1 Introduction
Single Image SuperResolution (SISR), which aims to restore a visually pleasing HighResolution (HR) image from its LowResolution (LR) version, is still a challenging task within computer vision research community [34, 36]. Since multiple solutions exist for the mapping from LR to HR space, SISR is highly illposed. To regularize the solution of SISR, various priors of natural images have been exploited, especially the current leading learningbased methods [39, 6, 22, 15, 16, 33, 32, 21, 1, 11, 20, 50] are proposed to directly learn the nonlinear LRHR mapping.
By modeling the sparse prior in natural images, the Sparse Coding (SC) based methods for SR [46, 47, 44] with strong theoretical support are widely used owing to their excellent performance. Considering the complexity in images, these methods divide the image into overlapping patches and aim to jointly train two overcomplete dictionaries for LR/HR patches. There are usually three steps in these methods’ framework. First, overlapping patches are extracted from input image. Then to reconstruct the HR patch, the sparse representation of LR patch can be applied to the HR dictionary with the assumption that LR/HR patch pair shares similar sparse representation. The final HR image is produced by aggregating the recovered HR patches.
Recently, with the development of Deep Learning (DL), many researchers attempt to combine the advantages of DL and SC for image SR. Dong
[6] firstly proposed the seminal CNN model for SR termed as SRCNN, which exploits a shallow convolutional neural network to learn a nonlinear LRHR mapping in an endtoend manner and dramatically overshadows conventional methods [47, 35]. However, sparse prior is ignored to a large extent in SRCNN for it adopts a generic architecture without considering the domain expertise. To address this issue, Wang [39] implemented a Sparse Coding based Network (SCN) for image SR, by combining the merits of sparse coding and deep learning, which fully exploits the approximation of sparse coding learned from the LISTA [9] based subnetwork.It’s worth to note that most of SC based methods utilize the sparse prior locally [28], , coping with overlapping image patches. Thus the consistency of pixels in overlapped patches has been ignored [10, 28]. To address this issue, CSC is proposed to serve sparse prior as a global prior [48, 28, 27] and it furnishes a way to fill the localglobal gap by working directly on the entire image by convolution operation. Consequently, CSC has attained much attention from researchers [48, 4, 13, 10, 30, 8]. However, very few studies focus on the validation of CSC for image SR [10], resulting in no work been reported that CSC based image SR can achieve stateoftheart performance. Can CSC based image SR show highly competitive results with recent stateoftheart methods [6, 22, 15, 16, 33, 32, 37, 21, 20, 50]? To answer this question, the following issues need to be considered:
Framework Issue. Compared with SC based image SR methods [46, 47], the lack of a unified framework has hindered progress towards improving the performance of CSC based image SR.
Optimization Issue. The previous CSC based image SR method [10] contains several steps and they are optimized independently. Hundreds of iterations are required to solve the CSC problem in each step.
Memory Issue. To solve the CSC problem, ADMM [3] is commonly employed [4, 42, 13, 43, 8], where the whole training set needs to be loaded in memory. As a consequence, it is not applicable to improve the performance by enlarging the training set.
MultiScale Issue. Training a single model for multiple scales is difficult for the previous CSC based image SR method [10].
Based on these considerations, in this paper, we attempt to answer the aforementioned question. Specifically, we exploit the advantages of CSC and the powerful learning ability of deep learning to address image SR problem. Moreover, massive theoretical foundations for CSC [28, 27, 7] make our proposed architectures interpretable and also enable to theoretically analyze our SR performance. In the rest of this paper, we first introduce CISTA, which can be naturally implemented using CNN architectures for solving the CSC problem. Then we develop a framework for CSC based image SR, which can address the Framework Issue
. Subsequently, CRNetA (CSC and Residual learning based Network) and CRNetB inspired by this framework are proposed for image SR. They are classified as pre and postupsampling models
[40]respectively, as the former takes Interpolated LR (ILR) images as input while the latter processes LR images directly. By adopting CNN architectures,
Optimization Issue and Memory Issue would be mitigated to some extent. For MultiScale Issue, with the help of the recently introduced scale augmentation [15, 16] or scalespecific multipath learning [21, 40] strategies, both of our models are capable of handling multiscale SR problem effectively, and achieve favorable performance against stateofthearts, as shown in Fig. 1.The main contributions of this paper include:

[noitemsep,topsep=0pt]

We introduce CISTA, which can be naturally implemented using CNN architectures for solving the CSC problem.

A novel framework for CSC based image SR is developed. Two models, CRNetA and CRNetB, inspired by this framework are proposed for image SR.
2 Related Work
2.1 Sparse Coding for Image SuperResolution
Sparse coding has been widely used in a variety of applications [51]. As for SISR, Yang [46] proposed a representative Sparse coding based SuperResolution (ScSR) method. In the training stage, ScSR attempts to learn the LR/HR overcomplete dictionary pair / jointly by given a group of LR/HR training patch pairs /. In the test stage, the HR patch is reconstructed from its LR version by assuming they share the same sparse code. Specifically, the optimal sparse code is obainted by minimizing the following sparsityinducing norm regularized objective function
(1) 
and then the HR patch is obtained by
. Finally, the HR image can be estimated by aggregating all the reconstructed HR patches. Inspired by ScSR, many SC based SR methods have been proposed by using various constraints on sparse code and dictionary
[45, 38].2.2 Convolutional Sparse Coding for Image SuperResolution
Traditional SC based SR algorithms usually process images in a patch based manner to reduce the burden of modeling and computation, resulting in the inconsistency problem [28]. As a special case of SC, CSC is inherently suitable for this issue [48]. CSC is proposed to avoid the inconsistency problem by representing the whole image directly. Specifically, an image can be represented as the summation of feature maps convolved with the corresponding filters : , where is the convolution operation.
Gu [10] proposed the CSCSR method and revealed the potential of CSC for image SR. In [10], CSCSR requires to solve the following CSC based optimization problem in both the training and testing phase:
(2) 
[10] solves this problem by alternatively optimizing the and subproblems [42]. The subproblem is a standard CSC problem. Hundreds of iterations are required to solve the CSC problem and the aforementioned Optimization Issue and Memory Issue cannot be completely avoided. Inspired by the success of deep learning based sparse coding [9], we exploit the natural connection between CSC and CNN to solve the CSC problem efficiently.
3 CISTA for solving CSC problem
CSC can be considered as a special case of conventional SC, due to the fact that convolution operation can be replaced with matrix multiplication, so the objective function of CSC can be formulated as:
(3) 
are in vectorized form and
is a sparse convolution matrix with the following attributes:(4)  
where and are following the notations of Zeiler [48], representing that array is flipped in left/right or up/down direction.
Iterative Soft Thresholding Algorithm (ISTA) [5] can be utilized to solve (3), at the iteration:
(5) 
where is the Lipschitz constant, and . Using the relation in (4) to replace the matrix multiplication with convolution operator, we can reformulate (5) as:
(6) 
where
is the identity matrix,
and . Note that identity matrix is also a sparse convolution matrix, so according to (4), there existing a filter satisfies:(7) 
so (6) becomes:
(8) 
where and . Even though (3) is for a single image with one channel, the extension to multiple channels (for both image and filters) and multiple images is mathematically straightforward. Thus for representing images of size with channels, (8) is still true with and .
As for , [26] reveals two important facts: (1) the expressiveness of the sparsity inspired model is not affected even by restricting the coefficients to be nonnegative; (2) the [25]activation function and the soft nonnegative thresholding operator are equal, that is:
(9) 
We set for simplicity. So the final form of (8) is:
(10) 
One can see that (10) is a convolutional form of (5), so we name it as CISTA. It provides the solution of (3) with theoretical guarantees [5]. Furthermore, this convolutional form can be implemented employing CNN architectures. So and in (10) would be trainable.
4 Proposed Method
In this section, our framework for CSC based image SR is first introduced. And then we implement it using CNN techniques. Since most of image SR methods can be attributed to two frameworks with different upsampling strategies, , preupsampling and postupsampling, we propose two models, CRNetA for preupsampling and CRNetB for postupsampling.
4.1 The framework for CSC based Image SR
Analogy to sparse coding based SR, we develop a framework for CSC based image SR. As shown in Fig. 3, LR feature maps are extracted from the input LR image using the learned LR filters. Then convolutional sparse codes of LR feature maps are obtained using CISTA with LR dictionary and shared parameter , as indicated in (10). Under the assumption that HR feature maps share the same convolutional sparse codes with LR feature maps, HR feature maps can be recovered by . Finally, the HR image is reconstructed by utilizing the learned HR filters.
In this work, we implement this framework using CNN techniques. However, when combining CSC with CNN, the characteristics of CNN itself must be considered. With more recursions used in CISTA, the network becomes deeper and tends to be bothered by the gradient vanishing/exploding problems. Residual learning [15, 16, 33] is such a useful tool that not only mitigates these difficulties, but helps network converge faster. In Fig. 4, residual/nonresidual networks with different recursions are compared experimentally and the residual network converges much faster and achieves better performance. Based on these observations, both of our proposed models adopt residual learning.
4.2 CRNetA Model for Preupsampling
As shown in Fig. 5, CRNetA takes the ILR image with channels as input, and predicts the output HR image as . Two convolution layers, consisting of filters of spatial size and containing filters of spatial size
are utilized for hierarchical features extraction from ILR image:
(11) 
The ILR features are then fed into a CISTA block to learn the convolutional sparse codes. As stated in (10), two convolutional layers and are needed:
(12) 
where is initialized to . The convolutional sparse codes are learned after recursions with shared across every recursion. When the convolutional sparse codes are obtained, it is then passed through a convolution layer to recover the HR feature maps. The last convolution layer is used as HR filters:
(13) 
Note that we pad zeros before all convolution operations to keep all the feature maps to have the same size, which is a common strategy used in a variety of methods
[15, 16, 33]. So the residual image has the same size as the input ILR image , and the final HR image would be reconstructed by:(14) 
Given ILRHR image patch pairs as a training set, our goal is to minimize the following objective function:
(15) 
where
denotes the learnable parameters. The network is optimized using the minibatch Stochastic Gradient Descent (SGD) with backpropagation
[18].4.3 CRNetB Model for Postupsampling
We extend CRNetA to its postupsampling version to further mine its potential. Notice that most postupsampling models [19, 37, 20, 50] need to train and store many scaledependent models for various scales without fully using the interscale correlation, so we adopt the scalespecific multipath learning strategy [40] presented in MDSR [21] with minor modifications to address this issue. The complete model is shown in Fig. 6
. The main branch is our CRNetA module. The preprocessing modules are used for reducing the variance from input images of different scales and only one residual unit with
kernels is used in each of the preprocessing module. At the end of CRNetB, upsampling modules are used for multiscale reconstruction.5 Experimental Results
Dataset  Scale  Bicubic  SRCNN [6]  RED30 [22]  VDSR [15]  DRCN [16]  DRRN [33]  MemNet [32]  CRNetA (ours) 
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Set14  /  /  /  /  /  /  /  /  
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B100  /  /  /  /  /  /  /  /  
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Urban100  /  /  /  /  /  /  /  /  
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Dataset  Scale  SRDenseNet [37]  MSRN [20]  DDBPN [11]  EDSR [21]  MDSR [21]  RDN [50]  CRNetB (ours)  CRNetB+ (ours) 
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B100  /  /  /  /  /  /  /  /  
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Urban100  /  /  /  /  /  /  /  /  
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Manga109  /  /  /  /  /  /  /  /  
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5.1 Datasets and metrics
Training Set By following [15, 33], the training set of CRNetA consists of 291 images, where of these images are from Yang [47] with the addition of images from Berkeley Segmentation Dataset [23]. For CRNetB, training images of DIV2K [34] are used for training.
Testing Set During testing, Set5 [2], Set14 [49], B100 [23] and Urban100 [14] are employed. As recent postupsampling methods [21, 20, 11, 50] also evaluate their performance on Manga109 [24], so does CRNetB.
Metrics Both PSNR and SSIM [41] on Y channel (, luminance) of transformed YCbCr space are calculated for evaluation.
5.2 Implementation details
CRNetA Data augmentation and scale augmentation [15, 16, 33] are used for training a single model for all different scales (, and ). Every convolution layer in CRNetA contains filters () of size while and have filters (). The network is optimized using SGD. The learning rate is initially set to and then decreased by a factor of every epochs. L2 loss is used for CRNetA, and we train a total of epochs.
CRNetB Every weight layer in CRNetB has filters () with the size of except and have filters (). CRNetB is updated using Adam [17]. The initial learning rate is and halved every epochs. We train CRNetB for epochs. Unlike CRNetA, CRNetB is trained using L1 loss for better convergence speed.
5.3 Comparison with CSCSR
We first compare our proposed models with the existing CSC based image SR method, , CSCSR [10]. Since CSCSR utilizes LR images as input image, it can be considered as a postupsampling method, thus CRNetB is used for comparison. Tab. 1 presents that our CRNetB clearly outperforms CSCSR by a large margin.
5.4 Comparison with State of the Arts
We now compare the proposed models with other stateofthearts in recent years. We compare CRNetA with preupsampling models (, SRCNN [6], RED30 [22], VDSR [15], DRCN [16], DRRN [33], MemNet [32]) while CRNetB with postupsampling architectures (, SRDenseNet [37], MSRN [20], DDBPN [11], EDSR/MDSR [21], RDN [50]). Similar to [21, 50], selfensemble strategy [21] is also adopted to further improve the performance of CRNetB, and we denote the selfensembled version as CRNetB+.
Tab. 2 and Tab. 3 show the quantitative comparisons on the benchmark testing sets. Both of our models achieve superior performance against the stateofthearts, which indicates the effectiveness of our models. Qualitative results are provided in Fig. 7. Our methods tend to produce shaper edges and more correct textures, while other images may be blurred or distorted. More visual comparisons are available in the supplementary material.
Fig. 11 shows the performance versus the number of parameters, our CRNetB and CRNetB+ achieve better results with fewer parameters than EDSR [21] and RDN [50]. It’s worth noting that EDSR/MDSR and RDN are far deeper than CRNetB (, ), but CRNetB is quite wider ( and have filters). As reported in [21], when increasing the number of filters to a certain level, , , the training procedure of EDSR (for ) without residual scaling [31, 21] is numerically unstable, as shown in Fig. ((a))(a). However, CRNetB is relieved from the residual scaling trick. The training loss of CRNetB is depicted in Fig. ((b))(b), it converges fast at the begining, then keeps decreasing and finally fluctuates at a certain range.
5.5 Parameter Study
The key parameters in both of our models are the number of filters () and recursions .
Number of Filters We set for CRNetA as stated in Section 5.2. In Fig. ((a))(a), CRNetA with different number of filters are tested (DRCN [16] is used for reference). We find that even is decreased from to , the performance is not affected greatly. On the other hand, if we decrease from to , the performance would suffer an obvious drop, but still better than DRCN [16]. Based on these observations, we set the parameters of CRNetB by making larger and smaller for the trade off between model size and performance. Specifically, we use for CRNetB. As shown in Fig. ((b))(b), the performance of CRNetB can be significantly boosted with larger (MDSR [21] and MSRN [20] are used for reference). Even with small , , , CRNetB still outperforms MSRN [20] with fewer parameters (M M).
Number of Recursions We also have trained and tested CRNetA with , , , recursions, so the depth of the these models are , , , respectively. The results are presented in Fig. ((a))(a). It’s clear that CRNetA with layers still outperforms DRCN with the same depth and increasing can promote the final performance. The results of using different recursions in CRNetB are shown in Fig. ((b))(b), which demonstrate that more recursions facilitate the performance improved.
6 Discussions
We discuss the differences between our proposed models and several recent CNN models for SR with recursive learning strategy, , DRRN [33], SCN [39] and DRCN [16]. Due to the fact that CRNetB is an extension of CRNetA, , the main part of CRNetB has the same structure as CRNetA, so we use CRNetA here for comparison. The simplified structures of these models are shown in Fig. 12, where the digits on the left of the recursion line represent the number of recursions.
Difference to DRRN. The main part of DRRN [33] is the recursive block structure, where several residual units with BN layers are stacked. On the other hand, guided by (10), CRNetA contains no BN layers. Coinciding with EDSR/MDSR [21], by normalizing features, BN layers get rid of range flexibility from networks. Furthermore, BN consumes much amount of GPU memory and increases computational complexity. Experimental results on benchmark datasets under commonused assessments demonstrate the superiority of CRNetA.
Difference to SCN. There are two main differences between CRNetA and SCN [39]: CISTA block and residual learning. Specifically, CRNetA takes consistency constraint into consideration with the help of CISTA block, while SCN uses linear layers and ignores the information from the consistency prior. On the other hand, CRNetA adopts residual learning, which is a powerful tool for training deeper networks. CRNetA ( layers) is much deeper than SCN ( layers). As indicated in [15], a deeper network has larger receptive fileds, so more contextual information in an image would be utilized to infer highfrequency details. In Fig. ((a))(a), we show that more recursions, , , can be used to achieve better performance.
Difference to DRCN. CRNetA differs with DRCN [16] in two aspects: recursive block and training techniques. In the recursive block, both local residual learning [33] and preactivation [12, 33] are utilized in CRNetA, which are demonstrated to be effective in [33]. As for training techniques, DRCN is not easy to train, so recursivesupervision is introduced to facilitate the network to converge. Moreover, an ensemble strategy (in Fig. ((c))(c), the final output is the weighted average of all intermediate predictions) is used to further improve the performance. CRNetA is relieved from these techniques and can be easily trained with more recursions.
7 Conclusions
In this work, we propose two effective CSC based image SR models, , CRNetA and CRNetB, for pre/postupsampling SR, respectively. By combining the merits of CSC and CNN, we achieve superior performance against recent stateofthearts. Furthermore, our framework and CISTA block are expected to be applicable in various CSC based tasks, though in this paper we focus on CSC based image SR.
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