Deep convolutional neural network for mixed random impulse and Gaussian noise reduction in digital images

IET Image ProcessingVolume 14, Issue 15 p. 3791-3801 Research ArticleFree Access Deep convolutional neural network for mixed random impulse and Gaussian noise reduction in digital images Mehdi Mafi, Corresponding Author Mehdi Mafi mmafi002@fiu.edu orcid.org/0000-0001-7223-2853 Center for Advanced Technology and Education, Department of Electrical and Computer Engineering, Florida International University, Miami, FL, USASearch for more papers by this authorWalter Izquierdo, Walter Izquierdo Center for Advanced Technology and Education, Department of Electrical and Computer Engineering, Florida International University, Miami, FL, USASearch for more papers by this authorHarold Martin, Harold Martin Center for Advanced Technology and Education, Department of Electrical and Computer Engineering, Florida International University, Miami, FL, USASearch for more papers by this authorMercedes Cabrerizo, Mercedes Cabrerizo Center for Advanced Technology and Education, Department of Electrical and Computer Engineering, Florida International University, Miami, FL, USASearch for more papers by this authorMalek Adjouadi, Malek Adjouadi Center for Advanced Technology and Education, Department of Electrical and Computer Engineering, Florida International University, Miami, FL, USASearch for more papers by this author Mehdi Mafi, Corresponding Author Mehdi Mafi mmafi002@fiu.edu orcid.org/0000-0001-7223-2853 Center for Advanced Technology and Education, Department of Electrical and Computer Engineering, Florida International University, Miami, FL, USASearch for more papers by this authorWalter Izquierdo, Walter Izquierdo Center for Advanced Technology and Education, Department of Electrical and Computer Engineering, Florida International University, Miami, FL, USASearch for more papers by this authorHarold Martin, Harold Martin Center for Advanced Technology and Education, Department of Electrical and Computer Engineering, Florida International University, Miami, FL, USASearch for more papers by this authorMercedes Cabrerizo, Mercedes Cabrerizo Center for Advanced Technology and Education, Department of Electrical and Computer Engineering, Florida International University, Miami, FL, USASearch for more papers by this authorMalek Adjouadi, Malek Adjouadi Center for Advanced Technology and Education, Department of Electrical and Computer Engineering, Florida International University, Miami, FL, USASearch for more papers by this author First published: 18 February 2021 https://doi.org/10.1049/iet-ipr.2019.0931Citations: 2AboutSectionsPDF 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 onFacebookTwitterLinked InRedditWechat Abstract This study utilises a deep convolutional neural network (CNN) implementing regularisation and batch normalisation for the removal of mixed, random, impulse, and Gaussian noise of various levels from digital images. This deep CNN achieves minimal loss of detail and yet yields an optimal estimation of structural metrics when dealing with both known and unknown noise mixtures. Moreover, a comprehensive comparison of denoising filters through the use of different structural metrics is provided to highlight the merits of the proposed approach. Optimal denoising results were obtained by using a 20-layer network with 40 × 40 patches trained on 400 180 × 180 images from the Berkeley segmentation data set (BSD) and tested on the BSD100 data set and an additional 12 images of general interest to the research community. The comparative results provide credence to the merits of the proposed filter and the comprehensive assessment of results highlights the novelty and performance of this CNN-based approach. 1 Introduction Noise remains a challenge in many image acquisition and transmission systems. The presence of noise, regardless of its nature or source of origin, often leads to image degradation that is as difficult to overcome as it is to formulate its underlying effects. This study focuses on images containing impulse noise (salt and pepper, and random value) and additive Gaussian noise. Impulse noise is caused by analogue-to-digital (A/D) converter saturation, transmission and/or memory errors, and faulty pixels in camera sensors [[1], [2]]. Gaussian noise, on the other hand, is caused by three common factors: amplifier noise, shot noise, and film grain noise [[3], [4]]. In order to remove or attenuate significantly these sources of noise, smoothing filters are often used to decrease the noise variance while attempting to preserve the important image details as much as possible. However, noise removal is further complicated when the images are degraded by mixtures of impulse and Gaussian noise, whose combination significantly alters the structural metrics of any given image. It thus becomes necessary to find a reliable process by which the challenging effects of such mixed noise could be attenuated. Therefore, a standard course of action is to perform a suitable smoothing technique to the image before some form of gradient could be applied to preserve the finer image details. Given that derivatives could amplify the effect of noise, a trade-off must be negotiated between the task of decreasing the noise variance and that of keeping all the relevant image details. This trade-off should consider the difficult challenge faced in any ensuing edge detection that needs to be performed on the resulting denoised image. Consequently, edge detection remains a difficult nontrivial problem and yet an essential preprocessing step for object identification, image segmentation, feature extraction, pattern recognition, and other relevant image processing tasks. When dealing with images, pertinent details can be useful for all types of real-world applications, but the concern has always been in delineating what really constitutes an edge with a high degree of similarity in contrast to other background pixels and noise data that could be misidentified as edges. The current literature provides two comprehensive reviews on the use of neural networks for image denoising [[5], [6]] along with several studies proposing different convolutional neural networks (CNNs) with various merits and different application domains such as hyperspectral image classification [[7], [8]], and Gaussian denoising [[9]-[12]]. While the authors of [[13]] produce other significant work on convolutional networks for eye fixation prediction by using a three-layer network with two convolutional layers and one fully-connected layer. Multi-label image classification is offered in [[14]] through an AlexNet-like network with two auxiliary layers and one additional input layer. Fine-grained image classification is discussed in [[15]] and multi-view multitask gaze estimation using n- and 1-fully connected layers is provided in [[16]]. Several impulse and Gaussian denoising methods have also been proposed, such as switching adaptive median and fixed weighted mean filter (SAMFWMF) [[17]], the proposed filter in [[18]] and CNN filters [[11], [12]]. Moreover, there are also several mixed impulse-Gaussian denoising methods, among them mixed noise filter (MNF) [[19]] (a combination of non-local mean [[20]] and bilateral filters [[21]]), adaptive median filter [[22]], total variation-based filters [[23], [24]], fuzzy-based filters [[25]-[30]], sparsity-based filter [[31]-[33]], low-rank approximation filter [[34], [35]], non-local similarity filter [[36]], maximum likelihood estimation filter (PARIGI method) [[37]], genetic programming filter [[38]], finite element filter [[39]], morphological-based filter [[40]], and the so-called framelet filter [[41]]. The methods uniquely used for either impulse or Gaussian denoising have shown good performance in the presence of the targeted noise, but such performance is degraded when the two noises are combined (i.e. present in a given image at the same time). Previously reported methods that have considered mixed impulse and Gaussian denoising filters are based on traditional methods that could not properly confront such problem when in the presence of high noise levels, leading to loss of image details and excessive blurring burdening further the edge detection process. For a fair assessment of the proposed method against others that have focused on mixed impulse and Gaussian noise under different intensity levels, a comparative study is provided contrasting the results obtained using the proposed deep learning algorithm against those from the most recent and effective denoising filters. In this comparative assessment, the focus is placed on the low-rank approximation algorithm (LSM-NLR) [[35]] which has proven to be effective for the removal of mixed impulse and Gaussian noise in images. The LSM-NLR method is based on Laplacian scale mixture (LSM) modelling and non-local low-rank regularisation. In order to model the impulse noise in LSM, a MAP estimator is defined by the authors in [[35]]. For non-local rank regularisation, a combination of the LSM model, the MAP estimator, and a low-rank regularisation model was used with the assumption that similar patches are interdependent, consequently resolving the denoising process by using an optimisation algorithm. The authors of this effective approach claim to obtain better result than the two-phase deblurring/denoising (TPD) method (or Cai1) [[42]], sparse and low-rank regularisation denoising (SLR) [[43]], the well-known BM3D [[44]], the non-locally centralised sparse representation [[45]], and the weighted encoding with sparse non-local regularisation (WESNR) [[33]]. It should be noted that the WESNR method does not have an impulse noise detection step due to its generated artefacts when in the presence of high noise levels. It defines the image as the product of sparse coding and a dictionary and in order to eliminate the noise, an optimal estimation of sparse code is calculated by encoding the noisy image over the dictionary. Because of the two different noise categories, the weight (close to 1 for pixels corrupted by Gaussian noise and smaller weights (w) for pixels corrupted by impulse noise) is assigned to residuals; and therefore, an optimal estimation for sparse coding is defined in the presence of mixed noise based on sparse regularisation. This method is claimed to outperform ROR-NLM [[46]], Cai [[47]], l 1 − l 0 [[31]], TF [[21]], and M + BM3D [[44]]. With all these challenges, the proposed study is motivated by three objectives: (i) To determine new ways for overcoming the persisting problems experienced by previously reported denoising methods in order to attenuate the effects of noise as much as possible while retaining more image details. (ii) To deblur the image in such a way as to yield an enhanced noise-free image were the absence of noise is visually appreciable. (iii) To preserve edge information and therefore yield sharper and continuous edge boundaries (one of the main aims of this study). Herein, a novel denoising filter is introduced to combine the ability to preserve more edge details with better prospects for achieving high structural similarity to the original (noise-free) image even in the presence of high levels of mixed impulse and Gaussian noise. The obtained results, as presented and discussed later, are contrasted to well-known denoising filters to gauge the performance of the proposed denoising method. 2 Proposed method The proposed method, as illustrated in Fig. 1, uses an end-to-end deep CNN to achieve optimal denoising of mixed impulse and Gaussian noise while, consequently, directly estimating the original noise-free image. Thereafter, batch normalisation is applied to speed up and improve the denoising process. Finally, the network is trained for both known and unknown noise levels. Known noise level is presumed when the training and testing data have similar noise levels. However, for unknown noise level, the network is trained based on a specific high noise intensity level and the testing data is assumed to have different levels of the noise intensity, which are presumed to be lower than those used in the training data. Fig. 1Open in figure viewerPowerPoint Network model 2.1 Noise models The normalised salt and pepper impulse noise model is expressed as follows: x c = X min Probability P p X max Probability P s X min < c < X min Probability 1 − P p − P s (1) In this type of representation, also used in [[48]], c denotes the uncorrupted value of a given pixel that could be affected by salt noise with probability Ps and by pepper noise with probability Pp. This normalised representation of the image assigns X min to be 0 and X max as 1. When using random value impulse noise as in [[49]], any given corrupted pixel has a random value between X min and X max . Gaussian noise is additive and independent. It can be the result of amplifiers noise, shot noise, film grain noise [[50]], and others. The model of Gaussian noise can thus be expressed as follows: x G i , j = x i , j + n i , j (2) where x G represents the noisy image containing Gaussian noise, x is the original noise-free image, and n is the additive Gaussian noise. 2.2 Evaluation metrics Standard structural metrics are computed to compare the performance of multiple filters against the proposed method and gauge the quality of the denoised image. The following are the metrics used in this study: Feature similarity index (FSIM) [[51]] measures the quality of the denoised image based on the human visual system (HVS). Peak signal-to-noise ratio (PSNR) measures the level of noise remaining in the denoised image. Equations (3)–(8) provide the formulations for FSIM and PSNR measures together with their respective related components. Hereafter, x i , j represent a pixel i , j in the original noise-free image, and y i , j represent a pixel i , j in the denoised image resulting from the filtering process. The FSIM between the noisy and the denoised images is measured as follows: FSIM = ∑ i ∈ Ω ⁡ S L i ⋅ P C m i ∑ i ∈ Ω P C m i (3) where Ω is a partial domain of the whole image with P C m i = max P C 1 i , P C 2 i , and where S L i is the similarity at the location i as given below: S L i = S PC i S G i (4) S PC i is the similarity measure between P C 1 and P C 2 as the extracted PC maps from the noisy and denoised images, respectively, as defined in the following equation: S PC i = 2 P C 1 i ⋅ P C 2 i + T 1 P C 1 i 2 + P C 2 i 2 + T 1 (5) where T 1 is a positive constant used to increase the stability of S PC i . S G i , defined in (6), represents the similarity measure between G 1 and G 2 representing the GM maps extracted from the noisy and denoised images, respectively, S G i = 2 G 1 i ⋅ G 2 i + T 2 G 1 i 2 + G 2 i 2 + T 2 (6) G M = G h 2 + G v 2 , where G h and G v are the partial derivatives in the horizontal and vertical directions. T 2 is a positive constant dependent on the dynamic range of GM. The PSNR measure is given as follows: PSNR = 10 log max x 2 MSE (7) where the MSE defines the mean square error as in (8) with max(x) expressing the maximum pixel intensity of image x. MSE = ∑ i = 0 M − 1 ∑ j = 0 N − 1 x i , j − y i , j 2 M × N (8) Here M and N define the x and y dimensions of the image. 2.3 Related work on the use of CNN for denoising images The following are some notable research endeavours that have used CNNs for image denoising. The work in [[52]] proposes a combination of sparse coding and deep neural networks, pre-trained with denoising auto-encoders (DAs), as an alternative to training pure DAs. The study in [[53]] uses a multilayer perceptron (MLP) for image denoising. While study in [[11]] presents a trainable non-linear reaction-diffusion (TRND) method and uses supervised training to construct a dynamic non-linear reaction-diffusion model with time-dependent parameters (linear filters and influence functions) for Gaussian denoising. This method in [[11]] is similar to the feedback CNN presented in [[54]], and in both cases, each iteration (stage) of the proposed diffusion process uses convolutional operations of a set of linear filters. The work in [[55]] introduced a super-resolution algorithm and CNN for mixed noise removal. The work in [[12]] introduces a deep CNN method (DnCNN) that can be seen as a generalisation of TRND [[11]] that (i) is easier to train, by replacing the influence function with a rectified linear units (ReLUs) [[56]]; (ii) increases architectural depth (number of convolution layers) to improve image modelling capacity; and (iii) incorporates batch normalisation [[57]] to improve performance. All the aforementioned networks are used solely for Gaussian denoising and all of them, except for [[12]], were used to remove known/predefined noise levels. As Gaussian noise is additive, the network in [[12]] removes the noise by using residual learning, i.e. learning the noise. However, when Gaussian and impulse noise are mixed the resulting interference is no longer additive and the network in [[12]] no longer provides adequate denoising results. It is for this reason, that the proposed approach did not use residual learning to directly estimate the denoised image. 2.4 Batch normalisation and network parameters During training, any change to a deep neural network layer's parameters causes a change in the distribution of the subsequent layer's input, referred to as an internal covariate shift. Batch normalisation [[57]] can alleviate internal covariate shift by learning the normalisation parameters of each part of the model and applying it to each mini-batch training. Empirically, it can be observed that the network's depth and patch size are dependent on the type and level of noise present. Specifically, larger patch sizes are shown to exhibit better performance in the presence of higher noise levels [[58]]. The network presented in [[11]] used ten convolutional layers (or five stages) with patches of 61 × 61 to remove a predefined level of Gaussian noise. The network in [[12]] uses 17 layers of 40 × 40 patches for known-noise-level denoising and 20 layers of 50 × 50 patches for unknown noise-level Gaussian denoising. 2.5 Network model The proposed CNN model is a modified version of the VGG [[59]] and DnCNN [[12]] models aimed at overcoming the challenge of mixed impulse and Gaussian noise removal. The network's input is a noisy image y i , produced by artificially injecting noise to a clean original image x i , and the network's output f y i is an estimate of the original noise-free image. The network's loss function is the summation of the squared error between the estimated and original noise-free images as formulated in (9). Finally, the network's parameters are updated by minimising this loss function as in [[6], [60]]. L = ∑ i = 1 N f y i − x i 2 2 (9) where N is the number of training image sets ( y i , x i ). As previously mentioned, the network used in this study resembles that used in [[12]] with few changes. It is composed of three different types of layers, where the first is a convolutional layer of 64 3 × 3 × 1 filters with ReLU non-linear activation functions [[9], [10], [13], [56]] used to create 64 feature maps, the second through second-to-last layers are batch normalised [[10], [57]] convolutional layers of 64 3 × 3 × 64 filters with ReLU activations [[56]]; and the last layer is composed of a single convolutional kernel of shape 3 × 3 × 64 used to output the reconstructed image. The use of ReLU activations [[56]] on convolutional layers separates the mixed noise from the noisy observations through the hidden layers. Finally, the input images are directly padded with zeros to reduce boundary artefact [[12]] resulting from size mismatches between different input images. Fig. 1 illustrates the proposed CNN-based network model. In contrast to [[12]], this network attempts to predict the noise-free image directly instead of obtaining it from subtracting the predicted residuals of the noisy image. This is a consequence of the non-additive nature of the noise types considered in this study. Although batch normalisation is used to prevent overfitting, additional steps are also considered to ensure the desired outcome. For example, every time the network starts a new epoch (a run through the training data) a new random seed is used to regenerate the noisy images. This extra step has proved to be a very helpful regulation technique as it prevents the network from seeing the same input image twice, or at least assign a very low probability to such event, allowing the network to better generalise. Through empirical evaluations, it is noted that optimal denoising results are obtained by using 20 layers with 40 × 40 patches for both known and unknown noise-level denoising. Although either, stochastic gradient descent (SGD) [[61]] and Adam gradient-based optimisation [[9], [62]] could have been used, previous CNNs [[7], [10], [12], [14]-[16]] have used SGD, and the performance of networks with and without batch normalisation for both SGD and Adam are assessed in [[12]]. This demonstrates that batch normalisation can significantly improve the PSNR for SGD by increasing the number of epochs. Therefore, in this study SGD [[61]] is used. Finally, all the source code developed and modified for this study is available at: 'https://github.com/wizquierdo/DnCNN' for other researchers to perform comparative assessments and explore any potential improvements. 3 Results In the implementation phase, 400 180 × 180 pixels images from the Berkeley segmentation dataset (BSD) are used to train the described network for both known and unknown noise-level removal, similar to [[11], [12], [35]]. While for testing, the data sets used include the BSD100 dataset and the additional 12 images shown in Fig. 2 not seen in the training. The study in [[35]] uses these same data sets. Fig. 2Open in figure viewerPowerPoint 12 test images considered As previously stated, the optimal results were obtained by using a 20-layer network with 40 × 40 patches for any noise-level. The SGD method is used with an additional learning rate of 0.1 (which decreased over subsequent epochs), weight decay of 0.0001, momentum of 0.09, and mini-batches of size 128 similar to studies reported in [[12]-[14], [16], [63]]. It should be noted that the type of noise mixture affected the numbers of epochs that the model needed to complete the training phase. All implementations were carried out in MATLAB 2017b using the MatConvNet package [[12], [20]] for CNNs on a PC with Nvidia Quadro M6000 GPU. The time required to train the network varied between 24 and 48 h depending on the noise mixture (as different mixtures required a different number of epochs). Once the network was trained, the results obtained from the proposed denoising method are compared to the results obtained from the WESNR [[33]] and the LSM-NLR [[35]] methods on different images and under the same mixed impulse and Gaussian noise intensities. Fig. 3 shows the comparative results when removing Gaussian noise with a standard deviation of 20 and 50% salt and pepper impulse noise from the test image 'Vase'. Fig. 4 shows the denoising comparison for Gaussian noise with a standard deviation of 20 and 30% random value impulse noise the from test image 'Flower'. Fig. 5 presents the performance in the presence of Gaussian noise with a standard deviation of 10, with 40% salt and pepper impulse noise, and 10% random value impulse noise from the test image 'Boat'. As seen from these figures, the proposed CNN attains better performance at preserving relevant image details than all other filters, with the highest similarity measure, and resulting in the least amount of noise residue. Moreover, these improvements lead to better edge tracking, especially when dealing with high-intensity mixtures of impulse and Gaussian noise. Fig. 3Open in figure viewerPowerPoint Denoising results of different filters on test image 'Vase' corrupted with Gaussian with standard deviation 20 and salt and pepper impulse noise with 50% (a) Original image, (b) Noisy image; images denoised by, (c) WESNR [[33]] (PSNR = 24.43 dB, FSIM = 0.9235), (d) LSM-NLR [[35]] (PSNR = 29.24 dB, FSIM = 0.9556), (e) Unknown noise-level proposed CNN (PSNR = 29.17 dB, FSIM = 0.9532), (f) Known noise-level proposed CNN (PSNR = 29.58 dB, FSIM = 0.9586) Fig. 4Open in figure viewerPowerPoint Denoising results of different filters on test image 'Flower' corrupted with Gaussian with standard deviation 20 and random value impulse noise with 30% (a) Original image, (b) Noisy image; images denoised by, (c) WESNR [[33]] (PSNR = 23.04 dB, FSIM = 0.8956), (d) l0-NLR [[35]] (PSNR = 23.51 dB, FSIM = 0.9071), (e) LSM–NLR [[35]] (PSNR = 24.36 dB, FSIM = 0.9156), (f) Unknown noise-level proposed CNN (PSNR = 27.07 dB, FSIM = 0.9482) Fig. 5Open in figure viewerPowerPoint Denoising results of different filters on test image 'Boat' corrupted with Gaussian with standard deviation 10, salt and pepper impulse noise with 40% and random value impulse noise with 10% (a) Original image, (b) Noisy image; images denoised by, (c) WESNR [[33]] (PSNR = 27.32 dB, FSIM = 92.75), (d) LSM-NLR [[35]] (PSNR = 28.89 dB, FSIM = 0.9482), (e) Unknown noise-level proposed CNN (PSNR = 30.97 dB, FSIM = 0.9620), (f) Known noise-level proposed CNN (PSNR = 30.82 dB, FSIM = 0.9646) Finally, Figs. 6-11 show the case the denoising results of the proposed CNN trained with varying degrees of known and unknown noise-levels in the presence of different intensity mixtures of Gaussian and impulse noises on the testing images. Fig. 6Open in figure viewerPowerPoint Denoising results of proposed CNN on test image 'Fruits' (a) Original test image, (b) Corrupted image with mixed Gaussian (s.d. = 10) and salt and pepper (50%), (c) Unknown noise-level denoising (PSNR = 33.28 dB, FSIM = 0.9736), (d) Known noise-level denoising (PSNR = 33.53 dB, FSIM = 0.9743) Fig. 7Open in figure viewerPowerPoint Denoising results of proposed CNN on test image 'Hill' (a) Original test image, (b) Corrupted image with mixed Gaussian (s.d. = 30) and salt and pepper (20%), (c) Unknown noise-level denoising (PSNR = 28.57 dB, FSIM = 0.9200), (d) Known noise-level denoising (PSNR = 28.77 dB, FSIM = 0.9236) Fig. 8Open in figure viewerPowerPoint Denoising results of proposed CNN on test image 'Boat' (a) Original test image, (b) Corrupted image with mixed Gaussian (s.d. = 30) and random value impulse noise (30%), (c) Unknown noise-level denoising (PSNR = 28.24 dB, FSIM = 0.9211) Fig. 9Open in figure viewerPowerPoint Denoising results of proposed CNN on test image 'Lena' (a) Original test image, (b) Corrupted image with mixed Gaussian (s.d. = 50) and random value impulse noise (50%), (c) Unknown noise-level denoising (PSNR = 27.31 dB, FSIM = 0.9107) Fig. 10Open in figure viewerPowerPoint Denoising results of proposed CNN on test image '385039 of BSD100 dataset' (a) Original test image, (b) Corrupted image with mixed Gaussian (s.d. = 10) and salt and pepper (40%) random value impulse noise (10%), (c) Unknown noise-level denoising (PSNR = 27.18 dB, FSIM = 0.8902), (d) Known noise-level denoising (PSNR = 27.09 dB, FSIM = 0.8962) Fig. 11Open in figure viewerPowerPoint Denoising results of proposed CNN on test image 'Couple' (a) Original test image, (b) Corrupted image with mixed Gaussian (s.d. = 20) and salt and pepper (10%) random value impulse noise (30%), (c) Unknown noise-level denoising (PSNR = 28.16 dB, FSIM = 0.9266), (d) Known noise-level denoising (PSNR = 28.55 dB, FSIM = 0.9303) Tables 1 and 2 show the results of known and unknown noise-levels removal for mixed Gaussian and impulse noise. They, respectively, show the results obtained from the average PSNR and the average FSIM [[51]] metrics from the 12 test images shown in Fig. 2 after the denoising process. Gaussian noise with a standard deviation of 10, 20, 30, and 50, and salt and pepper impulse noises with standard deviation 10, 20, 30, 40, and 50% were injected. Known noise-level denoising required 125 epochs while unknown noise-level denoising required 150; further increments of the number of training epochs yielded better results in both cases. The network was run 50 times for each noise level over the testing set and the means and standard deviations of the results were calculated. Table 1. PSNR (DB) comparison on 12 test images for unknown versus known noise-level denoising for mixed Gaussian and salt and pepper Gaussian noise Impulse noise, % WESNR [[33]] l1-NLR [[35]] l0-NLR [[35]] LSM-NLR [[35]] Proposed CNN (unknown) Proposed CNN (known) σ = 10 50 28.95 29.75 29.89 30.60 31.0823 ± 0.0095 31.4845 ± 0.0127 σ = 20 50 26.73 27.50 27.75 28.51 28.6711 ± 0.0096 28.9127 ± 0.0