1 Introduction
Image deblurring that aims at recovering a clear image from its blurry observation receives considerable research attention in decades. The blurry image is usually modeled as a convolution of clear image and blur kernel , i.e.,
(1) 
where denotes 2D convolution, and is additive noise. Thus, image deblurring is also well known as image deconvolution [Andrews and Hunt1977, Kundur and Hatzinakos1996]. When blur kernel is given, clear images can be recovered by deconvolution under the maximum a posterior (MAP) framework [Andrews and Hunt1977, Fergus et al.2006, Levin et al.2007, Krishnan and Fergus2009]:
(2) 
where is regularization term associated with image prior, and is a positive tradeoff parameter.
In conventional deconvolution methods, considerable research attention is paid on the study of regularization term for better describing natural image priors, including Total Variation (TV) [Wang et al.2008], hyperLaplacian [Krishnan and Fergus2009], dictionary sparsity [Zhang et al.2010, Hu, Huang, and Yang2010], nonlocal similarity [Dong, Shi, and Li2013], patchbased low rank prior [Ren et al.2016] and deep discriminative prior [Li et al.2018]. Note that alternative direction method of multipliers (ADMM) is often employed to efficiently solve these models. Besides, driven by the success of discriminative learning, the image priors can be learned from abundant training samples. With the halfquadratic splitting strategy, regression tree field [Jancsary et al.2012, Schmidt et al.2013] and shrinkage field [Schmidt and Roth2014] are proposed to model regularization term, and are effectively trained stagebystage. These learningbased methods have validated the superiority of discriminative learning over manually selected regularization term [Schmidt et al.2013, Schmidt and Roth2014].
Most recently, deep convolutional neural network (CNN), as a general approximator, has been successfully applied in low level vision tasks, e.g., image denoising [Zhang et al.2017], inpainting [Yang et al.2017], supperresolution [Dong et al.2014]. As for image deblurring, there are also several attempts, in which CNN is used to directly map blurry images to clear ones. In [Nah, Hyun Kim, and Mu Lee2017]
, a deep multiscale CNN is designed in image deblurring without explicit blur kernel estimation; as an upgrade, an recurrent unit is embedded into CNN such that multi scales share same CNN weight
[Tao et al.2018]. In [Kupyn et al.2018], a generative adversarial network (GAN) tries to train a ResNet [He et al.2016] supervised by the adversarial discriminator. However, these trained CNNbased models can only handle mildly blurry images, and usually fail in real cases, since practical blur trajectories are complex.When the blur kernel is known, CNNbased deconvolution has also been studied. On one hand, Xu et al xu2014deep. have validated that plain CNN cannot succeed in deconvolution. To make CNN work well for deconvolution, a specific blur kernel would be decomposed into inverse kernels [Xu, Tao, and Jia2014], which are then used to initialize CNN, inevitably limiting its practical applications. On the other hand, CNN is incorporated into conventional deconvolution algorithms under plugandplay strategy. In [Kruse, Rother, and Schmidt2017], CNN is employed to solve denoising subproblem in ADMM modulars. In [Zhang et al.2017], CNNbased Gaussian denoisers are trained offline, and are iteratively plugged under halfquadratic strategy. Although these methods empirically achieve satisfactory results, they are not trained endtoend, and some parameters need to be tuned for balancing CNN strength. As a summary, the effective CNN architecture for deconvolution still remains unsolved.
In this paper, we propose a novel concatenated residual convolutional network (CRCNet) for deconvolution. We first derive a closedform deconvolution solution driven by the minimum mean square error (MMSE)based discriminative learning. Then, using a power series expansion, we unfold MMSE solution into a sum of residual convolutions, which we name iterative residual deconvolution (IRD) algorithm. IRD is a very simple yet effective deconvolution scheme. As shown in Figure 2, the blur can be effectively removed with the increasing of iterations. Although IRD would magnify the noise in degraded image, the blur could still be significantly removed. Motivated by this observation, we design an effective CNN architecture for deconvolution, as shown in Figure 1. We adopt residual CNN unit to substitute the residual component in IRD. These residual CNN units are iteratively connected, and all intermediate outputs are concatenated and finally integrated, resulting in CRCNet. Interestingly, the developed nonlinear CRC model behaves efficient and robust. On test datasets, CRCNet can achieve higher quantitative metrics and recover more visually plausible texture details from compared with stateoftheart algorithms. We claim that effective CNN architecture plays the critical role in deconvolution, and CRCNet is one of the successful attempts. Our contributions are threefold:

We derive a closedform deconvolution solution driven by MMSEbased discriminative learning, and further unfold it into a seires, as a simple yet effective IRD algorithm.

Motivated by IRD algorithm, we propose a novel CRCNet for deconvolution. The CRCNet can be trained endtoend, but is not a plain CNN architecture and is well analyzed.

We discuss the contributions of CRCNet, and show the critical role of network architecture for deconvolution. Experimental results demonstrate the superiority of CRCNet over stateoftheart algorithms.
The reminder of this paper is organized as follows: Section 2 derives MMSEbased deconvolution, and then presents IRD algorithm. Section 3 designs CRCNet based on IRD, along with its training strategy. Section 4 demonstrates experimental results and Section 5 ends this paper with conclusions.
2 Iterative Residual Deconvolution
In this section, we first derive a deconvolution solution driven by minimum mean square error (MMSE) [Andrews and Hunt1977], which is then unfolded via series expansion, resulting in iterative residual deconvolution (IRD) algorithm. Finally we give an insightful analysis to IRD, and provide a potential CNN architecture for deconvolution.
2.1 MMSE Deconvolution
The convolution operation in Eqn. (1
) can be equivalently reformatted as a linear transform
(3) 
where is a Blocked Toeplitz Matrix [Andrews and Hunt1977] and
represents the columnwise expansion vector of matrix
. Then we aim to seek a linear transform to recover clear image(4) 
Let us assume a set of training image pairs . By minimizing MSE loss, we have
(5)  
where and are Gramm matrices of clear images and noises, respectively. represents the correlation between the th pixel and the th pixel of a sharp image and is similar.
On one hand, correlations among pixels of natural images are limited [Hu, Xue, and Zheng2012]
, thus eigenvalues of
can be deemed to be positive. Then, for any possible , we can always find an such that where represents the greatest eigenvalue. Hence, we have(6)  
where and .
On the other hand, is deemed to be zeromeaned gaussian noise with strength (assumed small in this work), thus approximates to . Hence, can be approximated as
(7) 
To now, we have obtained a closedform solution for deconvolution.
2.2 IRD Algorithm
For matrix with as , the following series expansion holds:
(8) 
As for the case in deconvolution, blur kernel is under two constraints [Kundur and Hatzinakos1996, Levin et al.2009, Perrone and Favaro2014]:
(9) 
and
(10) 
Under such constraints, the norm of degradation matrix is limited under 1. We also found empirically that eigenvalues of are generally positive.
Eqn. (11) can be implemented as an iterative algorithm. Matrix multiplications by and are equal to convolutions with and the flipped kernel .^{1}^{1}1The flip operation corresponds to in MatLab. The correlation matrix actually plays as the prior of clear images and is assured to be Toeplitz [Andrews and Hunt1977]. Hence, linear transform is equivalent to a convolution with a limited patch for pixels of clear images are correlated only in the vicinity. By assuming pixels in a clear image are independent [Hu, Xue, and Zheng2012, Ren et al.2018],
can be simplified as identity matrix
, i.e., . The detailed process is summarized as IRD Algorithm 1.The IRD algorithm is very simple yet effective for deconvolution. Figure 2 shows that the clear image can be satisfactorily restored from a noisefree blurry one after 1000 iterations. Although the noise is significantly magnified for a noisy blurry image, the blur can also be effectively removed.
To explore the significance of unfolded components, we extracted as shown in Figure 3. The energy of iterative residues attenuates but the component represents more detailed signals in higher frequency with increasing . Each iteration extracts residual information from the result of the previous iteration, and those components are finally summed to a clear image.
2.3 Comparison to Other Unfolded Methods
The proposed IRD is different from the existing unfolded algorithms. Previous unfolded methods focus on optimization to Eqn. (2). Specifically, ADMM introduces auxiliary variabel and augmented Lagrange multiplier into the original object function and optimizes each variable alternately. As another popular iterative deblur scheme, the accelerated proximal gradient (APG, also named as Fast Iterative Shrinkage/Thresholding Algorithm, FISTA) updates a dual variable to the proximal gradient mapping of (2), which significantly accelerates the convergence. A simplified L1regularized version of ADMM and APG is shown in Algorithm 2 and 3.^{2}^{2}2 is the soft shrinkage function at
In contrast, IRD reformulates the inverse process into residual convolutions and represents MMSE deconvolution as a sum of image components with gradually increasing frequency but lower energy. More interestingly, IRD provides a potential network structure for deconvolution. The iterative residual deconvolution pipeline reminds us the residual learning structure proposed by He et al. he2016deep. All convolutional parts in IRD can be learned as weights of a CNN. Such an analogy inspired us to propose the following network structure.
3 Concatenated Residual Convolutional Network
By imitating IRD algorithm, we designed a network as shown in Figure 1, which includes two main parts: the Iterative Residual Part and the Integrative Part, corresponding to and , respectively. For the first part, corresponds to the convdeconvminus structure of a Residual Unit. Considering that linear operator is symmetric , can be separated into . Note that operator and are equavalant to convolutions (see section 2), so their transposes correspond to transpose convolutions (also called deconvolutions in CNN). For the second, operator is implemented as conv layers on the concatenation of all residues with gradually decreased channels. Because a CNN manipulates convolutions channelwisely and sum the convolutions of all channels, this structure can sum all residues while adopting convolutions.
We take parametric rectified linear units (PReLU)
[He et al.2015] between conv or deconvlayers. The slope of PReLU on negative part is learned during backpropagation, which can be deemed as a nonlinear expansion to IRD.
Channel Expansion.
As the first step of CRCNet deblurring, the channel of input blurry image is mapped from 1 to through a 11 conv layer. The destination of channel expansion is to enhance the capability of conv/deconv weights and hence to improve network’s flexibility.
Residual Unit.
A Residual Unit (RU) calculates the difference between the input and the processed image to extract valid information. Formally,
(12) 
Compared to the Eqn. (11), in each RU, convolutional and deconvolutional (transpose convolutional) layers resemble . However, the autoencoderlike network can realize more complicated transforms by taking advantage of nonlinearity layers. Further, the weights of convolutional layers are learned from not only the blur but also clear images. Hence, an RU can extract information of images more efficiently.
Iterative Residual Part.
The intermediate output of a Residual Unit is fed to the next iteratively. As shown in Figure 1, represents the th RU, and is the output of . Formally,
(13) 
where is extended blurry input .
Integration.
The last part of our network is to integrate all extracted information from the blurry image. The input and all intermediate residues are concatenated and fed into an Integrative Unit (IU). IU takes three conv layers to play the role of and the channel dimension decreases gradually to 1 through convolutions as a weighted sum of unfolded components.
Loss Function.
An ideal deblurring model is expected to restore sufficient content of clear image and make the restored
looks sharp. Thus, the loss function of our network is designed to consist of a
content loss and an edge loss:(14) 
where is the smooth loss [Girshick2015]:
(15) 
which is more robust on outliers than MSE loss, and
(16) 
in which and represent horizential and vertical differential operator.
The edge loss constrains edges of to be close to those of . Our experiment showed that adding could speed up the convergence of the network efficiently and make restored edges sharp (See Figure 5).
4 Experimental Results
4.1 Training CRCNet
4.1.1 Training Dataset Preparation
Clear Image Set. Clear images are essential to train network weights. The dataset is expected to only consist of uniformly sharp and noisefree images with ample textures. We manually selected and clipped 860 RGB images from BSD500 [Martin et al.2001] and COCO [Lin et al.2014] dataset, during which we omitted all pictures with Bokeh Effect or motion blur.
Degradation Kernels. We randomly generated 10 degradation kernels by using code from [Chakrabarti2016] for training and testing. The generated kernels are shown in Figure 6.
Blurry images  HL  CSF  IRCNN  FDN  CRCNet 
4.1.2 Training details
We cropped training images into patches and used Adam [Kingma and Ba2014] optimizer with a minibatch of size 32 for training. The initial learning rate is and decay 0.8 per 1000 iterations. The network was only traine 20K iterations for each blur kernel to keep this process portable. In our experiment and . Ten clear images with ample details were selected for tests and the rest 850 were used for training.
In our experiments, expanded channel . Kernel sizes of and of each RU are . We take in iterative residual part and the dimension of the final concatenation before integration is 101.
The CRCNet was implemented in Python with PyTorch and tested on a computer with GeForce GTX 1080 Ti GPU and Intel Core i76700K CPU.
4.2 Comparison to states of the art
4.2.1 Test on synthetic blurry images.
Quantitative evaluations of average PSNR and SSIMof 10 test images with 10 kernels are shown in Table 1. Several test results are shown in Figure 7. Compared with stateofart approaches including traditional MAP method using HyperLaplacian priors (HL) [Krishnan and Fergus2009], learningbased method CSF [Schmidt and Roth2014] and CNNbased methods IRCNN [Zhang et al.2017] and FDN [Kruse, Rother, and Schmidt2017], CRCNet recovers more details in restored images, e.g., fur of Teddy bears and bright spray. Thus deblurred results of our method look more natural and vivid.
Specifically among contrasts, Schmidt and Roth schmidt2014shrinkage substitute the shrinkage mapping of halfquadratic (similar to ADMM but without the Lagrange multiplier) into a learned function constituted by multiple Gaussians, which could be deemed as a learning expansion to gradientbased unfolded method. CRCNet, as a derivative from IRD, achieves better performance. The current literature lacks an learningexpanded deblurring method of APG, thus we don’t list relative methods into comparison.
We also tested our network on benchmark Levin’s set [Levin et al.2009], which concludes 4 test images 8 kernels. The quantitative results are shown in Table 1 and visual comparisons are in Figure 8. Our approach achieves higher performance both visually and quantitatively.
4.2.2 Test on real blurry images.
We test proposed CRCNet and stateoftheart methods on realworld blurry images. These blurry images are produced by superposing 16 adjointing frames in motion captured using GoPro Hero 6. Blur kernels are estimated by [Zuo et al.2015]. Figure 10 shows that previous methods result in strong ringing effect and hence lower the image quality. In contrast, CRCNet remains plausible visual details while avoiding artifacts. We also take a quantitative perceptual scores proposed in [Ma et al.2017] on all methods; CRCNet obtains the highest (see Table 3).
Test set  Levin  Proposed 

HL  30.20 / 0.90  23.56 / 0.77 
CSF  33.53 / 0.93  27.24 / 0.85 
IRCNN  34.68 / 0.94  27.28 / 0.84 
FDN  35.08 / 0.96  29.37 / 0.89 
CRCNet  35.39 / 0.96  29.83 / 0.92 
Method  DCNN  CRCNet 

PSNR / SSIM  26.77 / 0.85  27.05 / 0.86 
param. amount  15M  0.4M 
Blurry  HL  CSF  IRCNN  FDN  CRCNet 
Blurry  DCNN  CRCNet 
4.2.3 Comparison with DCNN.
In the last part of experiments, we compare our method with previous nonblind deconvolution network DCNN on accompanying dataset in [Xu et al.2014] (see Table 2 and Figure 9). This comparison is listed seperatedly for only test code and trained weights on disk7 of DCNN are published. In this experiment, kernel is limitted as uniform disk of 7pixel radius and blurry images are extra degraded by saturation and lossy compression. We also list weight amounts of both networks. CRCNet obtains higher performance while taking much less network parameters. Further, DCNN requires specific initializations while CRCNet can be trained directly in endtoend way.
The implementation of this work and the clear image set are published at https://github.com/lisiyaoATbnu/crcnet.
Blurry images  HL  CSF  IRCNN  FDN  CRCNet 
Test image  lantern  model  chair 

Blurry  3.3  3.2  3.5 
HL  8.6  7.4  8.3 
CSF  8.0  6.4  7.7 
IRCNN  8.3  6.8  5.1 
FDN  8.4  6.3  6.3 
CRCNet  8.7  8.5  8.7 
4.3 Analysis to CRCNet
A question beyond the superior performance is whether the effectiveness of CRCNet depends on our proposed concatenated residual (CR) architecture or just a trivial ‘universal approximator’ relying on neural networks. To verify the contribution of CR structure, we give a discussion on relationship between IRD and CRCNet.
CRCNet plays a sequential nonlinear expansion of the iterative structure of IRD. Specifically, CRCNet realizes iteretive residues by several learned isolated conv/deconv layers rather than thausands of iterations using fixed shared weights and in IRD (see Figure 4). This iterativetosequential expansion enhances the flexibility and capacity of original method. In IRD algorithm, a large number of iterations are required for satisfactory deblurring quality; but in CRCNet, due to the powerful modeling capability of CNN, a very small number of layers can provide good restoration quality.
The iterative residual structure drives CRCNet to proccess images like IRD. To illustrate this point, we visualized intermediate outputs . Figure 11 shows that deep outputs of CRCNet contain highfrequency oscilations along edges in the image. That fact actually resembles IRD algorithm extracting highfreqency details after large amounts of iterations, as shown in Figure 3.
We in this paper claim that deep CNNbased model for deconvolution should be equipped with specific architecture instead of plain CNN, and our proposed CRCNet is one of the potential effective architectures.
5 Conclusions
In this paper, we proposed an effective deep architecture for deconvolution. By deriving the MMSEbased deconvolution solution, we first proposed an iterative residual deconvolution algorithm, which is simple yet effective. We further designed a concatenated residual convolutional network with the basic architecture of IRD algorithm. The restored results by CRCNet are more visually plausible compared with competing algorithms. The success of CRCNet shows that deep CNNbased restoration architecture should borrow ideas from conventional methods. In the future, we will develop more effective deep CNNbased restoration methods for other low level vision tasks.
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