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1 Signal Processing 92 (2012) Contents lists available at ScienceDirect Signal Processing journal homepage: Quaternion switching filter for impulse noise reduction in color image Xin Geng n, Xiaoguang Hu, Jin Xiao School of Automation Science and Electrical Engineering, Beihang University, Beijing , China article info Article history: Received 25 January 2011 Received in revised form 28 April 2011 Accepted 29 June 2011 Available online 7 July 2011 Keywords: Color image Impulse noise Quaternion unit transform Switching filter abstract A novel approach to impulse noise reduction in color image is introduced in this paper. By applying the quaternion unit transform theory, the difference between two color pixels can be represented in the quaternion form. Based on the difference mentioned above, an efficient filter that can switch between the vector median filter (VMF) and the identity filter (no filtering operation) is proposed. Extensive simulation result indicates that the proposed filter achieves a trade-off between noise suppression and detail preservation in both correlated and uncorrelated impulse noise scenarios when compared with other widely used filters. Furthermore, the computational complexity analysis shows that the proposed filter is quite efficient. & 2011 Elsevier B.V. All rights reserved. 1. Introduction Color images are often corrupted by impulse noise due to malfunctioning sensors, faulty memory locations in the hardware, aging of the storage material or transmission errors [1,2]. Although the traditional impulse noise reduction method in gray scale images can be applied to filtering the color channels separately, satisfactory results cannot be obtained due to the correlation of different channels. Many researches have shown that the nonlinear vector filtering technique, which treats the color sample pixels as three-dimensional vectors in RGB color domain, is more appropriate to suppress the color impulse noise [3]. One of the most important classes of nonlinear vector filters is based on the order-statistic theory, such as the vector median filter (VMF) and its extensions [4 9]. By means of a suitable distance to measure the similarity between two color vectors, these filters replace the center pixel of a filtering window withavectormedianandproducearobustperformance. n Corresponding author. Tel.: þ ; fax: þ address: gengxin213@hotmail.com (X. Geng). Besides the widely used Euclidean distance, many other distances or similarity measures can be employed in this class of filters, such as the Cosine distance, various fuzzy metrics and so on [10 12]. The main weakness of the traditional vector median filters is the loss of image details because the filters modify every pixel in the image regardless whether it is corrupted or not. One solution to this problem is to appropriately modify the distances by weighting coefficients to perform a weighted vector median operation. The filters based on this strategy produce much better results because it takes into account the importance of the specific samples in the filtering window [13 17]; however, these filters also process every pixel including the ones that are not corruptedbynoise. To avoid damage of noise-free pixels, a switching strategy, which detects the corrupted pixels is carried out before noise removal. If the pixel is detected as noise, it is replaced by the output of VMF or other filters; otherwise, it remains unchanged. Based on this strategy, many switching filters have been proposed, such as the adaptive vector median filter (AVMF) [18], the robust switching vector median filter (RSVMF) [19], the adaptive center-weighted vector median filter (ACWVMF) [20,21], the class of vector sigma filters [20,22], and so on. To decrease the number of misclassified /$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi: /j.sigpro

2 X. Geng et al. / Signal Processing 92 (2012) pixels, several improved noise detection techniques have been presented recently. In [1,23 26], the switching filters based on the peer group technique are introduced. In [27], a fast similarity-based vector filter (FSVF) is proposed based on the concept of similarity between pixels and non-parametric estimation. In [28,29], the fuzzy rank-ordered differences filter (FRF) and the improved FSVF are presented, which take advantage of the fuzzy metrics to detect noisy pixels. As an extension of two-dimensional complex numbers to four dimensions, the quaternion theory has been applied in color image processing [30 32]. In [33 35], three quaternionbased switching filters are introduced for impulse noise reduction. In conclusion, various switching filters have been proposed; however, the new distance measure and noise detection techniques are still being studied for higher detection accuracy. To improve the filtering results, this paper introduces a switching filter based on the quaternion theory. We represent a RGB color image as a pure quaternion form and measure both the intensity and chromaticity differences between two color pixels with the quaternion unit transform. Then we define a novel color pixel distance as the summation of the intensity and chromaticity differences. Consider a 3 3 filtering window, we calculate the color pixel differences in four directions and determine the center pixel to be noisy if the minimum of the four differences exceeds a threshold. This detection step is employed to determine the real noisy pixels and preserve the noise-free ones on the edges. Finally, the noisy pixels are replaced by the VMF output and the noise-free ones are unchanged. This paper is organized as follows. In the next section the basic knowledge and applications of quaternions are introduced. In Section 3 the new switching filter based on the quaternion theory is presented. Experiments are carried out in Section 4 to assess the proposed filter and compare it with other widely used filters. Finally conclusions are drawn in Section Properties and applications of quaternions in color images 2.1. Properties of quaternions The quaternions, discovered by Hamilton, are the extension of two-dimensional complex numbers [36]. A quaternion number q is a four-dimensional number, which consists of one real part and three imaginary parts. It can be represented as the hypercomplex form q ¼ aþbiþcjþdk, ð1þ where a, b, c and d are real; i, j and k are complex operators obeying the following rules: i 2 ¼ j 2 ¼ k 2 ¼ ijk ¼ 1, ij ¼ k, jk ¼ i, ki ¼ j, ji ¼ k, kj ¼ i, ik ¼ j: It can be seen from the rules that the multiplication of quaternions is not commutative. When a¼0, q is referred to as a pure quaternion. The modulus and conjugate of a ð2þ quaternion are defined as follows, respectively: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9q9 ¼ a 2 þb 2 þc 2 þd 2, ð3þ q ¼ a bi cj dk: ð4þ A quaternion with a unit modulus is referred to as a unit quaternion. Besides the hypercomplex form, there are many other representations of a quaternion as well, such as the polar form [32] q ¼ 9q9e ly ¼ 9q9ðcosyþlsinyÞ, where l is a unit pure quaternion; l and y are referred to as the eigenaxis and eigenangle of q, respectively Quaternion representation of color image Consider the widely used RGB color domain, a color image can be represented as the pure quaternion form q x,y ¼ r x,y iþg x,y jþb x,y k, ð6þ where x,y denote the coordinates and r,g,b 2f0,1, g. With the quaternion representation, a three-dimensional color pixel can be simply treated as a whole. To analyze the properties of the quaternion representation of a color p image, a unit pure quaternion is defined as l ¼ðiþjþkÞ= ffiffiffi 3. Thus any unit quaternion U can be represented as U ¼ 9U9e ly ¼ cosyþlsiny. We define the quaternion unit transform of a color pixel q as follows [30]: Y ¼ UqU ¼ cosyþlsiny ðriþgjþbkþ cosy lsiny ¼ðriþgjþbkÞcos2yþ p 2 ffiffiffi lðr þg þbþsin 2 yþ 1 pffiffiffi ðb gþi 3 3 þðr bþjþðg rþkšsin2y ¼ Y RGB þy I þy D, ð7þ p where Y RGB ¼ðriþgjþbkÞcos2y, Y I ¼ð2= ffiffiffi 3 Þlðr þg þbþsin 2 p y, Y D ¼ð1= ffiffiffi 3 Þ½ðb gþiþðr bþjþðg rþkšsin2y. YI represents the intensity of a color image. Y D is the projection of the tristimuli in Maxwell triangle rotated 901 and represents the p chromaticity [3,30]. When y ¼ p=4, T ¼ U9 y ¼ p=4 ¼ ð1= ffiffiffi pffiffiffi 2 Þþð1= 6 ÞðiþjþkÞ and we p obtain YRGB ¼0, Y I ¼ð1=3Þ ðrþg þbþðiþjþkþ and Y D ¼ð1= ffiffiffi 3 Þ½ðb gþiþðr bþjþðg rþkš. Similarly, Y* can be defined as follows: Y n ¼ TqT ¼ 1 ðr þg þbþðiþjþkþ 3 1 pffiffiffi ðb gþiþðr bþjþðg rþk ¼ YI Y D : ð8þ 3 Therefore we obtain Y I ¼ð1=2ÞðTqTþTqTÞ and Y D ¼ð1=2Þ ðtqt TqTÞ. Consider two color pixels q 1 and q 2, we define the intensity and chromaticity differences as d 1 ðq 1,q 2 Þ¼ ð1=2þ9ðtq 1 TþTq 1 TÞ ðtq 2 TþTq 2 TÞ9 and d 2 ðq 1,q 2 Þ¼ð1=2Þ9 ðtq 1 T Tq 1 TÞ ðtq 2 T Tq 2 TÞ9, respectively. Assume that the intensity values of q 1 and q 2 are similar, then d 1 approaches zero. Analogously d 2 is close to zero when the chromaticity values of two pixels are approximately equal. In our research, we consider the intensity and ð5þ

3 152 X. Geng et al. / Signal Processing 92 (2012) chromaticity as two vital elements for color pixel difference simultaneously. Therefore, we define the color pixel difference as d(q 1,q 2 )¼d 1 (q 1,q 2 )þd 2 (q 1,q 2 ), which is the summation of the intensity and the chromaticity differences. In fact, the proposed color pixel difference is a quaternion-based distance measure. Compared with others, such as the Euclidean distance, Cosine distance and fuzzy magnitude similarity measure [10], the proposed color pixel difference quantifies the intensity and chromaticity differences simultaneously. A quaternion-based chromaticity distance is presented in [33]. The quaternion unit transform is also applied in this distance but only the chromatic component Y D is used. Another quaternion-based distance utilizes the quaternion rotation theory [34,35]. The main weakness of this distance is that it cannot distinguish the different pixels on the gray line, where gray line represents an axis in RGB color space on which a color pixel is achromatic with values R¼G¼B. For example, the dark pixel (0,0,0) and white pixel (255,255,255) are significantly different; however, the computational result of this distance is 0. Therefore, we can still regard that only the chromaticity is taken into account in the design of this distance measure. The ordering criterion used in the directional-distance filter (DDF) combines the Minkowski and Cosine distances to incorporate information about both the vector magnitude and direction [10]. Note that this ordering criterion is only defined in a particular neighborhood, while the proposed color pixel difference is able to quantify the difference between two vectors. In addition, the fuzzy magnitudedirectional similarity measure in [10] also combines the measurement of intensity and chromaticity similarities; however, it employs a computationally expensive scheme to achieve this purpose as mentioned in Section Proposed switching filter In this section we propose a quaternion switching filter (QSF) based on the quaternion representation of color images and the color pixel difference mentioned in Section 2. The QSF employs the color pixel difference to detect whether the center pixel in a filtering window is noisy or not and switches to the VMF and identity operation, respectively. Noise detection is the backbone for the switching filter. The traditional switching filters usually employ the order statistics of pixels to detect noise. However, the pixels on edges are often mistakenly detected as noise in some scenarios since they have different intensity or chromaticity with their neighbors in the smooth area of the filtering window. Image details are lost after the filtering process due to the excessive smoothing. The significant characters of edges are structure and direction. The pixels on edges are arranged continuously in a specific direction, whereas the noisy pixels appear to be isolated. Consider an M N RGB color image, we can express it as a quaternion form according to Eq. (6). A 3 3 window W is applied as a filtering window (see Fig. 1) since a larger window size leads to a longer computational time. In the filtering window, we calculate the color pixel differences between the center pixel q 5 and other two neighboring pixels in four directions of 01, 451, 901 and For example, the color pixel difference in the direction of 01 is defined as V 1 ¼ 1 2 ðdðq 4,q 5 Þþdðq 5,q 6 ÞÞ: Similarly the other three pixel differences V 2, V 3 and V 4 associated with 451, 901 and 1351 are defined as follows: V 2 ¼ 1 2 ðdðq 3,q 5 Þþdðq 5,q 7 ÞÞ, V 3 ¼ 1 2 ðdðq 2,q 5 Þþdðq 5,q 8 ÞÞ, V 4 ¼ 1 2 ðdðq 1,q 5 Þþdðq 5,q 9 ÞÞ: ð9þ ð10þ ð11þ ð12þ when q 5 is on the edge, at least one value among V l (l¼1,2,3,4) will be extremely small theoretically. When q 5 is in the smooth area of an image, V l (l¼1,2,3,4) are all equal to zero approximately. However if q 5 is corrupted by impulse noise, all four values will turn to be significantly larger than zero even if the pixel is on the edge. Therefore we set a threshold T to determine whether the pixel is noisy or not. If V¼min{V l,(l¼1,2,3,4)} exceeds T, the center pixel is determined as noisy and replaced by the VMF output. Otherwise the center pixel is deemed as noise-free and kept unchanged. The VMF can be written in a quaternion form Fig. 1. Indexing convention for a 3 3 filtering window. q VMF x,y ¼ argmin X N q t 2fq 1,q 2,,q N g s ¼ 1 99q s q t 99, ð13þ Fig. 2. Test images.

4 X. Geng et al. / Signal Processing 92 (2012) Fig. 3. Dependence of the NCD on the parameter T for the test images contaminated by three kinds of impulse noise: (a) NM1 with p¼0.05, (b) NM1 with p¼0.15, (c) NM2 with p¼0.05, (d) NM2 with p¼0.15, (e) NM3 with j¼0.05 and (f) NM3 with j¼0.15.

5 154 X. Geng et al. / Signal Processing 92 (2012) where x and y denote the coordinates in the image, S and t index the pixels from the filtering window, N¼3 3 is the number of pixels in the window and {q 1,q 2,yq N } represents all the pixels in the window. As mentioned above, the proposed filter QSF can be written as follows: ( q QSF x,y ¼ qvmf x,y if V 4T, ð14þ q x,y otherwise : In the filtering window W, the neighbors of the center pixel may be corrupted by impulse noise, especially when the noise corruption probability is high. In this case, these noisy neighbors tend to reduce the noise detection accuracy. To detect noisy pixels precisely, the pixels in W should come from the filtered results as much as possible. Note that when filtering q 5, the pixels q 1, q 2, q 3 and q 4 have been filtered previously. Thus, when filtering q 5,we let q 1, q 2, q 3 and q 4 take the value of previously filtered results to reduce the adverse effects of the noisy neighbors, while q 5, q 6, q 7, q 8 and q 9 are the original pixels. 4. Experimental results and assessment 4.1. Noise model and objective measures The impulse noise corrupted color images can be found in many present-day applications. For example, television Table 1 Evaluation result of noise detection capability using four test images contaminated with 5% and 15% noise corruption probabilities of three impulsive noise models. Image NM1 NM2 NM3 E1 E2 E1 E2 E1 E2 Port 5% Port 15% Lena 5% Lena 15% Peppers 5% Peppers 15% Micro 5% Micro 15% images are corrupted by atmospheric interference and imperfection of the image data reception. In the application of digitized artworks, impulse noise is introduced by scanning damaged and granulated surfaces of the original artworks. Digital cameras may introduce impulse noise due to malfunctioning sensor, electronic interference and flaws in data transmission. Impulse noise is also introduced in cdna microarray images because of the nature of microarray technology [1,4,5,37]. In these real-life digital images, the form of impulse noise is varied. For example, the value of impulse noise caused by a malfunctioning sensor is expected to be fixed, while the value of noise may be random in other cases, such as electronic interference. Consequently, to simulate the real impulse noise, many models have been proposed [1,3,4,33]. In our experiments, three widely used color impulse noise models NM1, NM2 and NM3 are applied. NM1 and NM2 are uncorrelated impulse noise models [1,24] ( x k ¼ rk with probability p, o k ð15þ with probability 1 p, where k¼1,2,3 denotes the three channels in RGB color space; p is the channel corruption probability; O k and r k denote the original component and the contamination component, respectively. When r k equals 0 or 255 with equal probability, Eq. (15) is usually named as salt and pepper noise model denoted as NM1. If r k can choose any discrete value in [0,255], the random-valued noise model named as NM2 is obtained. In NM1 and NM2, the contamination of the color image components is uncorrelated. The noise is added in each of the color channels independently with probability p. Thus the overall contamination rate is j ¼ 1 ð1 pþ 3, which is much higher than p [1,24]. NM3 is correlated impulse noise model [3] 8 o with probability 1 j, >< fr 1,o 2,o 3 g with probability j 1 j, x k ¼ fo 1,r 2,o 3 g with probability j 2 j, ð16þ fo 1,o 2,r 3 g with probability j 3 j, >: fr 1,r 2,r 3 g with probability j 0 j, where r k (k¼1,2,3) equals 0 or 255 with equal probability; j is the sample corruption probability; j 1, j 2 and j 3 are the channel corruption probabilities and j 0 ¼ 1 ðj 1 þj 2 þj 3 Þ. Table 2 Comparison result when filtering the Port image corrupted with p¼5%, 15% and 30% of NM1. Filter 5% 15% 30% MAE MSE NCD 100 MAE MSE NCD 100 MAE MSE NCD 100 IF VMF FPGF ASVMF RSVMF ACWVMF FRF IPGSVF QSVMF ESF QSF

6 X. Geng et al. / Signal Processing 92 (2012) In our experiments, j 1 ¼j 2 ¼j 3 ¼0.25 is used. Based on the above introduction, we can see that the three noise models cover a wide range of the real-life impulse noise, including both the fixed and random-valued impulse noise with the channel correlated and uncorrelated contamination mechanism. Consequently, these models, which approximate the real-life noise mechanism, are employed in this section to assess the proposed filter. Three objective measures are employed to assess the performance of the filters. They are mean absolute error (MAE), mean squared error (MSE) and normalized color distance (NCD). The MAE and MSE are utilized to evaluate the detail preservation and noise suppression capability of a filter, respectively. The NCD measures the perceptual error in the CIELab color space [3,33]. In the next experiments, the color images Port, Lena, Peppers and Table 3 Comparison result when filtering the Lena image corrupted with p¼5%, 15% and 30% of NM2. Filter 5% 15% 30% MAE MSE NCD 100 MAE MSE NCD 100 MAE MSE NCD 100 IF VMF FPGF ASVMF RSVMF ACWVMF FRF IPGSVF QSVMF ESF QSF Table 4 Comparison result when filtering the Peppers image corrupted with j¼5%, 15% and 30% of NM3. Filter 5% 15% 30% MAE MSE NCD 100 MAE MSE NCD 100 MAE MSE NCD 100 IF VMF FPGF ASVMF RSVMF ACWVMF FRF IPGSVF QSVMF ESF QSF Table 5 Comparison result when filtering the Micro image corrupted with j¼5%, 15% and 30% of NM3. Filter 5% 15% 30% MAE MSE NCD 100 MAE MSE NCD 100 MAE MSE NCD 100 IF VMF FPGF ASVMF RSVMF ACWVMF FRF IPGSVF QSVMF ESF QSF

7 156 X. Geng et al. / Signal Processing 92 (2012) Micro in Fig. 2 have been used to evaluate the performance of the proposed filter QSF Parameter selection There is a single parameter T in the QSF. Fig. 3 shows the dependence of the NCD on the parameter T for the test Table 6 Average rank of performance in terms of NCD achieved by each considered filter in the comparison experiments. Filter QSF 1.33 QSVMF 4.25 FRF 4.25 ACWVMF 4.33 ESF 4.83 IPGSVF 5.08 FPGF 5.33 RSVMF 6.83 ASVMF 9.00 VMF 9.33 Average rank images contaminated by the three kinds of impulse noise. As T approaches 0, the QSF turns into the VMF and has a better capability of noise suppression. On the other hand, as T is large enough, the QSF is similar to the identity filter and no filtering operation is performed. It can be seen from Fig. 3 that the optimal value of T is different for each image, noise model and corruption probability. Nearoptimal experimental results can be achieved by the general setting of T¼35 in all the cases. Thus T¼35 is utilized in the next experiments Evaluation of the noise detection capability Because noise detection is the most significant step in a switching filter, we first evaluate the performance of noise detection capability. Assume that O 1 denotes the set of pixels, which are detected as noise in the filtering process and O 2 denotes the set of real corrupted pixels in the image, we define n 0 is cardinality of O 1 \ O 2, n 1 is the cardinality of O 1 and n 2 is the cardinality of O 2. Then two measures of E 1 ¼n 0 /n 1 and E 2 ¼n 0 /n 2, which represent the accuracy rate and detection rate, respectively, are used to evaluate the noise detection capability [38]. Fig. 4. Visual comparison using the Port image: (a) an enlarged part of Port image corrupted with p¼15% of NM1 (b) VMF output, (c) FPGF output, (d) IPGSVF output, (e) ESF output, (f) ACWVMF output, (g) FRF output, (h) QSVMF output and (i) QSF output.

8 X. Geng et al. / Signal Processing 92 (2012) In our experiment the four test images contaminated with 5% and 15% noise corruption probabilities of three impulsive noise models are used to evaluate the noise detection capability (p¼5% and 15% for NM1 and NM2, j¼5% and 15% for NM3). It can be seen from Table 1 that both E1 and E2 exceed 0.9 for NM1 and NM3 noise models, which means that the noise detection method is effective. For NM2 noise model, the noisy pixels are more difficult to be detected because of the following reasons: first, the random value of impulse noise leads to minor differences between the noisy pixels and their neighbors; second, the NM2 noise model is uncorrelated so that the noise corruption probability is much higher than the channel corruption probability p. Even in that case, more than 80% noisy pixels can be detected and the accuracy rate still exceeds Performance comparison In the next experiments the QSF is compared with the classical VMF and other recent proposed switching filters. The filters used in this paper are the identity filter (named as IF), the vector median filter (VMF) [3], the fast peer group vector median filter (FPGF) [24], the adaptive sigma vector median filter (ASVMF) [22], the robust switching vector median filter (RSVMF) [19], the adaptive center-weighted vector median filter (ACWVMF) [20], the fuzzy rank-ordered differences statistic filter (FRF) [28], the iterative peer group switching vector filter with M k1k2 similarity measure (IPGSVF) [26] and the quaternion switching vector median filter (QSVMF) [35]. Moreover, to see the gain of the proposed color pixel difference, we implement a version of the proposed filter without the quaternion unit transform. That is, we replace the proposed color pixel difference in the QSF with Euclidean distance. This filter is named as the Euclidean switching filter (ESF) and the threshold is determined as 30 in the experiments. Tables 2 5 show the comparison of the objective measures using the test images contaminated with different impulse noise models. To summarize these results, for each considered case in Tables 2 5, the filters are sorted in increasing order of NCD performance. Then we compute the average of the ranks achieved by the considered filter in the experiments and sort the filters in increasing order of average rank in Table 6. The visual Fig. 5. Visual comparison using the Lena image: (a) an enlarged part of Lena image corrupted with p¼15% of NM2 (b) VMF output, (c) FPGF output, (d) IPGSVF output, (e) ESF output, (f) ACWVMF output, (g) FRF output, (h) QSVMF output and (i) QSF output.

9 158 X. Geng et al. / Signal Processing 92 (2012) comparisons of the QSF with other good performance filters are provided in Figs These filters are selected in the visual comparison experiments because of the following reasons: the VMF is the most well known vector filter and it is of common reference for other vector filters; the QSVMF, FRF, ACWVMF, ESF, IPGSVF and FPGF are the recent effective switching filters and occupy the top six in Table 6 except the QSF. An enlarged part of the filtering result is shown in Figs. 4 6 in order to illustrate the filtering result more clearly. After analyzing Tables 2 6, we can see that the proposed filter QSF has the best overall performance compared with the other filters in terms of the objective measures for all three noise models; on the other hand, the QSF is able to achieve a good balance between noise suppression and detail preservation in Figs It can be also observed from Tables 2 5 that the QSF and IPGSVF have a better performance than the other switching filters with high noise corruption probability (p¼30% for NM1 and NM2, j¼30% for NM3). Specifically, when the noise corruption probability is high, the QSF outperforms the IPGSVF for NM3 noise model according to the MAE, MSE and NCD. For NM1 and NM2 noise models, the IPGSVF has lower values of MSE but higher values of MAE and NCD than the QSF. This means the noise suppression capability of IPGSVF outperforms the QSF, while the detail preservation capability and the perceptual error performance of IPGSVF are inferior to the QSF in the high noise corruption probability occasion. Thus, when the noise corruption probability is high, we can make a conclusion that the QSF outperforms the IPGSVF for NM3 and exhibits a competitive performance of MAE and NCD with the IPGSVF for NM1 and NM2. Another quaternion-based filter QSVMF also achieves a good performance in Tables 2 4; however, it is necessary to notice that the performance of QSVMF decreases for Micro image (see Table 5 and Fig. 6). As analyzed in Section 2, the quaternion-based color pixel difference used in the QSVMF cannot distinguish the achromatic pixels. Notice that the pixels on the Micro image are close to achromatic ones, that is, the three components of these pixels are approximately equal to each other. Thus, we can see that the dark and white impulses are unable to be removed. In addition, we can notice that the performance of the QSF is much better than the ESF. This means Fig. 6. Visual comparison using the Micro image: (a) an enlarged part of Micro image corrupted with j¼15% of NM3 (b) VMF output, (c) FPGF output, (d) IPGSVF output, (e) ESF output, (f) ACWVMF output, (g) FRF output, (h) QSVMF output and (i) QSF output.

10 X. Geng et al. / Signal Processing 92 (2012) the proposed color pixel difference based on the quaternion unit transform outperforms the traditional Euclidean distance. As a conclusion, we can say that the QSF has a better performance of noise suppression and detail preservation than the other switching filters and produces a robust filtering result in each noise corrupted case. As we know, additive Gaussian noise is another common source of noise, which is often introduced during image acquisition. Unlike impulse noise, additive Gaussian noise is characterized by adding to each image pixel a value from a zero-mean Gaussian distribution. The switching structure used in the impulse noise reduction filter is unable to remove the additive Gaussian noise, because there may be no noise-free pixels in the presence of additive Gaussian noise. Most filters for Gaussian noise reduction are designed to take advantage of the zeromean property and try to suppress it by locally averaging pixel channel values, such as the arithmetic mean filter (AMF) and the Gaussian filter [37]. A simple method to remove the mixed additive Gaussian noise and impulse noise is to use the impulse noise reduction filter followed by an additive Gaussian noise reduction filter. Thus it is necessary to see the performance of the proposed filter when the image is corrupted by the mixed noise. Fig. 7 shows the comparison of the switching filters when filtering a Peppers image corrupted with mixed NM3 impulse noise and additive Gaussian noise. It can be seen that the QSF is able to suppress impulse noise even in the presence of additive Gaussian noise. To demonstrate how to deal with the mixed noise, we use the AMF to process the output of the QSF and show the filtering result in Fig. 7(i). Consequently, the QSF can be used as a preprocessor for Gaussian noise-driven filters when the images are corrupted by mixed noise. The real noisy image is also used to make the visual comparison. Fig. 8 shows the experimental results when a real noisy image is filtered. The image used in the experiment is a part of digitized painting. The impulse noise is introduced by scanning the damaged and granulated surface of the original artwork. It can be seen from Fig. 8 that the trade-off between noise suppression and detail preservation can be achieved by the QSF. This result further confirms that the proposed filter is suitable for real noisy images. Fig. 7. Visual comparison using the Peppers image: (a) an enlarged part of Peppers image corrupted with mixed j¼30% NM3 impulse noise and s¼20% additive Gaussian noise, (b) FPGF output, (c) IPGSVF output, (d) ESF output, (e) ACWVMF output, (f) FRF output, (g) QSVMF output, (h) QSF output and (i) Output of the combination of QSF and AMF.

11 160 X. Geng et al. / Signal Processing 92 (2012) Fig. 8. Visual comparison using the real noisy image: (a) a part of digitalized painting, (b) VMF output, (c) FPGF output, (d) IPGSVF output, (e) ESF output, (f) ACWVMF output, (g) FRF output, (h) QSVMF output and (i) QSF output Computational complexity analysis In this section we first compare the computational complexity of the QSF and the classical filter VMF. Some assumptions are necessary before we analyze the computational complexity. We define the computational complexity as the total number of elementary operations needed to filter a pixel. The elementary operations are SQRT, MULT, ADD and COMP, where SQRT is the scalar square roots, MULT is the scalar multiplications or divisions, ADD is the scalar additions or subtractions and COMP means comparisons. Note that the division and subtraction can be regarded as MULT and ADD, respectively, in the calculation process. Firstly, we analyze the computational complexity of the QSF. As aforementioned, the color pixel difference consists of intensity and chromaticity differences. Consider two color pixels represented in the quaternion form q 1 ¼ r 1 iþg 1 jþb 1 k and q 2 ¼ r 2 iþg 2 jþb 2 k, the intensity difference can be written as d 1 ðq 1,q 2 Þ¼ 1 2 9ðTq 1TþTq 1 TÞ ðtq 2 TþTq 2 TÞ9 ¼ p 1 ffiffiffi 9r 1 þg 1 þb 1 r 2 g 2 b 2 9, ð17þ 3 which means that the calculation of an intensity difference needs 5 ADDs and 1 MULT. The chromaticity difference between q 1 and q 2 is d 2 ðq 1,q 2 Þ¼ 1 2 9ðTq 1T Tq 1 TÞ ðtq 2 T Tq 2 TÞ9 ¼ 1 pffiffiffi 9 ðb 1 g 1 Þ ðb 2 g 2 Þ iþ ðr1 b 1 Þ ðr 2 b 2 Þ j 3 þ½ðg 1 r 1 Þ ðg 2 r 2 ÞŠk9, ð18þ which means that the calculation of a chromaticity difference needs 11 ADDs, 4 MULTs and 1 SQRT. Thus the computation of color pixel difference d needs 17 ADDs, 5 MULTs and 1 SQRT. The numbers of elementary operations for the traditional and recent distance measures have been summarized in [10]. As mentioned in Section 2, the fuzzy magnitude-directional similarity measure is another distance measure that combines intensity and chromaticity similarities. The computation of this similarity measure needs 6 COMPs, ADDs, 23 MULTs and 2 SQRTs [10]. Obviously, the proposed color pixel difference outperforms it in terms of the computational complexity. Consider a 3 3 window and four directional noise detections, we have the computational complexity T QSF as follows: T QSF ¼ 8SQRTþ44MULTþ140ADDþ3COMPþjT VMF, ð19þ

12 X. Geng et al. / Signal Processing 92 (2012) where j is the noise corruption probability and T VMF is the computational complexity of the VMF. Secondly, we analyze the computational complexity of the VMF with the same method mentioned above. The computation of Euclidean distance between two pixels needs 5 ADDs, 3 MULTs and 1 SQRT. Consider a 3 3 window, the VMF requires at least 9(9-1)/2 Euclidean distances and (9-1) COMPs. Thus to filter a pixel in a 3 3 window, the computational complexity of the VMF is T VMF ¼ 36SQRTþ108MULTþ180ADDþ8COMP: ð20þ It can be seen from Eqs. (19) and (20) that the computational complexity of the QSF is clearly less than that of the VMF because the QSF employs fewer elementary operations to filter a pixel if j is not very high. The total time required to complete an operation is proportional to the normalized total number of equivalent scalar operations [3] Time ¼ k ð25 ðsqrtþþ4 ðmultþþðaddþþ: ð21þ The weights used in Eq. (21) do not pertain to any particular machine and can be considered as mean values of those coefficients commonly encountered [3]. Thus an approximate comparison between T QSF and T VMF can be made if we omit the COMP operations since the number of COMP operation is fewer when compared with other operations. Then we can conclude that the QSF is faster than the VMF when j is less than 65%. Thirdly we define the speed gain g¼t VMF /T i for the further study of the computational complexity, where T i is the running time of the QSF, QSVMF, FRF and ACWVMF. These filters are selected because they have the best filtering performance according to Table 6. The running time is tested in MATLAB using a computer with Intel Pentium GHz CPU. Fig. 9 shows the comparison result of the speed gain. This figure indicates that the QSF has the lowest computational complexity compared with other filters, while the ACWVMF employs a computationally expensive scheme to remove noisy pixels. Fig. 9. 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