Deep CNN for removal of salt and pepper noise

IET Image ProcessingVolume 13, Issue 9 p. 1550-1560 Research ArticleFree Access Deep CNN for removal of salt and pepper noise Yan Xing, Corresponding Author Yan Xing xy1128@126.com orcid.org/0000-0002-8076-0805 School of Mathematics, Hefei University of Technology, Hefei, 230009 People's Republic of ChinaSearch for more papers by this authorJian Xu, Jian Xu School of Mathematics, Hefei University of Technology, Hefei, 230009 People's Republic of ChinaSearch for more papers by this authorJieqing Tan, Jieqing Tan School of Mathematics, Hefei University of Technology, Hefei, 230009 People's Republic of ChinaSearch for more papers by this authorDaolun Li, Daolun Li School of Mathematics, Hefei University of Technology, Hefei, 230009 People's Republic of ChinaSearch for more papers by this authorWenshu Zha, Wenshu Zha School of Mathematics, Hefei University of Technology, Hefei, 230009 People's Republic of ChinaSearch for more papers by this author Yan Xing, Corresponding Author Yan Xing xy1128@126.com orcid.org/0000-0002-8076-0805 School of Mathematics, Hefei University of Technology, Hefei, 230009 People's Republic of ChinaSearch for more papers by this authorJian Xu, Jian Xu School of Mathematics, Hefei University of Technology, Hefei, 230009 People's Republic of ChinaSearch for more papers by this authorJieqing Tan, Jieqing Tan School of Mathematics, Hefei University of Technology, Hefei, 230009 People's Republic of ChinaSearch for more papers by this authorDaolun Li, Daolun Li School of Mathematics, Hefei University of Technology, Hefei, 230009 People's Republic of ChinaSearch for more papers by this authorWenshu Zha, Wenshu Zha School of Mathematics, Hefei University of Technology, Hefei, 230009 People's Republic of ChinaSearch for more papers by this author First published: 19 June 2019 https://doi.org/10.1049/iet-ipr.2018.6004Citations: 7AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Image denoising is a common problem during image processing. Salt and pepper noise may contaminate an image by randomly converting some pixel values into 255 or 0. The traditional image denoising algorithm is based on filter design or interpolation algorithm. There exists no work using the convolutional neural network (CNN) to directly remove salt and pepper noise to the authors' knowledge. In this study, they utilise CNN with the multi-layer structure for the removal of salt and pepper noise, which contains padding, batch normalisation and rectified linear unit. In training, they divide images into three parts: training set, validation set and test set. Experimental results demonstrate that the architecture can effectively remove salt and pepper noise for the various noisy images. In addition, their model can remove high-density noise well due to the extensive local receptive fields of the deep neural networks. Finally, extensive experimental results show that their denoiser is effective for those images with a large number of interference pixels which may cause misjudgement. In a word, they generalise the application of CNN to salt and pepper noise removal and obtain competitive results. 1 Introduction During image acquisition and transmission, impulse noise can arise due to a faulty sensor, camera miss-focus, interference in the channel, atmospheric turbulence etc. Salt and pepper noise in an image is a typical impulsive noise that sets the corrupted pixel value to the maximum or minimum of the pixel variation range. The noise deteriorates the image quality and affects the subsequent image processing. Therefore, we need to remove noise to get high-quality pictures. Many non-linear filtering methods including median filtering (MF) have been proposed for salt and pepper denoising task. However, we often encounter weight optimisation problems, in which we have to manually choose parameters based on prior knowledge, which is very time-consuming. In recent years, deep learning has gradually become an influential method for image classification. AlexNet [1], which includes eight layers, has been proved effective for extracting high-level image features in the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) competition in 2012. In 2014, deep models such as VGGNet [2] and InceptionNet [3], which used 19 layers and 22 layers, respectively, had achieved a 7.3 and 6.7% error rates in ILSVRC. By increasing the number of hidden layers, ResNet [4] of 152 layers reduced the error rate of the 1000-category image classification task to 3.57% in 2015. We can see that a better approximation can be achieved by increasing the number of hidden layers. With the development of hardware, people have begun to apply the deep learning model to the other image processing fields including image fusion [5] and image denoising [6]. The latest papers and experimental results [7] show that the residual learning is also an effective Gaussian denoising method. Inspired by this, we use deep convolutional neural network (CNN) to evaluate contaminated pixels by means of clean pixels, instead of the traditional MF method and different interpolation algorithms. Our network architecture uses 17 layers to learn how to remove salt and pepper noise and estimate the values of contaminated pixels. Extensive experiments show that the proposed method can not only remove low-level salt and pepper noise, but also effectively remove up to 90% high-level noise. In addition, our model is robust to those images with a large number of original pixel values of 0 and 255. The contributions of this work are as follows: (i) We successfully generalise the application of deep learning for the removal of salt and pepper noise. Here, the black box of CNN can be regarded as a simulation of the generalised interpolation function. The experimental results show that our method outperforms state-of-the-art filtering or interpolation methods. (ii) The denoiser is an end-to-end trainable deep CNN. The training of the denoiser can converge quickly. Good denoising results can be obtained even after a few epochs. The rest of this paper organised is as follows: a brief survey of related work is reviewed in Section 2. The network architecture and the denoising algorithm are presented in Section 3. Experimental results and analysis are given in Section 4. The conclusions are drawn in Section 5. 2 Related work Salt and pepper noise, a kind of bipolar impulse noise, randomly contaminates a certain percentage of pixels in an image to two extremes. Many methods have already been proposed to reduce the noise. 2.1 Median filter for salt and pepper noise removal Filtering is a very effective image denoising method. Researchers have designed a variety of filters to make the image look smoother and more realistic. The filters can be divided into linear filters and non-linear filters. The linear filter is easy to design and implement due to its simple mathematical expression and mature theory. Some improved linear filters have achieved advanced results in common image denoising such as in [8]. However, people find that the linear filter will destroy the image edge and it is not suitable for salt and pepper noise. After that, a variety of non-linear filters have been proposed. Traditional non-linear filtering methods include MF methods [9] and adaptive filtering methods [10]. The median filter is the most popular non-linear filter for removing salt and pepper noise because of its good denoising power and computational efficiency [11]. To achieve better visual effects, many kinds of improved median filters have been proposed such as adaptive centre-weighted median filters (ACWMFs) [12], noise adaptive fuzzy switching median (NAFSM) filters [13], decision-based unsymmetric trimmed median filters (DBUTMFs) [14], adaptive weighted mean filter (AWMF) [15], efficient weighted-average filter (EWAF) [16], different applied median filter (DAMF) [17] and so on. 2.2 New non-linear filtering algorithm From a mathematical point of view, image denoising, being a special case of image restoration, can be regarded as a function approximation problem, in which the denoised result is the approximation of a clean image. The wavelet denoising method maps the actual signal space to the wavelet transform domain for the best recovery of the original signal. Joshi et al. [18] presented the joint scheme of wavelet transform and MF to remove salt and pepper noise. Some methods such as decision-based algorithm (DBA) [19] and fuzzy-based decision algorithm (FBDA) [20] used optimisation algorithms to estimate the noisy pixel values and some other methods denoise by interpolation. Bai and Tan [21] designed the Newton–Thiele filter (NTF) for salt and pepper noise based on Newton–Thiele interpolation algorithms. Mohanty et al. [22] utilised the B-splines interpolation to remove the salt and pepper noise for noise levels ranging from 15 to 95%. A neural network is more and more widely used in image processing. However, experiments have shown that simple neural network systems such as back-propagation (BP) networks are difficult to remove noise directly and effectively. Therefore, some special neural networks [23] were designed to remove the salt and pepper noise. Faro et al. [24] proposed a denoising scheme combining multi-layer perceptron networks and fuzzy logic. Faro et al. [25] presented Bayesian networks to remove the salt and pepper noise; meanwhile, preserving edges. Xiao-Ling et al. [26] use a BP neural network to design a noise detector as preprocessing. 2.3 Deep learning model in image denoising With the progress of deep learning, there have been several attempts to handle the denoising problem by deep neural networks [27-29]. The feed-forward denoising CNN (DnCNN) model was proposed in [7] and obtained promising results in Gaussian denoising by adopting residual learning strategy and batch normalisation (BN). [30] trained a set of CNN denoisers and integrated them to the model-based optimisation method to remove the image Gaussian noise. Lately, more and more image denoising methods based on deep learning models have emerged. Bae et al. [31] proposed a novel feature space deep residual learning algorithm that outperforms the existing residual learning method in Gaussian denoising. Since Block-Matching and 3D filtering (BM3D) is generally considered as a baseline for image denoising, Yang and Sun [32] designed BM3D-Net for image denoising by combining BM3D with CNN. Mccann et al. [33] reviewed recent uses of CNNs for inverse problems in imaging, which is inspiring and meaningful. However, to the best of our knowledge, there is no work to remove salt and pepper noise directly by using CNN. 3 Proposed method In this section, instead of traditional median filters, interpolation-based filters and optimisation-based methods, we will design CNN-based denoisers according to the characteristics of images corrupted by salt and pepper noise. In the following, we will elaborate our method from three aspects: (i) the construction of network architecture for salt and pepper noise removal; (ii) training of deep learning model; and (iii) selection of denoisers. In general, image denoising is to recover the latent clean image f from the degraded observation y. It is an ill-posed inverse problem, so people need a priori, i.e. regularisation, to constrain the solution space. It can be formulated as follows: (1) (2)where is the noisy image; is the clean image; and is the denoised output, that is, an approximation of . In (2), is a fidelity term, is a regularisation term and is a weight directly controlling the significance of . In supervised learning network, the training set includes many noisy-clean image pairs. So the corresponding fidelity term is . The model in [7] is to learn the residual mapping (3)It just changes the fidelity term in (2) to (4)According to the characteristics of salt and pepper noise, our denoising network is designed as learning identity mapping . 3.1 Network architecture Inspired by DnCNN [7] and Visual Geometry Group (VGG),University of Oxford [2], the structure ofthe deep neural network is illustrated in Fig. 1. The network depth is set to 17 so that we can extract morelow-level and semantic features from the original image. In Fig. 1, the first layer is a convolutional layerwith 64 convolution kernels plus rectified linear units (ReLUs). Fig. 1Open in figure viewerPowerPoint Architecture of the salt and pepper denoising network; 0 and255 in the input are the possibly corrupted points About 64 convolution kernels of size are used to convolve the input to produce64 feature maps. The bias of the convolution layer is initialised to 0. Theconvolutional sliding stride is 1, and we use the simple zero paddings beforeevery convolution to make each feature output of the same size as the input. Thenetwork structure of the middle layers from 2 to 16 is exactly the same: 64convolution kernels of size +BN+ReLU. We only use convolution kernels because smallconvolutional kernels of size can achieve promising results compared withother sizes, as pointed out in [2]. Thelast layer consists of a convolution layer with one kernel of size . We use the loss function (5) where are noisy-clean training image pairs and is the set of parameters in the network.The clean image is the label for training and is the training output image of thedenoising network. The gradients on both sides of the traditional sigmoid function tend to 0 and the maximum gradient is a quarter at 0. This will cause the back-propagated error to decay exponentially as it passes through the layers. Once the network is back-propagating in multiple layers, the gradient will continue decaying till vanishing. Therefore, all our activation functions use the ReLUs proposed in [34]. In addition, in order to increase the learning rate of the network and prevent the occurrence of overfitting, we have added BN [35] in the middle layers. We set the batch size to 128 so that the network can learn the accurate batch statistic estimation. A wide local receptive field can be acquired due to the deep network structure, and more pixel information can be used to estimate the contaminated points, as shown in Fig. 2. Fig. 2Open in figure viewerPowerPoint Local receptive field of our neural network 3.2 Training of deep learning model In the case of a grey-scale image, impulse noise may be represented by randomly distributed fixed values (FVs), i.e. 255 (white pixel, salt) and 0 (black pixel, pepper). The salt and pepper noise dealt with in our paper is in FV form. We trained 400 images of in Train400 following [7]. Before training, each image was continuously rotated by a certain angle () four times, flipped at each angle and then cropped into patches so that we can get more training examples. The clipping window slides by step 20, so an image can be cropped into patches of size . We take the clean patches as labels and add different levels of noises artificially as input. Fig. 3 shows the clean patch and noisy patch cropped from barbara. Fig. 3Open in figure viewerPowerPoint Left is the original patch and the right is the noisy patch From a mathematical point of view, removing salt and pepper noise can be regarded as using noise-free points to estimate noisy points. We do not expect the noisy pixels to affect the training process. So all the points in an image are divided into two categories: (i) noise-free points whose values are neither 0 nor 255 and (ii) possibly contaminated points whose values are 0 or 255. In the training input, we set the points of type (ii) to 0. This is because, on the one hand, the neural network learns to remove unipolar noise more easily and faster than bipolar noise; on the other hand, it improves the network denoising performance. During the process of convolution, the weights in the convolution kernel are initialised to normally distributed random numbers, and the computation in one sliding window is as follows: (6)where is the weight in the convolutional kernel and is the corresponding pixel value in the image. We do not expect the noisy pixel values to affect the training process. So we set the points of type (ii) to 0 as follows: (7)This leads to the convolution procedure containing only the noise-free points of type (i) and the noisy pixels do not impact the output. The experiment in Fig. 4 shows that our network has a strong ability to remove salt and pepper noise just after a few epochs. For the experiment, we can see that during the first 40 batches the loss function reduces rapidly, and the network quickly learns the denoising rule and adjust the parameters adaptively. This is mainly attributed to the adopted network architecture, where the BN layer leads to a larger learning rate and Adam optimisation algorithms throughout the training. Fig. 4Open in figure viewerPowerPoint Graph of loss function after the first 700 batch training during the first epoch for an image with noise density 90% We have trained the network using lots of images with all levels of noises (density from 10 to 90% with interval 10). We save all the trained parameters including weights and biases after 50 epochs and generate different denoisers for different levels of noises d, which are denoted by denoiser (d). 3.3 Selection of denoisers During the validation phase, we use 12 images as a validation set from Set68 [7] in order to prevent overfitting (see Fig. 5). We set the values of pixels of type (ii) in the image to 0. Although during the process we possibly misclassify noise-free pixels, whose original values are 255 or 0, into noisy pixels, we can still get satisfactory results when the proper denoiser is chosen. Fig. 5Open in figure viewerPowerPoint 12 Images as a validation set in order to prevent overfitting For those clean images with few pixels 0 or 255, extensive experiments show that the noisy images with noise density below 50% can be well-denoised by Denoiser (0.50), the noisy images with noise density 50–70% can be nicely denoised by Denoiser (0.70), and Denoiser (0.90) is suitable for noise density of 70–90%. Why have we chosen three denoisers for various levels of noise densities? First, we desire to find proper denoisers as few as possible to deal with salt and pepper noise of various densities and improve the denoiser's generalisation capability. Second, we, by extensive experiments, find that Denoiser (), Denoiser (0.70) and Denoiser (0.90) have good ability of denoising generalisation for intervals [1%, 50%], (50%, 70%] and (70%, 90%], respectively. The experiments in the next section show that the selection of the three denoisers is reasonable, and they have good performance and robustness. For those clean images with many misclassified pixels, the denoiser selected by the interval, in which the estimated value of noise density lies in is not the best choice. Once we set the uncontaminated pixels whose values are 255 to 0, it will affect the final output of the network. The loss of the information of the points, whose original values are 255 in the input, degrades the evaluation of the polluted neighbour pixel value. Then, we have made a large number of related experiments and fully measured the denoising generalisation capability of Denoiser(0.50), Denoiser(0.70) and Denoiser(0.90). We find that Denoiser (0.90) has a better quantitative index in denoising the images with a large number of misclassified points than the state-of-the-art methods when the noise density is larger than 30%. The usage of the proposed denoisers is in three steps. The first step is to acquire the input of denoiser. Detect the points whose values are 255, and replace the value of these points with 0 in order to make the output unaffected by the contaminated pixels and accelerate the denoising speed. The second step is to choose the appropriate denoiser according to the estimated noise density of the noisy image by the formula: (8)where n is the number of pixels with value 0 or 255 and M and N indicate the length and width of the image, respectively. In the last step, an output is generated by the chosen denoiser and modified by backfilling the pixels that are neither 0 nor 255. Our denoising algorithm is presented in Algorithm 1 (Fig. 6). To fully demonstrate the effectiveness and robustness of our algorithm, we test it on more datasets and use the mean of peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) as metrics in Section 4. Fig. 6Open in figure viewerPowerPoint Algorithm 1: Denoise 4 Experiments and analysis The performance of our method is tested with different grey images whose dynamic ranges are [0, 255]. We use GTX 1080 graphic processor in our experiments and the Nvidia cuDNN-v5.1 deep learning library is employed to accelerate the computation during the training. We conduct experiments with the matconvnet-1.0-beta25 package [36]. It took about 14 h to train the denoising network for 50 epochs. The performance of our method is quantitatively evaluated by PSNR, which is defined as follows: (9) (10)where f is the clean image; Y is the denoised image; and is the size of the image. In addition to PSNR, we also use SSIM as the other quality metric because the well known and commonly used PSNR can sometimes produce unreliable results. SSIM is defined as follows: (11) (12) (13)where and are the averages of the images X and Y, respectively; and are the variances of the images X and Y, respectively; and is the cross-covariance. and are usually constant, generally 0.01 and 0.03, respectively. L = 255 is for 8 bit grey-scale images. 4.1 Experiments In the following, we will show that our algorithm is superior to the existing methods in subjective visual effect and objective quality assessment. For those images with few misclassified points, our method performs better than the existing algorithms: MF, ACWMF, DBA, NAFSM, FDBA, DBUTMF, NTF, AWMF, EWAF and DAMF. We take lena, baboon and boat as examples, and the experimental results are listed in Tables 1 and 2. Our method outperforms the state-of-the-art methods in PSNR and SSIM. Fig. 7 compares the edge details of the restored images by various methods from 'lena' corrupted by 80% salt and pepper noise. The edges in images are well-preserved by our method. Table 1. PSNR of the denoised images of 'lena', 'baboon' and 'boat' by different methods Image Algorithms 10% 20% 30% 40% 50% 60% 70% 80% 90% lena MF 28.18 26.21 25.21 24.27 23.07 21.57 17.72 12.57 8.04 ACWMF 39.26 30.35 23.78 18.67 14.85 18.67 9.62 7.77 6.24 DBA 41.27 37.25 34.40 32.02 29.80 27.58 25.24 22.54 19.17 NAFSM 42.66 38.85 36.32 34.35 32.95 31.37 29.59 27.56 23.49 FBDA 43.34 38.67 35.57 33.52 31.48 29.11 25.78 20.14 16.38 DBUTMF 43.64 39.68 37.12 34.53 31.71 29.82 26.10 25.43 24.65 NTF 43.51 40.81 38.56 36.74 35.03 33.39 31.73 29.51 26.91 AWMF 39.01 37.37 36.03 34.85 33.64 32.19 30.53 28.52 26.21 DAMF 43.17 39.24 36.74 34.92 33.36 31.77 30.23 28.48 25.96 EWAF 42.63 39.27 37.34 35.72 34.33 32.93 31.37 29.65 27.01 our method (50 epoch) 46.72 43.44 41.47 39.83 38.18 36.70 34.83 32.22 29.36 baboon MF 20.25 19.96 19.50 19.13 18.65 17.33 15.38 11.09 7.08 ACWMF 27.31 24.36 20.65 16.81 13.53 10.77 8.58 6.76 5.32 DBA 28.75 26.59 24.96 23.51 22.20 21.01 19.72 18.36 16.83 NAFSM 31.95 28.59 26.65 25.08 23.80 22.57 21.42 20.18 18.29 FBDA 32.69 29.22 26.87 25.41 24.11 22.91 21.40 19.41 16.61 DBUTMF 33.11 29.83 27.21 25.73 24.57 23.61 21.69 20.69 20.33 NTF 32.91 30.16 27.94 26.55 25.43 24.39 23.39 22.45 21.29 AWMF 31.50 29.59 28.06 26.80 25.56 24.43 23.24 21.98 20.48 DAMF 33.51 30.14 28.14 26.61 25.31 24.09 23.02 21.83 20.38 EWAF 33.68 30.46 28.55 27.15 25.89 24.78 23.67 22.41 20.95 our method (50 epoch) 37.46 33.88 31.71 29.87 28.27 26.84 25.41 23.74 22.20 boat MF 24.92 24.18 23.41 22.51 21.85 20.47 17.37 12.69 8.46 ACWMF 34.03 28.42 33.49 18.80 15.11 12.23 9.96 8.14 6.67 DBA 37.31 33.59 31.38 29.32 27.24 25.46 23.5 21.22 18.61 NAFSM 38.90 35.18 33.03 31.24 29.59 28.27 26.94 25.20 22.22 FBDA 38.64 35.01 31.96 30.15 28.23 26.08 22.21 18.16 14.32 DBUTMF 39.33 35.67 33.42 31.09 29.8 28.59 26.88 25.65 23.67 NTF 41.48 38.03 36.74 34.15 32.57 30.93 29.03 27.01 24.66 AWMF 35.32 33.78 32.50 31.13 29.93 28.61 27.25 25.54 23.40 DAMF 39.06 35.49 33.00 31.15 29.66 28.27 26.96 25.38 23.21 EWAF 38.33 35.20 33.39 31.83 30.54 29.46 27.88 26.22 24.29 our method (50 epoch) 42.95 39.72 37.65 36.18 34.62 32.95 31.37 28.97 26.30 Bold values indicate the larger PSNR/SSIM is, the better the denoising result is, and the closer the denoised image is to the original image. Table 2. SSIM of the denoised images of 'lena', 'baboon' and 'boat' by different methods Image Algorithms 10% 20% 30% 40% 50% 60% 70% 80% 90% lena MF 0.8149 0.8098 0.8025 0.7925 0.7752 0.7187 0.5297 0.2031 0.0346 ACWMF 0.9858 0.9252 0.7386 0.4717 0.2397 0.1133 0.0547 0.0268 0.0128 DBA 0.9889 0.9742 0.9550 0.9313 0.9000 0.8545 0.7922 0.7085 0.5764 NAFSM 0.9903 0.9787 0.9648 0.9494 0.9301 0.9062 0.8753 0.8258 0.7003 AWMF 0.9821 0.9738 0.9630 0.9502 0.9343 0.9128 0.8833 0.8398 0.7720 DAMF 0.9904 0.9789 0.9652 0.9498 0.9314 0.9078 0.8769 0.8359 0.7633 EWAF 0.9904 0.9799 0.9683 0.9554 0.9397 0.9209 0.8964 0.8602 0.7988 our method (50 epoch) 0.9923 0.9848 0.9766 0.9680 0.9571 0.9443 0.9271 0.8950 0.8482 baboon MF 0.3848 0.3810 0.3776 0.3723 0.3593 0.3331 0.2544 0.1113 0.0288 ACWMF 0.9188 0.8542 0.7194 0.5191 0.3138 0.1686 0.0857 0.0417 0.0173 DBA 0.9675 0.9285 0.8798 0.8226 0.7551 0.6694 0.5700 0.4514 0.3097 NAFSM 0.9706 0.9375 0.8987 0.8538 0.8034 0.7424 0.6708 0.5742 0.4220 AWMF 0.9571 0.9314 0.9002 0.8625 0.8174 0.7628 0.6921 0.6039 0.4744 DAMF 0.9722 0.9393 0.9025 0.8601 0.8118 0.7518 0.6860 0.5971 0.4682 EWAF 0.9737 0.9435 0.9096 0.8720 0.8267 0.7756 0.7091 0.6228 0.4977 our method (50 epoch) 0.9857 0.9680 0.9471 0.9204 0.8872 0.8415 0.7789 0.6762 0.5391 boat MF 0.6597 0.6549 0.6497 0.6374 0.6206 0.5755 0.4213 0.1729 0.0351 ACWMF 0.9681 0.9014 0.7415 0.4956 0.2632 0.1332 0.0646 0.0329 0.0152 DBA 0.9803 0.9571 0.9270 0.8886 0.8378 0.7798 0.6928 0.5893 0.4482 NAFSM 0.9822 0.9622 0.9400 0.9128 0.8811 0.8450 0.7981 0.7301 0.5976 AWMF 0.9707 0.9559 0.9384 0.9155 0.8886 0.8531 0.8089 0.7465 0.6494 DAMF 0.9722 0.9393 0.9025 0.8601 0.8118 0.7518 0.686 0.5971 0.4682 EWAF 0.9831 0.9651 0.9456 0.9230 0.8969 0.8691 0.8249 0.7700 0.6834 our method (50 epoch) 0.9870 0.9736 0.9593 0.9455 0.9276 0.9057 0.8816 0.8319 0.7521 Bold values indicate the larger PSNR/SSIM is, the better the denoising result is, and the closer the denoised image is to the original image. Fig. 7Open in figure viewerPowerPoint Visual comparison of the edge detail restored from 'lena' corrupted by 80% salt and pepper noise For those images with lots of misclassified points, our method behaves competitively. We took bridge as an example, which has a large number of interference points: 1416 px with original value 0 and 410 px with original value 255. Fig. 8 gives the visual comparison of the denoised images by different methods. As shown in Fig. 9, when the noise density is larger than 30%, Denoiser (0.90) can achieve the best quantitative index among the existing methods. Although Denoiser (0.90) shows adaptability for the noise of density from 30 to 90%, we cannot get the best results at density 10–30%. To get the good denoised results for those images which have a large quantity of black and white points of their own, and are contaminated by salt and pepper noise of low density 10–30%, Denoiser (0.10) is selected to denoise them, as described in Algorithm 1(see Fig. 6). Table 3 shows excellent effects. Table 3. Comparison of PSNR and SSIM of the denoised results by different methods for image 'bridge' contaminated by different densities of salt and pepper noise Bridge Algorithms 10% 20% 30% 40% 50% 60% 70% 80% 90% PSNR MF 22.54 21.98 21.55 21.02 20.43 19.29 16.64 12.30 8.14 ACWMF 30.59 26.55 22.16 18.03 14.58 11.83 9.66 7.89 6.39 DBA 33.10 29.97 27.98 26.13 24.45 22.66 21.20 19.19 16.63 NAFSM 34.24 31.00 28.88 27.37 25.94 24.45 23.3 21.93 19.52 AWMF 31.09 29.82 28.75 27.62 26.47 25.36 24.09 22.73 20.82 DAMF 33.00 30.34 28.43 26.97 25.60 24.51 23.31 22.12 20.22 EWAF 34.06 31.46 29.63 28.18 26.97 25.82 24.58 23.22 21.40 our method (50 epoch) 35.92 33.10 29.96 28.3 27.40 26.47 25.5 24.16 22.47 SSIM MF 0.4946 0.4884 0.4818 0.471 0.4528 0.4195 0.3184 0.1542 0.0364 ACWMF 0.9478 0.8806 0.7458 0.5359 0.3234 0.1770 0.0904 0.0442 0.0165 DBA 0.9748 0.9425 0.9033 0.8517 0.7865 0.7056 0.6007 0.4719 0.3019 NAFSM 0.9766 0.9504 0.921 0.8661 0.8439 0.7931 0.7311 0.6444 0.4966 AWMF 0.9568 0.9386 0.9156 0.8859 0.8482 0.8016 0.7419 0.6592 0.5194 DAMF 0.9768 0.9498 0.9195 0.8851 0.8446 0.7949 0.7326 0.6499 0.5144 EWAF 0.9772 0.9542 0.9271 0.8969 0.8597 0.8183 0.7581 0.6806 0.5515 our method (50 epoch) 0.9855 0.9681 0.9447 0.9022 0.8771 0.8445 0.7974 0.7208 0.5982 Bold values indicate the larger PSNR/SSIM is, the better the denoising result is, and the closer the denoised image is to the original