membership functions of fuzzy sets Ai and Bj , be Ai and Bj , respectively, where i = 1,2, and j = 1,2. Then, the five layers of the ANFIS structure are defined as follows.
min,ω med,ω max,ω 3. If Di,j < Di,j < Di,j , then go to step 5, else set to = to + 2. 4. If ω ≤ ωmax , then go to step 2, else the center pixel c(i, j) is judged as a corrupted pixel. min,ω max,ω < c(i, j) < Di,j , then the center pixel c(i, j) is an uncor5. If Di,j rupted pixel, else c(i, j) is judged as a corrupted pixel.
According to the theory of the adaptive median filter, the extreme values in the detection region will be detected as the corrupted pixels, some of them may be uncorrupted pixels, in order to improve accurate rate of noise detection, the first detection results
Layer 1:
Ai (x) = max min
x−a
Bj (y) = max
i
bi − ai
min
,
y − aj
ci − x ci − bi
cj − y
, bj − aj cj − bj
,0
(1)
,0
(2)
where Ai (x) and Bj (y) were chosen as triangular membership functions with the parameters of a, b and c [24]. Layer 2: ωij = Ai (x)Bj (y)
(3)
where ωij denotes the logical operation and which has been used as product for all the rules [24]. Layer 3: ω ¯ ij =
ωij ω11 + ω12 + ω21 + ω22
where ω ¯ ij is the normalized value of ωij . Layer 4:
(4)
Fig. 2. MSE curves of all algorithms at different noise densities for Lena image.
X. Wang et al. / Int. J. Electron. Commun. (AEÜ) 65 (2011) 429–434
431
Fig. 4. PSNR curves of all algorithms at different noise densities for Lena image.
Fig. 3. MSE curves of all algorithms at different noise densities for peppers image.
were judged again by local fuzzy membership function, which was obtained as follows: Let ω ¯ ijω be the mean of the uncorrupted pixels in the block of size ω × ω with the center pixel c(i, j), we have ω ¯ ijω =
1 Nijω
ω
c(i + k, j + k)
(7)
k=−ω
where Nijω indicates the total number of uncorrupted pixels determined in the first step in the block, c(i + k, j + k) denotes the uncorrupted pixels. The square deviation of the uncorrupted pixels is defined as 2 =
1 Nijω
ω
[c(i + k, j + k) − ¯ω ] ij
2
(8)
k=−ω
The fuzzy membership degree (c(i, j)) of the center pixel c(i, j) in the block is defined as ) ) (c(i, j)) = exp(−(c(i, j) − ¯ω ij 2
(9)
The noise candidates were judged again based on the local fuzzy membership function using following rule:
fussy model (i.e., ANFIS) yields good interpolation and extrapolation characteristics in surface fitting [25,26], it can be used as a surface interpolator to restore the corrupted pixels based on their neighbor uncorrupted pixels. The restoration of the corrupted pixels is detailed as follows: (1) The corrupted pixels could be found based on the results of the double noise detector as explained in Section 3.1. (2) For each corrupted pixel C(xc , yc ), find an uncorrupted pixel P which is at the closest distance to the corrupted pixel C(xc , yc ). (3) Find a total of k uncorrupted pixels that are at the closest distance to the uncorrupted pixel P. Then determine the (x, y, g)k values of these pixels where (x, y)k and gk denote the coordinates and the gray value of the related uncorrupted pixels, respectively. (4) Continually train the ANFIS structures until one of the criteria given below has been realized. The trained-ANFIS is denoted as fp (x, y): (a) Train the ANFIS structure for a maximum of 20 epochs (This criterion has been used to limit the runtime required to train the ANFIS structure. Therefore, the computational complexity of the proposed filter has been limited). (b) Stop training if the error value computed by using the output value of ANFIS is less than 0.001. (5) Restore the gray value gc of the corrupted pixel by using the equation of gc = f(xc , yc ).
c(i, j) is a corrupted pixel, (c(i, j)) ≥ T c(i, j) is not a corrupted pixel, (c(i, j)) < T The noise candidates are very likely corrupted when the block is highly corrupted by the noise, the threshold value (T) may be larger, on the contrary, when the block is lowly corrupted by the noise and the threshold value (T) may be smaller, therefore, the threshold value (T) is defined as: ω T (1 − ri,j )·
ω where ri,j =
(10) ω Ni,j
ω×ω
.
3.2. The restoration of the corrupted pixels In real images, the intensity level of a pixel is more related with the neighbor pixels than distant ones [2]. Therefore, surface fitting is a useful tool to restore the corrupted pixels. Since the TSK
Fig. 5. PSNR curves of all algorithms at different noise densities for peppers image.
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Fig. 6. ˇ curves of all algorithms at different noise densities for Lena image.
4. Experiments The proposed method has been implemented using Matlab 7.5. Two images including Lena and peppers as shown in Figs. 8(a) and 9(a) were considered in the computer implementation, which are 256 × 256 pixels sized and 8 bit per pixel images.
Fig. 7. ˇ curves of all algorithms at different noise densities for peppers image.
The RUSSO [3], IMF [16], PSM [16], SDROM [17], SDROMR [17], IRF [18], ACWM [19], ACWMR [19], YÜKSEL [20], CWM [21], and SMF [22], have been simulated as well for performance comparison. The major improvement achieved by the proposed filter has been demonstrated with the extensive simulations of the mentioned test images corrupted with impulse noise at different densities.
Fig. 8. Images of Lena filtered with the proposed approach: (a) Lena with 40% of noise, (b) filtered image of (a), (c) Lena with 60% of noise, (d) filtered image of (c), (e) Lena with 80% of noise, (g) filtered image of (e), (f) original Lena image.
X. Wang et al. / Int. J. Electron. Commun. (AEÜ) 65 (2011) 429–434
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Fig. 9. Images of peppers filtered with the proposed approach: (a) peppers with 40% of noise, (b) filtered image of (a), (c) peppers with 60% of noise, (d) filtered image of (c), (e) peppers with 80% of noise, (g) filtered image of (e), (f) original peppers image.
Restoration performances are objectively evaluated by using the mean-squared-error (MSE) and peak signal-to-noise rate (PSNR), which are defined as follows:
MSE =
1 Ni,j
MN
M N
(O(x, y) − R(x, y))2
255 √ MSE
¯ I − I¯ ) (O − O, ¯ O − O) ¯ · (I − I¯ , I − I¯ ) (O − O,
−1 8 −1
−1 −1 −1
(14)
¯ and I¯ are the mean value of O and I, respectively. O Function is defined as
(12)
(I, O) =
N M
Ii,j Oi,j
(17)
i=1 j=1
where O(x, y) and R(x, y) denote the gray values of the original noisefree image pixels and the restored image pixels, respectively. On the other hand, it is of great importance to evaluate the detail preservation performances of these methods, another measure ˇ for detail preservation is also considered, which is defined as [27].
ˇ=
−1 −1 −1
(11)
x=1 y=1
PSNR = 20 log
Laplacian operator given by
(13)
where O and I¯ are the high-pass filtered images of O(x, y) and R(x, y) obtained with a 3 × 3 pixel standard approximation of the
For Lena and peppers images, all three quantities including MSE, PSNR and ˇ at different noise densities are calculated and summarized in Figs. 2–7. Seen from these MSE, PSNR and P curves, the performance of the proposed algorithm is excellent at all noise situations in noise suppression and detail preservation, especially at high noise density values. For visual comparison, the images obtained for Lena and pepper images with 40%, 60%, and 80% noise density are shown in Figs. 8 and 9. From these figures, we note that our filter gives reasonably good visual image, which are clear, especially on hair and eyes of the girl. It is also shown that the noise suppression and detail preservation are satisfactorily achieved by using the proposed filter even if the noise density is high (i.e., 90%).
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5. Conclusion In this paper, a novel impulsive noise filter is proposed, which is composed of a double noise detector and restoration of the corrupted pixels using ANFIS. During the noise detection, noise pixels are identified with the noise detection algorithm of the adaptive median filter, and judged again by local fuzzy membership function. The intensity value of the corrupted pixel is updated with the value estimated by the ANFIS. Experimental results show that the new approach is effective in all test items. Acknowledgements This work was supported by the Natural Science Foundation of Shaanxi province, China under Grant No. 2009JM8003; the National Natural Science Foundation of China under Grant No. 90204012; the National High-Tech Research and Development Plan of China under Grant No. 2002AA143021. References [1] Albovik R. Handbook of image and video processing. Academic Press; 2000. [2] Bovik AC, Huang T, Munson DC. A generalization of median filtering using linear combination of order statistics. IEEE Trans Acoustic Speech Signal Process ASSP 1983;31:1342–50. [3] Russo R, Ramponi G. A fuzzy filter for images corrupted by impulse noise. IEEE Signal Process Lett 1996;6:168–70. [4] Xu Y, Lae EM. Restoration of images contaminated by mixed Gaussian and impulse noise using a recursive minimum–maximum method. Vision Image Signal Proc IEEE Proc 1998;145:264–70. [5] Huang TSG, Yang J, Tang GY. Fast two-dimensional median filtering algorithm. IEEE Trans Acoustics Speech Signal Process 1979;1:13–8. [6] Umbaugh SE. Computer vision and image processing. Prentice-Hall International Inc.; 1998. [7] Nodes TA, Gallagher NC. The output distribution of median type filters. IEEE Trans Commun 1984. COM-32. [8] Sun T, Neuvo Y. Detail-preserving median based filters in image processing. Pattern Recogn Lett 1994;15:341–7. [9] Hwang H, Haddad RA. Adaptive median filters: new algorithms and results. IEEE Trans Image Process 1995;4:499–502. [10] Chen T, Wu HR. Space variant median filters for the restoration of impulse noise corrupted images. IEEE Trans Circuits Syst II: Analog Digital Signal Process 2001;48:784–9. [11] Eng HL, Ma KK. Noise adaptive soft-switching median filter. IEEE Trans Image Process 2001;10:242–51. [12] Pok G, Liu J, Nair A. Selective removal of impulse noise based on homogeneity level information. IEEE Trans Image Process 2003;12:85–92. [13] Schulte S, Nachtegael M, Witte VD, Kerre EE. A fuzzy impulse noise detection and reduction method. IEEE Trans Image Process 2006;15:1153–62. [14] Zhang S, Karim M. A new impulse detector for switching median filters. IEEE Signal Process Lett 2002;9:360–3. [15] Huang Z. A median filter based on judging impulse noise by statistic and adaptive threshold. Congr Image Signal Process 2008:207–10. [16] Wang Z, Zhang D. Progressive switching median filter for the removal of impulse noise from highly corrupted images. IEEE Trans Circuits Systems-0: Analog Digital Signal Process 1999;46:78–80. [17] Abreu E, Lighstone M, Mitra SK, Arakawa K. A new efficient approach for the removal of impulse noise from highly corrupted images. IEEE Trans Image Process 1996;5:1012–25. [18] Chen T, Wu HR. A new class of median based impulse rejecting filters. In: IEEE International Conference on Image Processing. 2000. p. 916–9. [19] Chen T, Wu HR. Adaptive impulse detection using center weighted median filters. IEEE Signal Process Lett 2001;8:1–3. [20] Yuksel M, Bastrk A. Efficient removal of impulse noise from highly corrupted digital images by a simple neuro-fuzzy operator. AEU Int J Electron Commun 2003;57:214–9.
[21] Muneyasu M, Nishi N, Hinamoto T. A new adaptive center weighted median filter using counter propagation networks. IEEE Trans Image Process 2000;337:631–9. [22] Tukey JW. Nonlinear methods for smoothing data. In: Proc Conf Rec EASCON. 1974. p. 673. [23] Besdok E, Civicioglu P, Alci M. Using an adaptive neuro-fuzzy inference system based interpolant for impulsive noise suppression from highly distorted images. Fuzzy Sets Syst 2005;150:525–43. [24] Jang J. Anfis: adaptive-network-based fuzzy inference system. IEEE Trans Systems Man Cybern 1993;23:665–85. [25] MathWorks: Matlab, fuzzy logic toolbox, user’s guide. New York: The Math Works Inc.; 2002. [26] Yen J, Wang L. Improving the interpretability of TSK fuzzy models by combining global learning and local learning. IEEE Trans Fuzzy Syst 1998;6:530–7. [27] Chen SS, Yang X, Cao G. Impulse noise suppression with an augmentation of ordered difference noise detector and an adaptive variational method. Pattern Recogn Lett 2009;30:460–7. Xuan Wang was born in 1966. He received the B.S. and M.S. degrees in Electrical Engineering from Shaanxi Normal University Xi’An, China in 1983 and 1987, and received Ph.D. degree in the key laboratory of the ministry of education for Computer Networks and Information Security. He is currently professor and head of School of Physics and Information Technology at Shaanxi Normal University. His research interests include image processing and pattern recognition.
Xue-Qin Zhao was born in 1985. She is currently pursuing the M.S. degree at Shaanxi Normal University. Her research interests include image processing.
Fang-Xia Guo was born in 1965. She received the M.S. degree in Electrical Engineering from Shaanxi Normal University. She is currently vice professor of Shaanxi Normal University and pursuing the Ph.D. degree at Shaanxi Normal University. Her research interests include image processing and pattern recognition.
Jian-Feng Ma was born in 1961. He received the Ph.D. degree in communication and electronic systems from Xidian University. He is currently a professor and the Dean of School of Computer in Xidian University. He is also a member of IEEE. His research interests include image processing and information and network security.
Contents lists available at ScienceDirect
International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.de/aeue
Impulsive noise detection by double noise detector and removal using adaptive neural-fuzzy inference system Xuan Wang a,∗ , Xue-qing Zhao a , Fang-xia Guo a , Jian-feng Ma b a b
School of Physics and Information Technology, Shaanxi Normal University, Xi’An 710062, China The Key Laboratory of the Ministry of Education for Computer Networks and Information Security Xidian University, Xi’An 710071, China
a r t i c l e
i n f o
Article history: Received 21 May 2009 Accepted 5 June 2010 Keywords: Double noise detector Nonlinear filtering ANFIS Impulsive noise
a b s t r a c t A novel impulsive noise elimination filter combining a double noise detector with an adaptive neuralfuzzy inference system (ANFIS) is proposed. The proposed filter has a two-phase scheme for removing impulse noise: detecting noisy pixels and removing noise. In the detecting noisy pixels phase, noise candidates identified with the noise detection algorithm of the adaptive median filter are judged again by local fuzzy membership function to improve the accurate rate of noise detection. In the removing noise phase, the corrupted pixels are restored using adaptive neural-fuzzy inference system based on their neighbor uncorrupted pixels. Experimental results show that the proposed filter yields a better performance than the recent filters in terms of details preservation and noise suppression, and our scheme can remove impulsive noise with noise level as high as 90%. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction Efficient removal of noise from image date is of key importance because the performances of subsequent image processing tasks are strictly dependent on the results of the noise removal operation [1]. Two common types of image noise are the impulse noise and the Gaussian noise. There are many works on the restoration of images corrupted by the two types of noise [2–4]. Median filter is a nonlinear filtering technique widely used for removal of impulse noise [5]; it yields a reasonable noise removal performance with the cost of introducing undesirable blurring effects into image details [6,7]. In order to alleviate the undesirable blurring effects, switching median filters were introduced [8], e.g., the adaptive median filter [9], the multistate median filter [10], the median filter based on homogeneity information [11,12], and the fuzzy median filter [13]. Since these switching filters only replace the identified possible noisy pixels with the median values of their filtering windows, and leaving all other pixels unchanged, these filters present a better performance than that of the conventional median filters. Their main drawback is that the noisy pixels are replaced with the median values of their filtering windows without taking into account local features such as the presence of edges. Hence, the edges and details of images are not recovered satisfactorily.
∗ Corresponding author. Tel.: +86 29 85303867; fax: +86 29 85303867. E-mail address: [email protected] (X. Wang). 1434-8411/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2010.06.004
Recently, some variations of switching median filters were proposed to achieve good performances by improving impulse noise detection accuracy [14,15] or restoring the noisy pixels based on their local features such as the presence of edges and details [16–18]. Nonlinear approaches based on fuzzy systems and artificial neural networks [19,20] have emerged as attractive alternatives to classical noise suppression techniques due to their ability to handle information in the form of rules that emulate human thinking and decision making. There are two problems to be considered in constructing a switching median filter, one is how to detect noisy pixels accurately, and the other is how to restore the noisy pixels efficiently. Poor noise detection leads to some uncorrupted pixels are treated as the noisy pixels or some noisy pixels are not restored, and thus the performance of the constructed switching median filter is depressed. To detect noisy pixels accurately, and restore details of images satisfactorily, a new switching median filter combining a double noise detector with an adaptive neural-fuzzy inference system is proposed in this paper. Experimental results show that it yields a better performance than the recent filters including Progressive switching median filter (PSM) [16], two-state recursive signal-dependent rank order mean filter (SDROMR) [17], recursive adaptive-center weighted median filter (ACWMR) [19], Yüksel’s fuzzy filter [20], Russo’s if-then-else fuzzy reasoning filter (RUSSO) [3], etc. The remaining of the paper is organized as follows. An adaptive neural-fuzzy inference system is given in Section 2. In Section 3, we introduce our proposed filter. The experimental results and conclusions are presented in Sections 4 and 5, respectively.
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X. Wang et al. / Int. J. Electron. Commun. (AEÜ) 65 (2011) 429–434
Every node in this layer is a square node with a linear function [24] ij as ij = ω ¯ ij fij = ω ¯ ij (pij x + qij y + rij )
(5)
Layer 5: The single node in this layer is labeled with , which computes the overall output, g, as the summation of all incoming signals, i.e., g=
2 2
i,j
(6)
i=1 j=1
3. The proposed method Fig. 1. The ANFIS structure, which has been used in this paper.
2. The adaptive neural-fuzzy inference system (ANFIS) ANFIS is a fuzzy inference system implemented in the framework of adaptive networks [23–25], which serves as a basis for constructing a set of fuzzy if-then-else rules with appropriate membership functions to generate the stipulated input–output pairs. The ANFIS structure used in this method is illustrated in Fig. 1, which possesses two inputs (x, y) and one output (g). Here, (x, y) and (g) denote the spatial positions and the intensity gray values of the pixels, respectively. Each input of the ANFIS structure has two different triangular membership functions and the rule base contains a total of 4 rules, which are detailed as follows:
In the proposed method, a novel double noise detector was proposed to improve the accurate rate of noise detection; the corrupted pixels identified by the double noise detector were restored using an adaptive neural-fuzzy inference system based on their neighbor uncorrupted pixels. 3.1. The double noise detector The double noise detector consists of two steps. In the first step, noise candidates were identified with the noise detection algorithm ω of the adaptive median filter, which was detailed as follows: Let Di,j indicate a block of size ω × ω pixels, c(i, j) be the center pixel of the block, and ωmax × ωmax be the maximum block. The center pixel c(i, j) was handled by following steps: 1. Set the initial parameter value ω = 3. max,ω 2. Compute the minimum gray value Di,j , median gray value
Rule 1: if x is A1 and y is B1 , then f11 = p11 x + q11 y + r11 Rule 2: if x is A1 and y is B2 , then f12 = p12 x + q12 y + r12 Rule 3: if x is A2 and y is B1 , then f21 = p21 x + q21 y + r21 Rule 4: if x is A2 and y is B2 , then f22 = p22 x + q22 y + r22
med,ω min,ω Di,j and the maximum gray value Di,j of pixels in the block ω Di,j
where p, q and r denote the consequent parameters [24] .A combination of least-squares and back-propagation gradient descent method has been used at the training phase of the fuzzy structure. In order to compute the parameters of the membership functions, which were used to model given set of inputs (x, y) and singleoutput data (g), the details are given as follows. Let the membership functions of fuzzy sets Ai and Bj , be Ai and Bj , respectively, where i = 1,2, and j = 1,2. Then, the five layers of the ANFIS structure are defined as follows.
min,ω med,ω max,ω 3. If Di,j < Di,j < Di,j , then go to step 5, else set to = to + 2. 4. If ω ≤ ωmax , then go to step 2, else the center pixel c(i, j) is judged as a corrupted pixel. min,ω max,ω < c(i, j) < Di,j , then the center pixel c(i, j) is an uncor5. If Di,j rupted pixel, else c(i, j) is judged as a corrupted pixel.
According to the theory of the adaptive median filter, the extreme values in the detection region will be detected as the corrupted pixels, some of them may be uncorrupted pixels, in order to improve accurate rate of noise detection, the first detection results
Layer 1:
Ai (x) = max min
x−a
Bj (y) = max
i
bi − ai
min
,
y − aj
ci − x ci − bi
cj − y
, bj − aj cj − bj
,0
(1)
,0
(2)
where Ai (x) and Bj (y) were chosen as triangular membership functions with the parameters of a, b and c [24]. Layer 2: ωij = Ai (x)Bj (y)
(3)
where ωij denotes the logical operation and which has been used as product for all the rules [24]. Layer 3: ω ¯ ij =
ωij ω11 + ω12 + ω21 + ω22
where ω ¯ ij is the normalized value of ωij . Layer 4:
(4)
Fig. 2. MSE curves of all algorithms at different noise densities for Lena image.
X. Wang et al. / Int. J. Electron. Commun. (AEÜ) 65 (2011) 429–434
431
Fig. 4. PSNR curves of all algorithms at different noise densities for Lena image.
Fig. 3. MSE curves of all algorithms at different noise densities for peppers image.
were judged again by local fuzzy membership function, which was obtained as follows: Let ω ¯ ijω be the mean of the uncorrupted pixels in the block of size ω × ω with the center pixel c(i, j), we have ω ¯ ijω =
1 Nijω
ω
c(i + k, j + k)
(7)
k=−ω
where Nijω indicates the total number of uncorrupted pixels determined in the first step in the block, c(i + k, j + k) denotes the uncorrupted pixels. The square deviation of the uncorrupted pixels is defined as 2 =
1 Nijω
ω
[c(i + k, j + k) − ¯ω ] ij
2
(8)
k=−ω
The fuzzy membership degree (c(i, j)) of the center pixel c(i, j) in the block is defined as ) ) (c(i, j)) = exp(−(c(i, j) − ¯ω ij 2
(9)
The noise candidates were judged again based on the local fuzzy membership function using following rule:
fussy model (i.e., ANFIS) yields good interpolation and extrapolation characteristics in surface fitting [25,26], it can be used as a surface interpolator to restore the corrupted pixels based on their neighbor uncorrupted pixels. The restoration of the corrupted pixels is detailed as follows: (1) The corrupted pixels could be found based on the results of the double noise detector as explained in Section 3.1. (2) For each corrupted pixel C(xc , yc ), find an uncorrupted pixel P which is at the closest distance to the corrupted pixel C(xc , yc ). (3) Find a total of k uncorrupted pixels that are at the closest distance to the uncorrupted pixel P. Then determine the (x, y, g)k values of these pixels where (x, y)k and gk denote the coordinates and the gray value of the related uncorrupted pixels, respectively. (4) Continually train the ANFIS structures until one of the criteria given below has been realized. The trained-ANFIS is denoted as fp (x, y): (a) Train the ANFIS structure for a maximum of 20 epochs (This criterion has been used to limit the runtime required to train the ANFIS structure. Therefore, the computational complexity of the proposed filter has been limited). (b) Stop training if the error value computed by using the output value of ANFIS is less than 0.001. (5) Restore the gray value gc of the corrupted pixel by using the equation of gc = f(xc , yc ).
c(i, j) is a corrupted pixel, (c(i, j)) ≥ T c(i, j) is not a corrupted pixel, (c(i, j)) < T The noise candidates are very likely corrupted when the block is highly corrupted by the noise, the threshold value (T) may be larger, on the contrary, when the block is lowly corrupted by the noise and the threshold value (T) may be smaller, therefore, the threshold value (T) is defined as: ω T (1 − ri,j )·
ω where ri,j =
(10) ω Ni,j
ω×ω
.
3.2. The restoration of the corrupted pixels In real images, the intensity level of a pixel is more related with the neighbor pixels than distant ones [2]. Therefore, surface fitting is a useful tool to restore the corrupted pixels. Since the TSK
Fig. 5. PSNR curves of all algorithms at different noise densities for peppers image.
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Fig. 6. ˇ curves of all algorithms at different noise densities for Lena image.
4. Experiments The proposed method has been implemented using Matlab 7.5. Two images including Lena and peppers as shown in Figs. 8(a) and 9(a) were considered in the computer implementation, which are 256 × 256 pixels sized and 8 bit per pixel images.
Fig. 7. ˇ curves of all algorithms at different noise densities for peppers image.
The RUSSO [3], IMF [16], PSM [16], SDROM [17], SDROMR [17], IRF [18], ACWM [19], ACWMR [19], YÜKSEL [20], CWM [21], and SMF [22], have been simulated as well for performance comparison. The major improvement achieved by the proposed filter has been demonstrated with the extensive simulations of the mentioned test images corrupted with impulse noise at different densities.
Fig. 8. Images of Lena filtered with the proposed approach: (a) Lena with 40% of noise, (b) filtered image of (a), (c) Lena with 60% of noise, (d) filtered image of (c), (e) Lena with 80% of noise, (g) filtered image of (e), (f) original Lena image.
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Fig. 9. Images of peppers filtered with the proposed approach: (a) peppers with 40% of noise, (b) filtered image of (a), (c) peppers with 60% of noise, (d) filtered image of (c), (e) peppers with 80% of noise, (g) filtered image of (e), (f) original peppers image.
Restoration performances are objectively evaluated by using the mean-squared-error (MSE) and peak signal-to-noise rate (PSNR), which are defined as follows:
MSE =
1 Ni,j
MN
M N
(O(x, y) − R(x, y))2
255 √ MSE
¯ I − I¯ ) (O − O, ¯ O − O) ¯ · (I − I¯ , I − I¯ ) (O − O,
−1 8 −1
−1 −1 −1
(14)
¯ and I¯ are the mean value of O and I, respectively. O Function is defined as
(12)
(I, O) =
N M
Ii,j Oi,j
(17)
i=1 j=1
where O(x, y) and R(x, y) denote the gray values of the original noisefree image pixels and the restored image pixels, respectively. On the other hand, it is of great importance to evaluate the detail preservation performances of these methods, another measure ˇ for detail preservation is also considered, which is defined as [27].
ˇ=
−1 −1 −1
(11)
x=1 y=1
PSNR = 20 log
Laplacian operator given by
(13)
where O and I¯ are the high-pass filtered images of O(x, y) and R(x, y) obtained with a 3 × 3 pixel standard approximation of the
For Lena and peppers images, all three quantities including MSE, PSNR and ˇ at different noise densities are calculated and summarized in Figs. 2–7. Seen from these MSE, PSNR and P curves, the performance of the proposed algorithm is excellent at all noise situations in noise suppression and detail preservation, especially at high noise density values. For visual comparison, the images obtained for Lena and pepper images with 40%, 60%, and 80% noise density are shown in Figs. 8 and 9. From these figures, we note that our filter gives reasonably good visual image, which are clear, especially on hair and eyes of the girl. It is also shown that the noise suppression and detail preservation are satisfactorily achieved by using the proposed filter even if the noise density is high (i.e., 90%).
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5. Conclusion In this paper, a novel impulsive noise filter is proposed, which is composed of a double noise detector and restoration of the corrupted pixels using ANFIS. During the noise detection, noise pixels are identified with the noise detection algorithm of the adaptive median filter, and judged again by local fuzzy membership function. The intensity value of the corrupted pixel is updated with the value estimated by the ANFIS. Experimental results show that the new approach is effective in all test items. Acknowledgements This work was supported by the Natural Science Foundation of Shaanxi province, China under Grant No. 2009JM8003; the National Natural Science Foundation of China under Grant No. 90204012; the National High-Tech Research and Development Plan of China under Grant No. 2002AA143021. References [1] Albovik R. Handbook of image and video processing. Academic Press; 2000. [2] Bovik AC, Huang T, Munson DC. A generalization of median filtering using linear combination of order statistics. IEEE Trans Acoustic Speech Signal Process ASSP 1983;31:1342–50. [3] Russo R, Ramponi G. A fuzzy filter for images corrupted by impulse noise. IEEE Signal Process Lett 1996;6:168–70. [4] Xu Y, Lae EM. Restoration of images contaminated by mixed Gaussian and impulse noise using a recursive minimum–maximum method. Vision Image Signal Proc IEEE Proc 1998;145:264–70. [5] Huang TSG, Yang J, Tang GY. Fast two-dimensional median filtering algorithm. IEEE Trans Acoustics Speech Signal Process 1979;1:13–8. [6] Umbaugh SE. Computer vision and image processing. Prentice-Hall International Inc.; 1998. [7] Nodes TA, Gallagher NC. The output distribution of median type filters. IEEE Trans Commun 1984. COM-32. [8] Sun T, Neuvo Y. Detail-preserving median based filters in image processing. Pattern Recogn Lett 1994;15:341–7. [9] Hwang H, Haddad RA. Adaptive median filters: new algorithms and results. IEEE Trans Image Process 1995;4:499–502. [10] Chen T, Wu HR. Space variant median filters for the restoration of impulse noise corrupted images. IEEE Trans Circuits Syst II: Analog Digital Signal Process 2001;48:784–9. [11] Eng HL, Ma KK. Noise adaptive soft-switching median filter. IEEE Trans Image Process 2001;10:242–51. [12] Pok G, Liu J, Nair A. Selective removal of impulse noise based on homogeneity level information. IEEE Trans Image Process 2003;12:85–92. [13] Schulte S, Nachtegael M, Witte VD, Kerre EE. A fuzzy impulse noise detection and reduction method. IEEE Trans Image Process 2006;15:1153–62. [14] Zhang S, Karim M. A new impulse detector for switching median filters. IEEE Signal Process Lett 2002;9:360–3. [15] Huang Z. A median filter based on judging impulse noise by statistic and adaptive threshold. Congr Image Signal Process 2008:207–10. [16] Wang Z, Zhang D. Progressive switching median filter for the removal of impulse noise from highly corrupted images. IEEE Trans Circuits Systems-0: Analog Digital Signal Process 1999;46:78–80. [17] Abreu E, Lighstone M, Mitra SK, Arakawa K. A new efficient approach for the removal of impulse noise from highly corrupted images. IEEE Trans Image Process 1996;5:1012–25. [18] Chen T, Wu HR. A new class of median based impulse rejecting filters. In: IEEE International Conference on Image Processing. 2000. p. 916–9. [19] Chen T, Wu HR. Adaptive impulse detection using center weighted median filters. IEEE Signal Process Lett 2001;8:1–3. [20] Yuksel M, Bastrk A. Efficient removal of impulse noise from highly corrupted digital images by a simple neuro-fuzzy operator. AEU Int J Electron Commun 2003;57:214–9.
[21] Muneyasu M, Nishi N, Hinamoto T. A new adaptive center weighted median filter using counter propagation networks. IEEE Trans Image Process 2000;337:631–9. [22] Tukey JW. Nonlinear methods for smoothing data. In: Proc Conf Rec EASCON. 1974. p. 673. [23] Besdok E, Civicioglu P, Alci M. Using an adaptive neuro-fuzzy inference system based interpolant for impulsive noise suppression from highly distorted images. Fuzzy Sets Syst 2005;150:525–43. [24] Jang J. Anfis: adaptive-network-based fuzzy inference system. IEEE Trans Systems Man Cybern 1993;23:665–85. [25] MathWorks: Matlab, fuzzy logic toolbox, user’s guide. New York: The Math Works Inc.; 2002. [26] Yen J, Wang L. Improving the interpretability of TSK fuzzy models by combining global learning and local learning. IEEE Trans Fuzzy Syst 1998;6:530–7. [27] Chen SS, Yang X, Cao G. Impulse noise suppression with an augmentation of ordered difference noise detector and an adaptive variational method. Pattern Recogn Lett 2009;30:460–7. Xuan Wang was born in 1966. He received the B.S. and M.S. degrees in Electrical Engineering from Shaanxi Normal University Xi’An, China in 1983 and 1987, and received Ph.D. degree in the key laboratory of the ministry of education for Computer Networks and Information Security. He is currently professor and head of School of Physics and Information Technology at Shaanxi Normal University. His research interests include image processing and pattern recognition.
Xue-Qin Zhao was born in 1985. She is currently pursuing the M.S. degree at Shaanxi Normal University. Her research interests include image processing.
Fang-Xia Guo was born in 1965. She received the M.S. degree in Electrical Engineering from Shaanxi Normal University. She is currently vice professor of Shaanxi Normal University and pursuing the Ph.D. degree at Shaanxi Normal University. Her research interests include image processing and pattern recognition.
Jian-Feng Ma was born in 1961. He received the Ph.D. degree in communication and electronic systems from Xidian University. He is currently a professor and the Dean of School of Computer in Xidian University. He is also a member of IEEE. His research interests include image processing and information and network security.