Blind deblurring and denoising via a learning deep CNN denoiser prior and an adaptive <i>L</i> <sub>0</sub> ‐regularised gradient prior for passive millimetre‐wave images

IET Image ProcessingVolume 14, Issue 17 p. 4774-4784 Research ArticleOpen Access Blind deblurring and denoising via a learning deep CNN denoiser prior and an adaptive L0-regularised gradient prior for passive millimetre-wave images Dianjun Sun, Dianjun Sun orcid.org/0000-0001-9200-0320 School of Electrical and Information Engineering, Wuhan Institute of Technology, Wuhan, Hubei, 430074 People's Republic of China Hubei Key Laboratory of Optical Information and Pattern Recognition, Wuhan, Hubei, 430074 People's Republic of China Hubei Research Center of Video Image and High Definition Projection, Wuhan, Hubei, 430074 People's Republic of ChinaSearch for more papers by this authorYu Shi, Corresponding Author Yu Shi shiyu0125@wit.edu.cn School of Electrical and Information Engineering, Wuhan Institute of Technology, Wuhan, Hubei, 430074 People's Republic of China Hubei Key Laboratory of Optical Information and Pattern Recognition, Wuhan, Hubei, 430074 People's Republic of China Hubei Research Center of Video Image and High Definition Projection, Wuhan, Hubei, 430074 People's Republic of ChinaSearch for more papers by this authorYayuan Feng, Yayuan Feng School of Electrical and Information Engineering, Wuhan Institute of Technology, Wuhan, Hubei, 430074 People's Republic of China Hubei Key Laboratory of Optical Information and Pattern Recognition, Wuhan, Hubei, 430074 People's Republic of China Hubei Research Center of Video Image and High Definition Projection, Wuhan, Hubei, 430074 People's Republic of ChinaSearch for more papers by this author Dianjun Sun, Dianjun Sun orcid.org/0000-0001-9200-0320 School of Electrical and Information Engineering, Wuhan Institute of Technology, Wuhan, Hubei, 430074 People's Republic of China Hubei Key Laboratory of Optical Information and Pattern Recognition, Wuhan, Hubei, 430074 People's Republic of China Hubei Research Center of Video Image and High Definition Projection, Wuhan, Hubei, 430074 People's Republic of ChinaSearch for more papers by this authorYu Shi, Corresponding Author Yu Shi shiyu0125@wit.edu.cn School of Electrical and Information Engineering, Wuhan Institute of Technology, Wuhan, Hubei, 430074 People's Republic of China Hubei Key Laboratory of Optical Information and Pattern Recognition, Wuhan, Hubei, 430074 People's Republic of China Hubei Research Center of Video Image and High Definition Projection, Wuhan, Hubei, 430074 People's Republic of ChinaSearch for more papers by this authorYayuan Feng, Yayuan Feng School of Electrical and Information Engineering, Wuhan Institute of Technology, Wuhan, Hubei, 430074 People's Republic of China Hubei Key Laboratory of Optical Information and Pattern Recognition, Wuhan, Hubei, 430074 People's Republic of China Hubei Research Center of Video Image and High Definition Projection, Wuhan, Hubei, 430074 People's Republic of ChinaSearch for more papers by this author First published: 18 March 2021 https://doi.org/10.1049/iet-ipr.2020.1193AboutSectionsPDF 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 Passive millimetre-wave (PMMW) imaging frequently suffers from blurring and low resolution due to the long wavelengths. In addition, the observed images are inevitably disturbed by noise. Traditional image deblurring methods are sensitive to image noise, even a small amount of which will greatly reduce the quality of the point spread function (PSF) estimation. In this paper, we propose a blind deblurring and denoising method via a learning deep denoising convolutional neural networks (DnCNN) denoiser prior and an adaptive -regularized gradient prior for passive millimetre-wave images. First, a blind deblurring restoration model based on the DnCNN denoising prior constraint is established. Second, an adaptive -regularized gradient prior is incorporated into the model to estimate the latent clear image, and the PSF is estimated in the gradient domain. In a multi-scale framework, alternate iterative denoising and deblurring are used to obtain the final PSF estimation and noise estimation. Ultimately, the final clear image is restored by non-blind deconvolution. The experimental results show that the algorithm used in this paper not only has good detail recovery ability but is also more stable to different noise levels. The proposed method is superior to state-of-the-art methods in terms of both subjective measure and visual quality. 1 Introduction Passive millimetre-wave (PMMW) radiation is a potentially powerful tool for remote sensing detection. The waves lie between the infrared and microwave bands and have characteristics not found in those bands. On account of their long wavelength, passive millimetre waves can penetrate fog and heavy rain and are less affected by meteorological conditions. Due to these unique characteristics, PMMW imaging systems work around the clock and are of great significance in monitoring the earth's environment and exploring resources, as well as in transportation applications, military security and other applications [1]. In these applications, the main problem with PMMW imaging systems is poor spatial resolution. The spatial resolution is characterised by the wavelength and the size of the aperture of the antenna. Compared with infrared or optical systems, PMMW systems have a longer wavelength, thus resulting in lower image quality. There are many ways to improve the spatial resolution. For example, using a larger antenna to increase the aperture size or using a synthetic aperture with an antenna array may help improve resolution. However, due to the limitation of the platform and the requirements in many practical applications, the size of the antenna aperture is usually limited. Consequently, the acquired images become severely blurred, and the spatial resolution becomes low. A more economical method is to use image restoration or super-resolution technology to improve image resolution. The observed images are inevitably disturbed by the system and natural clutter noise, which greatly increases the difficulty of image restoration because the image deblurring method is very sensitive to noise, and even low-level system noise will greatly reduce the quality of the point spread function (PSF) estimation [2]. Therefore, determining how to economically estimate the PSF as accurately as possible to restore an image and improve the resolution of the image has become a hot topic. The restoration of PMMW images is very important for target detection, recognition and monitoring. Many image enhancement techniques have been developed [3-9] to improve the spatial resolution of PMMW images. In these methods, the resolution of PMMW images is usually improved by blind image deconvolution, which is a severely ill-posed inverse problem [10, 11]. Therefore, prior knowledge must be introduced to obtain a reasonable and stable solution. In [6], the authors assume that the PSF of the PMMW imaging system is a Gaussian model and then perform regularisation through scalar shrinkage in the Fourier and wavelet domain, thereby effectively solving the deconvolution problem. In [12], the authors used a blind image deconvolution algorithm with spatially adaptive total variation regularisation. The authors also used Gaussian blur to simulate the PSF of the PMWW imaging system. The spatial information is incorporated into the regularisation term, thereby effectively improving image recovery performance and the resolution of the image. In [7], the authors proposed a blind image deconvolution regularisation framework based on a compressed sensing imaging system; this framework solves the image compression problem through a series of unconstrained sub-problems. The Lp matrix norm is used as the regularisation term of the image. The smooth modelling can be considered to be a generalisation of Gaussian models (such as the models in [3, 9]) and Gauss-like models. However, in practical applications, the parameter models of the PMMW imaging system PSF are uncertain, so these methods of parameterising PSF models cannot solve practical problems well. Recently, many scholars have proposed image deconvolution methods for non-parametric PSF models [13, 14]. Levin et al. [13] believe that in the maximum a posteriori estimation blind deconvolution framework, the PSF is estimated, the latent image is alternately iterated, and the covariance of the latent image estimate is considered when the PSF is updated. In [14], Krishnan et al. performed image deconvolution in the sparse gradient domain. The method separately estimates the PSF in a multi-scale framework and then performs a non-blind deconvolution to restore the image, thereby providing a good idea for improving the resolution of PMMW images. In [15], Alexandru and Adrian proposed a PSF estimation technique based on adaptive filtering. The method begins with an estimation of the PSF obtained from two gyro sensors and uses a pair of underexposed images together with the blurred image to adaptively improve it. In [16], Joshi et al. proposed an algorithm that estimates non-parametric, spatially-varying blur functions at subpixel resolution from a single image. In [17] Agrawal and Xu proposed an efficient method to estimate the PSF of a blurred image using image gradients spatial correlation. These blind deconvolution methods can obtain good deblurring results, but the deblurring results are sensitive to noise interference, and in practical situations, PMMW imaging systems inevitably have noise interference, which greatly affects the accuracy of the PSF estimation. In PMMW imaging systems, there will always be some unavoidable system and natural clutter noise. These noises greatly affect the estimation of the PSF. Zhong [2] proposed applying directional filters with different directions to the input image, using a Radon transform to accurately estimate the PSF for each filtered image and finally, using a Radon inverse transform to reconstruct the PSF. This method effectively reduces the noise in the estimation of the PSF. However, this method is suitable for the restoration of motion-blurred images, while PMMW images with noise interference have a Gaussian-like blur. Fang and Shi [18] proposed an iteratively weighted blind deconvolution method that combines data fidelity and regularisation terms into a blind deconvolution framework and adds appropriate smoothing constraints. This method is effective for PMMW image restoration with low-level noise interference, but this method neither analyses the impact of high-level noise on PSF estimation nor considers PMMW image restoration with high-level noise interference. With the development of deep learning, many deep learning methods for image denoising have been introduced. Zhang et al. [19] proposed using residual learning and batch normalisation based on feedforward denoising convolutional neural networks (DnCNNs) to speed up training and improve the denoising performance. Tripathi and Bag [20] proposed a novel convolutional neural network (CNN), viz. CNN-DMRI, for denoising of magnetic resonance images (MRIs) scans. The proposed network can denoise MRI images effectively without losing crucial image details. In [21, 22], this framework generalises recent work on training neural nets from noisy images and on cross-validation for matrix factorisation. In [23, 24], neural network denoising achieved good results. However, for blurred and degraded images with noise interference, directly applying image denoising methods usually reduces edge or detail information, thereby leading to a biased PSF estimation [2]. Image denoising and image deblurring are two opposing issues. Image denoising blurs the edges of the image, while image deblurring enhances image edges. The direct application of image deblurring methods usually enhances the edges and makes the noise more noticeable. The direct application of image denoising methods usually destroys the blurred information in the input image, thereby leading to a biased PSF estimation. Neither of these two types of methods alone can achieve good experimental results. To solve the problems of low spatial resolution and noise interference in PMMW imaging systems, this paper presents a multi-scale PMMW image blind restoration method based on DnCNN denoising prior and an adaptive L0 constrained gradient prior. The multi-scale framework is used to estimate the latent image and the PSF, and the obtained PSF is used to obtain the final restored image through non-blind deconvolution. This multi-scale coarse-to-fine mechanism can avoid the local minimum caused by the blind deconvolution algorithm. The authors in [2, 4, 18] pointed out that in the presence of noise, image deconvolution is not a simple problem, and denoising is very important in blind deconvolution tasks. To better suppress the noise in PMMW images, a blind deblurring restoration model based on DnCNN denoising prior constraints is first established, a gradient prior constraint with an adaptive L0 constraint is used to estimate the latent clear image, and the PSF is estimated in the gradient domain to obtain better quality PSF estimates. Using the obtained initial PSF estimation, the non-blind deconvolution method is used to deconvolve the blurred noise image, and after upsampling, the latent image is substituted into the denoising network. Alternate iterative denoising and blurring are used to accurately estimate the PSF. Finally, the final restored image is restored by non-blind deconvolution. Experiments show that compared with current representative methods, the proposed method has better image restoration performance according to quantitative and qualitative evaluation indicators. The remaining part of this paper is organised as follows. The proposed method is introduced in Section 2. Extensive experimental results are reported in Section 3. Section 4 concludes this paper. The main contributions of this work are as follows: In this paper, we develop a new blind restoration method for PMMW images under low signal-to-noise ratio (SNR). Applying a learning deep CNN denoiser prior and an adaptive L0 regularised gradient prior to obtain denoising and deblurring results simultaneously. In order to improve denoising performance, we add a skip connection between the input and output, express the output as a linear superposition of a non-linear transformation of the input and output. An adaptive L0 regularised gradient prior is incorporated into the model to estimate the latent clear image, and the PSF is estimated in the gradient domain. The adaptive prior makes the PSF estimation more accurate. In a multi-scale framework, denoising and deblurring are iterated alternately to obtain the ultimately accurate PSF estimation and noise removal. 2 Proposed method Fig. 1 summarises the main steps of the method used in this paper. In a multi-scale framework, first, a blind deblurring restoration model based on DnCNN denoising prior constraints is established. Secondly, an adaptive L0-constrained gradient prior is introduced to estimate the latent image, and the PSF is estimated in the gradient domain to obtain a better PSF estimation. Using the obtained initial PSF estimation, the non-blind deconvolution method is used to deconvolve the blurred noise image, and after upsampling, the latent image is substituted into the denoising network. Alternate iterative denoising and blurring are used to accurately estimate the PSF. Finally, the final restored image is restored by non-blind deconvolution based on the estimated PSF. Fig. 1Open in figure viewerPowerPoint Flowchart of a blind restoration of a blurred and noisy image 2.1 Proposed model Generally, the degradation of PMMW imaging is modelled by: g = x ⊗ h + n(1) where g is the observed PMMW image, x is the high-resolution image, h is the PSF, n is the noise, and ⊗ is the convolution operator. Denoising and deblurring the PMMW degraded image involves estimating the high-resolution image x and PSF h from a single image. This estimation is an ill-posed inverse problem, for which there are infinite solutions. Generally, we impose a prior regularisation term to constrain the high-resolution image x and the PSF h. It is suggested that the minimisation problem of solving x and h is as follows: min x , h 1 2 σ 2 ∥ g − x ⊗ h ∥ 2 2 + ξ ⋅ Φ x + λ ∥ h ∥ 2 2 (2) where ( 1 / 2 σ 2 ) ∥ g − x ⊗ h ∥ 2 2 is a data fidelity term determined by the degraded model, σ is a weight parameter, Φ x (which is introduced here) is a regularisation term, ξ is a regularisation parameter, and λ is a weight parameter. The regularisation term Φ x is non-convex and indecomposable. To solve (2), the auxiliary variable z is introduced to obtain the following equivalent constraint optimisation formula: x , h = arg min x , h 1 2 σ 2 ∥ g − z ⊗ h ∥ 2 2 + ξ ⋅ Φ x + λ ∥ h ∥ 2 2 . (3) Then, the half-quadratic splitting (HQS) method tries to minimise the following cost function: Γ μ x , z , h = 1 2 σ 2 ∥ g − z ⊗ h ∥ 2 2 + ξ ⋅ Φ x + μ 2 ∥ z − x ∥ 2 + λ ∥ h ∥ 2 2 + R z (4) where μ is the penalty parameter and the fourth term R z is the L0-constrained regularisation term for the image gradient, which can be solved by the following iterative scheme: z , h = arg min z , h ⁡ ∥ g − z ⊗ h ∥ 2 2 + μ σ 2 ∥ z − x ∥ 2 + λ ∥ h ∥ 2 2 + R z x = arg min x μ 2 ∥ z − x ∥ 2 + ξ ∙ Φ x .(5) As we can see, the fidelity and regularisation terms are divided into two separate sub-problems. The first problem is the deblurring problem, and the second problem is the denoising problem. These two sub-problems can be solved independently, but unlike those combined methods of denoising and deblurring first, the solution to the second sub-problem is the data fidelity term of the first sub-problem, thereby making the experimental results approximate the actual results. 2.2 Gradient prior R z based on an adaptive L 0 constraint The prior constraint regularisation term R z of the variable z is key to solving the first sub-problem of (5). The sparseness of the clear image gradient is often used to constrain the variable z, where the gradient of the image z is denoted as ∇ z . In [25], since the number of non-zero elements of the clear image gradient is less than the number of non-zero elements of the degraded image gradient, the prior constraint of the L 0 norm of the image gradient is used as the regularisation term, denoted as M ∥ ∇ z ∥ 0 , where M is a positive parameter and is used to adjust the weight of the regularisation term penalty. If the parameter M is set too small, the image noise cannot be suppressed well, while if the parameter M is set too large, the image details cannot be well preserved. To solve this problem, we propose a gradient prior based on adaptive L 0 constraints. First, we use the edge indicator [26] of the difference eigenvalue D g to distinguish the edge from the flat area. The difference eigenvalue D g edge indicator is defined as D g = κ 0 ω 1 − ω 2 ω 1 (6) where κ 0 is a weighting factor, and ω 1 and ω 2 are two eigenvalues of the Hessian matrix of the image z, where ω 1 is the larger eigenvalue and ω 2 is the smaller eigenvalue. The eigenvalues of the image Hessian matrix characterise the concavity and convexity in the direction of the feature vector near the local area. The larger the eigenvalue is, the stronger the convexity. The difference eigenvalue D g in the edge area of the image is much larger than the D g value in the flat area. For the edge region of the image, we give a small penalty term to the prior constraint regularisation term of the L 0 norm of the image gradient to preserve image details. For the flat area of the image, we give a large penalty term to the prior constrained regularisation term of the L 0 norm of the image gradient to suppress image noise. Therefore, we introduce the gradient a priori ( γ / ( 1 + β D g ) ) ∥ ∇ z ∥ 0 with an adaptive L 0 constraint to the first problem of (5). The first sub-question of rewriting (5) is min z , h ⁡ ∥ z ⊗ h − g ∥ 2 2 + μ σ 2 ∥ z − x ∥ 2 + γ 1 + β D g ∥ ∇ z ∥ 0 (7) where ( γ / ( 1 + β D g ) ) ∥ ∇ z ∥ 0 is the regularisation term of image gradient and γ and β are weight parameters. For the L0 regularisation term in (7), we use HQS. The L0 minimisation method introduces the gradient P=(Ph, Pv)T, which corresponds to the latent clear image in the horizontal and vertical directions, to solve (7). Equation (7) can be rewritten as min z , h ⁡ ∥ z ⊗ h − g ∥ 2 2 + μ σ 2 ∥ z − x ∥ 2 + α ∥ ∇ ( z ) − P ∥ 2 2 + γ 1 + β D ( g ) ∥ P ∥ 0 (8) where α is the penalty parameter. When α is close to infinity, the solution of (8) is close to the solution of (7) [27]. We can minimise z and P by fixing other variables to solve (8) effectively. The value of P is initialised to zero. In each iteration, fixed variables x, h, P, and z are obtained by solving the following formula: min z ⁡ ∥ z ⊗ h − g ∥ 2 2 + μ σ 2 ∥ z − x ∥ 2 + α ∥ ∇ ( z ) − P ∥ 2 2 (9) Since (9) is a least-squares problem with an L0-norm constraint, the partial function of the objective function for the variable z is zero, and an analytical solution for z is obtained. To speed up the calculation, the fast Fourier transform (FFT) is used to transform the convolution in the spatial domain to the frequency domain. The solution of z is as follows: z = F − 1 F F h ¯ F g + ( γ / ( 1 + β D g ) ) F P + μ σ 2 F x F h ¯ F h + ( γ / ( 1 + β D g ) ) F ∇ ¯ f ∇ + μ σ 2 (10) where F ⋅ represents the FFT, F − 1 ⋅ represents the inverse Fourier transform, and F ⋅ ¯ represents a conjugate complex number of F ⋅ . Given z, Р can be expressed as min P ⁡ α ∥ ∇ ( z ) − P ∥ 2 2 + γ 1 + β D ( g ) ∥ P ∥ 0 (11) Equation (11) is a classic L0-norm minimisation problem [28], which can be solved pixel by pixel. P can be solved as P = ∇ ( z ) , | ∇ ( z ) | 2 ≥ γ α ( 1 + β D ( g ) ) , 0 otherwise , (12) Given z, the PSF estimation in (5) is a least-squares problem. It is usually more accurate to estimate the PSF in the gradient domain [29]. Therefore, we can estimate the PSF h by min h ⁡ ∥ ∇ ( z ) ⊗ h − ∇ g ∥ 2 2 + λ ∥ h ∥ 2 2 (13) We obtain the solution of (13) by using the FFT. After obtaining PSF h, the negative element of h is set to 0, and h is normalised so that its element sum is 1. These steps are taken so that h satisfies our definition of PSF. Similar to state-of-the-art methods, the PSF estimation is performed in a coarse-to-fine manner by using an image pyramid strategy. 2.3 Deep learning denoising prior The deep CNN designed in this paper learns an image denoising prior. The input is a noisy image y = x + n, and the latent clear image is predicted by learning the mapping function F y = x. The DnCNN trains the residual image R y ≃ n through the residual network and then obtains a clear image x = y − R y . The loss function of the network is set to: L Θ = 1 2 N ∑ i = 1 N ⁡ ∥ R y i ; Θ − y i − x i ∥ 2 2 (14) where y i − x i i = 1 N represents N noise-clear image pairs and Θ represents a trainable network parameter. The structure of the DnCNN is shown in Fig. 2. Fig. 2Open in figure viewerPowerPoint Architecture of the DnCNN network Network structure: the DnCNN has 17 layers, and there are three types of network layers, as shown in Fig. 2. (i) Conv + ReLU: For the first layer, 64 convolution kernels with a size of 3 × 3 × c are used to generate 64 feature maps, and then, rectified linear units (ReLUs) are used as the activation function. Here, c represents the number of image channels, that is, for a greyscale image c = 1 and for a colour image c = 3. (ii) Conv + BN + ReLU: Layers 2 to 16 use 64 convolution kernels with a size of 3 × 3 × 64, and batch normalisation is added between the convolution and activation functions. (iii) Conv: For the last layer, a 3 × 3 × 64 convolution kernel is used to reconstruct the output. The DnCNN model has two main features: It uses the residual learning formula to learn R(y), and it uses batch normalisation to speed up training and improve the denoising performance. Additionally, we add a skip connection between the input and output and express the output as a linear superposition of a non-linear transformation of the input and output; consequently, the network actually understands the difference between the input and the output. By combining convolution with the ReLU, the DnCNN can gradually separate image structure from the noisy observation through the hidden layers. This mechanism is similar to the iterative noise removal strategies used in methods such as Expected Patch Log Likelihood (EPLL) and Weighted Nuclear Norm Minimization (WNNM) [30], but DnCNN is trained in an end-to-end fashion. A series of feature maps is obtained from the noisy image through the convolution layer. These feature maps enter the next convolution layer to obtain deeper features through non-linear mapping and finally output a denoised image. To solve the second sub-problem of (5), rewrite the equation as follows: x = arg min x 1 2 1 / μ 2 ∥ z − x ∥ 2 + ξ ⋅ Φ x . (15) According to Bayesian probability, (15) is equivalent to denoising the image z through a Gaussian denoising filter with a noise level 1 / μ . To solve this problem, we rewrite (15) by x = DnCNN ( z , 1 / μ ) .(16) According to (15) and (16), the image regularisation term Φ x can be replaced with the DnCNN denoising prior term. There are several advantages to this replacement. First, the unknown image prior Φ x is difficult to solve when solving (15). Secondly, different image denoising priors can be used to solve a specific problem together. Due to (5), both sub-problems are relatively easy to solve. After multiple iterations, an accurate PSF and a denoised process map are obtained. The non-blind deconvolution is used to obtain a restored image. 3 Experimental results and discussion 3.1 Comparative experiments In this section, we will demonstrate the performance of the proposed algorithm through simulated and real blurred PMMW images with different noise levels. As shown in Fig. 3, the following eight test images were simulated: Cameraman, Satellite, Butterfly, Boat, Jet, House, Pepper, and Bridge. All test images are normalised to 0 to 1. We set σ = 1, λ = 2, μ = 0.04, γ = 0.02, β = 2, and α max = 10 5 . The noise level of denoiser 1 / μ should be set from large to small. In our experimental settings, the denoiser is decayed exponentially from 49 to a value in [1, 31], depending on the noise level. The number of iterations is set to 30, as we find that it is large enough to obtain a satisfactory performance. We add different levels of noise to the blurred image and use different methods to restore the blurred noise image. At the same time, in order to obtain the best comparison effect, we will also adjust the parameters of the three comparison methods to the best. The specific parameters are set as follows. In [14] Krishnan's method we set regularisation parameter λ = 20, maximum outer iterations M = 2, ISTA threshold t = 0.001. In [32] Levin's method we set the Laplacian prior α = 0.5, box filters of size λ = 1. In [18] Fang's method we set P = 2, c0 = 0.6, and ∈ = 0.0001. Fig. 3Open in figure viewerPowerPoint Eight test images for simulation. Top (from left to right): Cameraman, Satellite and Butterfly. Bottom (from left to right): Jet, House and Pepper Training and testing data: For Gaussian denoising with either known or unknown noise level, we follow [33] to use 400 images of size 180 × 180 for training. To train DnCNN for Gaussian denoising with the known noise level, we consider three noise levels, i.e. 0.007, 0.015, and 0.04. We set the patch size as 40 × 40, and crop 128 × 1, 600 patches to train the model. We use stochastic gradient descent (SGD) with weight decay of 0.0001, a momentum of 0.9 and a mini-batch size of 128. We train 50 epochs for DnCNN models. The learning rate was decayed exponentially from 1×10−1 to 1×10−4 for the 50 epochs. All the experiments are carried out in the Spyder (Python 3.6) environment running on a PC with Intel(R) Core (TM) i7-8700 Central Processing Unit (CPU) 3.00 GHz and an Nvidia GeForce GTX 1080 Graphics Processing Unit (GPU). We compare the proposed algorithm with the methods proposed by Krishnan et al. [14], Levin et al. [32] and Fang and Shi [18]; both of these methods are implemented within a multi-scale framework. The algorithm proposed by Levin et al. [32] is a very representative method based on variational Bayesian blind image deconvolution. Without adjusting parameters, this method is regarded as the benchmark method for experimental comparison. The method proposed by Krishnan et al. [14] uses a normalised sparseness measurement image and sparse measurement of PSF; this use