Image super-resolution is currently a hotspot issue in natural imaging [1, 2, 3], medical imaging , surveillance [5, 6] and security . It allows to recover High-Resolution Images () with better visual quality and refined details from the corresponding Low-Resolution Images . Magnetic Resonance Imaging (MRI) provides powerful support for disease diagnosis and treatment [8, 9]. On the other hand, high resolution MRI is expensive and subject to artifacts due to the elaborate hardware . Moreover, the longtime data acquisition and breath holding, combined with the unconscious or autonomous movement of patients, often lead to the missing of key information and motion artifacts in images . Super-resolution reconstruction of MRI images thus offers new promises for mitigating the costs of high resolution MRI technology. It simplifies the MRI scanning process effectively, shortens the scanning time, reduces the use of MRI contrast agents, and is also safer for patients . By restoring high-resolution images, pathological lesions can be detected with high precision, enabling doctors to carry out more accurate diagnoses. Thus, the reconstruction of MRI data requires higher textural detail than traditional super-resolution tasks in clinical diagnosis [13, 14, 15].
Conventional image super-resolution approach restores the original by fusing multiple of the same scene. It is an ill-posed inverse problem  due to the deficiency of , ill-conditioned registration and the absence of blurring operators. Specifically, numerous produce the same after resolution degradation, making it difficult to restore the image details accurately . For medical images, even small changes in textural details can affect a doctor’s diagnose . Prior knowledge is therefore often exploited to normalize the Super-Resolution Image () generation process . In traditional methods, this prior information can be learned from several pairs of high and low-resolution images . In addition, different methods have been proposed to stabilize the inversion of the ill-posed problem, such as prediction-based method , edge-based method  and sparse representation method [21, 18]. But these methods often over-smooth images because of ringing and jagged artifacts [22, 18].
Deep learning can learn the mutual dependency between input and output data, which has been widely explore to restore the image details for precise super resolution [23, 24, 25, 26]. The deep learning-based super-resolution aims to directly learn the end-to-end mapping function of to
through a neural network. Specifically, it extracts higher-level abstract features by multi-layer nonlinear transformation and learns the implying rules from data via the powerful capability of data fitting. Such learning capability empowers it to make reasonable decision or prediction for new data. The deep learning based super-resolution consists of 4 key steps: 1) Collect original images as and obtain by down-projection; 2) Feed the into convolutional networks to extract feature; 3) Reconstruct the by deconvolutional layer or up-projection; 4) Calculate the loss between and to optimize the super-resolution networks.
Although great progress has been made in medical image super-resolution in recent years, several challenges remain. The first is about the curse of network layers - too few network layers often degrade the super-resolution performance due to the insufficient model capacity whereas too many layers often make optimization challenging and also introduce high computational costs . The second is about the monotonous structure of most existing network structures which makes it less effective to improve the super-resolution by increasing network layers. The third is about new noises which many existing super-resolution algorithms tend to introduce during up-projection processes. This directly leads to unrealistic image details that could mislead the doctor’s diagnosis seriously.
In this paper, we propose an innovative network for MRI super-solution. The contributions can be summarized in two major points. First, a Coupled-Projection Residual Network (CPRN), contains a shallow network and a deep network for effective MRI image super-solution. Specifically, the shallow network exploits coupled-projection to calculate the errors of repeated up-projection and down-projection for preserving more interrelations between and . Such coupled-projection helps retain enriched image details as more as possible at the early stage of the network, and improves the alignment of the content of reconstructed images and the . The deep network inherits features of the shallow network but learns high-frequency differences between and by cascading the residuals. Second, a step-wise connection module termed by CPRN_S is designed for refining the MRI image super-resolution. By fusing the feature maps from the down-projection of the shallow network and the corresponding output of the residual blocks, it improves the MRI image super-resolution effectively by relating the information from the deep and shallow networks and strengthening the feature propagation. Experiments show that CPRN_S saves up to 30% of network parameters but achieves superior super-resolution performance, more detailes to be discussed in Experiments.
The rest of this paper is organized as follows. Section II briefly described related works on image super-resolution. Section III presents our proposed CPRN in details. Section IV then describes experimental results. Finally, a few concluding remarks are drawn in Section V.
Ii Related work
Ii-a DNN based Image Super-Resolution
Image super-resolution has been studied by interpolation and statistics based techniques[27, 28, 29] in early years that aim to reconstruct and restore image details and realistic textures. For example, the relationship between and can be learned from the correspondence function that is obtained via neighbor embedding and sparse coding techniques [25, 30, 31, 32, 33]. In recent years, deep neural networks (DNNs) have been widely studied for image super-resolution and much improved super-resolution performance has been obtained [34, 35, 36]. For example, 
proposes a Super-Resolution Convolutional Neural Network (SRCNN) that first uses bicubic interpolation to enlarge anto the target size and then produces via nonlinear mapping as fitted by a three-layer convolutional network.  proposes the Faster-SRCNN that speeds up the SRCNN by adopting a smaller kernel size and sharing the mapping layers.  proposes Efficient Sub-Pixel CNN (ESPCN) that reduce computational complexity by extracting features from of original size directly. Based on CNN, Oktay et al. proposed de-CNN that improves reconstruction by using multiple images acquired from different viewing planes . SRCNN3D generates brain from input by using three-dimensional CNN (3DCNN) and the patches of other brain .  presents an Anti-Aliasing (AA) self-super resolution (SSR) algorithm that exploits high-frequency information of in-plane MRI slices and is capable of reconstructing without external training data. On the other hand, all aforementioned methods are monotonous which miss to exploit features of various structures sufficiently. A multi-structure network is desired to capture richer features for more effective image super-resolution.
Generative Adversarial Network (GAN) has recently been applied to different tasks such as image recognition, style transfer as well as super-resolution.  first applies GAN for image super-resolution which labels the discriminator by and feeds to the generator to compute .  presents a novel GAN for video super-resolution where the GAN is enhanced by a distance loss in feature and pixel spaces.  combines GAN and 3DCNN for image super-resolution at multiple levels.  presents a GAN-based cascade refinement network that integrates a content loss and an adversarial loss to reconstruct phase contrast microscopy images. Although GAN-based medical image super-resolution methods achieve very promising PSNR, they tend to hallucinate image fine details which is extremely unfavorable for medical images .
Ii-B Feedback Mechanism
The feedback mechanism decomposes the prediction process into multiple steps that implement feedback by estimating and correcting the current estimation iteratively[42, 43, 44, 45]. It has been widely used in human pose estimation and image segmentation [42, 43]. For example,  presents an Iterative Error Feedback (IEF) network that improving the initial solution gradually by back-progagating the error prediction and learning richer data representations.  makes predictions for image segmentation by taking advantage of the implicit structure underlying the output space.  applies feedback mechanism to super-resolution where a Deep Back-Projection Network (DBPN) was designed for projecting errors at each stage. Back-projection helps to reduce the reconstruction error effectively via up-sampling and calculating reconstruction errors iteratively.
Our proposed super-resolution network extends the feedback mechanism by introducing coupled-projection blocks which exploits alternative up-projection and down-projection for computing reconstruction errors and preserving more interrelations between and . The content of reconstructed images can be aligned with the by the coupled-projection.
Skip-connection of network, such as residual-networks and dense-based networks, enables the flexible transmission, combination and reuse of features. For residual-based networks, Very Deep Super-Resolution (VDSR) exploits residual idea to include more network layers to expand the receptive field 
, where zero padding is implemented to keep the size of feature maps and final output images unchanged. and  achieve multi-scale super-resolution within a single network and improve the computational efficiency greatly. 
presents an Enhanced Deep Super-Resolution network (EDSR) that removes redundant network modules and is more suitable for low-level computer vision problems. presents a Deeply-Recursive Convolutional Network (DRCN) that deepens the network structure by combining recursive neural networks and skip connection in residual learning.  describes a symmetric encoder-decoder network (RED) that introduces a deconvolutional layer for each convolutional layer, where the convolutional layers capture abstract image contents and the deconvolutional layers enlarge feature sizes and restore image details.  presents a Deep Recursive Residual Network (DRRN) that adopts multi-path recursive model by local and global residual learning and multi-weight recursive learning.  presents a Laplacian Pyramid Super-Resolution Network (LapSRN) that can obtain intermediate reconstruction via progressive up-sampling to finer levels.
Dense-based network feeds the features of each layer in the dense block to all subsequent layers, concatenating the features of all layers instead of adding them directly like residual networks. In , Super-Resolution using DenseNet (SRDenseNet) applied dense block structures and improved super-resolution clearly by fusing complementary low-level and high-level features.  connects convolutional layers densely through Residual Dense Block (RDB) which better exploits the hierarchical features from all layers. Inspired by the skip-connection, our proposed network constructs step-wise connection to enhance feature propagation in image super-resolution. The step-wise connection from simple to complex is perfectly aligned with the human cognitive process and it also helps flexible embedding in other backbone networks.
The residual-based connection is often too simple and limited without exploiting cross-layer features sufficiently. The dense-based connection improves the cross-layer feature propagation significantly, but it requires a large amount of memory due to channel stacking. Different from the two types of skip-connection approaches, our proposed step-wise connection improves the feature propagation by connecting the corresponding layers of the shallow and deep sub-networks. This design is close to the human from-simple-to-complex cognitive process which helps connect and align cross-layer features effectively. It also reduces memory consumption, more details to be discussed in Experiment part.
In this section, we first present the framework of our method. Then we describe two parts of the proposed CPRN for super-resolution, the deep and shallow network. Finally, we modify and refine our CPRN to strengthen feature propagation.
Iii-a Framework Overview
Here we present the framework of CPRN in detail. First, in deep learning, we find that shallow networks are good at extracting simple, low-level features (edge features), so as to make the content of reconstructed images consistent with [53, 54]. Inspired by Dense DBPN (DDBPN) , a coupled-projection approach is adopted in our shallow network. We build interdependent up- and down-projection blocks, each representing different levels of image degradation and high-resolution components. The shallow network provides a feed-back mechanism for projection errors, so that the network can retain more detail information when generating deeper features. This iterative feedback can help reduce the newly-added noise during the up-sampling stage. Thus, the reconstructed approximates the , which is helpful for avoiding consequent misdiagnose due to unreal details generated by noise. Second, since the low-frequency information of and is similar, we only need to obtain the high-frequency difference by the residual network, while inheriting features of shallow networks. A number of residual blocks are cascaded to construct our deep network. The network structure aims to obtain only the residuals so as to ensure good reconstruction. Particularly, the features obtained from the shallow and deep layers are merged, for purpose of getting the final . In this way, the could retain more details. Third, to contact the shallow and deep network effectively, we refine them with step-wise connection and make it consistent with human cognitive processes (from simple to complex). This step-wise connection network achieves a competitive performance while with fewer convolutional layers, and can be flexibility embedded in other backbone networks.
We describe the pipeline of the proposed network CPRN in Fig 1. The shallow network is illustrated in Part A. We cascade multiple coupled-projection blocks to encourage feature reuse. The coupled-projection is described in Fig. 2, which couples an up-projection module and a down-projection module. Herein is used to denote feature map. images are fed into the convolutional layers to obtain with a size of . Then, and with different sizes are obtained after a deconvolutional layer and convolutional layer. is calculated using the residual errors of and , and then send into a Deconv(up) layer to obtain . We integrate with to get . The down-projection block is similar to the up-projection block, except for the following points: 1) the down-projection has two convolutional layers and one deconvolutional layer; 2) the size of is and the size of the final is . The deep network is illustrated in Part B. We feed the final feature map from Part A into several convolutional layers to obtain .
residual blocks without Batch Normalization (BN) layers are cascaded behind. The end of Part B is a convolutional layer with an up-scaling factor , which recovers the super-resolution image. Specially, we utilize the methodology of residual learning to merge the outputs of shallow and deep layers, and then reconstruct the final .
Iii-B Shallow Network
The overall process of the shallow network is described in Algorithm 1. We use to represent the -th deconvolutional layer, parameterized by , and use to represent the -th convolutional layer, parameterized by . represents the feature of the -th convolutional layer, and represents the feature of the -th deconvolutional layer. represents the batch size. We sample low resolution images and high-resolution images from the and datasets, respectively. The previously computed feature map serves as the input feature. First, the feature map is obtained by mapping through the deconvolutional layer . Then, the feature map is obtained from the convolutional layer . Afterwards, the residuals of and are mapped to the deconvolutional layer so as to get . The final can be obtained by combining with . This process is defined as an up-projection block. The down-projection is similar, but the process is in reverse. We firstly map through the convolutional layer to obtain the feature map , and then get the feature map by way of the deconvolutional layer . Afterwards, the is obtained by mapping the residuals of and to the convolutional layer. Finally, we combine and for the purpose of obtaining
. Moreover, to encourage feature reuse and avoid the vanishing-gradient problem, we link the previous up (or down)-projections to the input of each up (or down)-projection. The procedure of shallow network is illustrated in Part A of Fig.1.
Iii-C Deep Network
The deep network of the proposed CPRN architecture can be described as follows:
where is the feature map from the shallow network, and and are the intermediate feature maps. is the scaling factor, ( and ) are the up- and down-sampling operators, respectively, () is the invariant scale; is the spatial convolution operator; is the convolutional layer and is the residual block. First, we send the feature map obtained from the shallow network to the convolutional layer to obtain the new feature map . Note that the size of is smaller than . Then, we send to 16 residual blocks to get . The final is obtained by Eq.(3). Note that we reconstruct the final image by merging the features obtained from shallow and deep layers.
The residual network was first proposed to solve a high-level computer vision problem . Because super-resolution is a low-level computer vision problem, the network structure of ResNet is not completely applicable. To have the network structure satisfy the needs of super-resolution as much as possible, we slim down the network and delete unnecessary modules. The space saved can be used to expand the capacity and depth of the network. In this work, BN layers are removed to prevent the range flexibility of the features from being reduced . Because the dropout layer discards many features which might be important for super-resolution, we reconstruct the without incorporating the dropout technique .
Iii-D Refinement using Step-wise Connection
As the depth of the network increases, the capacitive of the model will gradually increase, and the features will become more abstract. Therefore, it is necessary to contact the shallow and deep network effectively, by which the details in the shallow work can be well preserved. Based on this insight, we refine CPRN with Step-wise connection to make it consistent with human cognitive processes (from simple to complex), named CPRN_S. The pipeline of our step-wise network CPRN_S is described in Fig. 3. We fuse the feature map which outputted by each down projection in shallow network with the corresponded output of the residual blocks. When the number of residual blocks increases, the incoming features of deep networks become more diverse. Besides, such progressive propagation approach could prevent the input of excessive abstract features and increasing the difficulty of network optimization, when the network capacitive is insufficient. It can be expressed as following:
where represents the features from -th down-projection in shallow network, represents the features from -th residual block in shallow network, and represents the input of the -th residual block. The improved network not only increases the feature fusion between shallow and deep networks, but also reduces the number of parameters caused by the network depth to a certain extent. And the number of residual block in CPRN_S is about of the CPRN network. In general, CPRN_S possesses the advantages of complexity reduction and network distillation.
This section presents experimental results including comparisons with the state-of-the-art and detailed analysis of our proposed super-resolution network.
Iv-a Datasets and Evaluation Metrics
Our proposed super-resolution network CPRN was evaluated over three public brain MRI datasets: Brats, ATLAS_native, and ATLAS_standardized. Brats is obtained from the cancer genomics program ‘The Cancer Genome Atlas’ (TCGA, https://www.cancer.gov/about-nci/organization/ccg/research/structural-genomics/tcga), which consists of preoperative and multimodal images of glioblastoma and lower grade glioma. The whole dataset contains 102 T1-weighted MRI samples and each sample has 155 image slices. The size of each image is 240240. We read the image from the .nii file of each patient and selected the 60-th, 80-th, 100-th, 120-th and 140-th slices as the data to obtain sufficient brain imaging area. ATLAS_native and ATLAS_standardized datasets are obtained from the ‘Anatomical Tracings of Lesions After Stroke’ (ATLAS, https://github.com/npnl/ATLAS/), which have 304 and 229 T1-weighted MRI samples, respectively. Since the number of image slices in each sample is different, we segmented the image slices of each sample into six groups, and extract an image slice as the from each segmentation point. The is obtained by down-sampling images to , of the original resolutions, respectively, via bicubic interpolation.
Two widely used metrics were utilized in evaluations including peak signal-to-noise ratio (PSNR) and structural similarity (SSIM). Specifically, PSNR evaluates the discrepancy between corresponding pixel points, and SSIM measures image similarity from three aspects including brightness, contrast and structure.
Iv-B Implementation Details
Our model was implemented on PyTorch 1.0. For the standard CPRN, we used 6 coupled-projection blocks and 16 residual blocks. But for the step-wise CPRN_S, we used 6 coupled-projection blocks and 6 residual blocks. In the shallow sub-network, the kernel sizes were set to 6, 8 on the images of two different sizes, while the strides were set to 2, 4, and padding was set to 2. In the deep sub-network, the kernel size, stride, and padding of the residual blocks were set at 3, 1, and 1, respectively, to keep the size of feature maps unchanged. The number of channels was set at 32 and 64 in the shallow and deep sub-network. During training, the initial learning rate was set at 1e-4 and batch size at 16. The L1 loss function was adopted with Adam where the momentum is 0.9 and the weight decay was 1e-4. All models are trained with 300 epochs and patch size is 4848. The proposed networks were evaluated over 10, 30, and 30 images from the Brats, ATLAS_native, and ATLAS_standardized, respectively.
Iv-C Comparison with baseline methods
We compare our networks with several state-of-the-art networks SRCNN , VDSR , EDSR , and DDBPN , and the comparison is based on two up-scaling by and . Table I shows the super-resolution ( and ) for the dataset Brats, ATLAS_native and ATLAS_standardized dataset. As Table I shows, our networks achieve better PSNR and SSIM consistently under both up-scaling cases, and this largely attribute to the complementary shallow and deep sub-networks that help retain fine-detailed features in the high-resolution images. Specifically, CPRN achieves comparable PSNR and SSIM with other monotonous models, revealing the robustness of our two-stage network in medical image super-resolution. VDSR and SRCNN obtain relatively poorer performance, largely due to their redundant structures where some unnecessary network layers such as BN could reduce the range flexibility of the features. In addition, VDSR, EDSR, and DDBPN show lower consistency across the three datasets, e.g. DDBPN performs better than EDSR on Brats and ATLAS_native but worse than EDSR on ATLAS_standardized for the 2 up-scaling.
Fig. 4 shows the super-resolution images by our network and the compared ones over the three dataset. As Fig. 4 show, the super-resolution images are well aligned with the quantitative PSNR and SSIM where our CPRN produces clearer image reconstruction with better details. Specifically, the blood vessels in by our network retain higher consistency with the ground-truth to a large extent. The boundary between the blood vessels and gray matter is clearer with high similarity to the ground-truth . All these show that our CPRN preserves more high-resolution components than other networks and reconstructs quality image with more detailed features. As a comparison, SRCNN, VDSR, EDSR, and DDBPN tend to generate misleading information in several cases.Specifically, the EDSR produces a stripe pattern in its reconstructed images over the Brats dataset, and its reconstructed images over the other two datasets are also vague and blurry. The Pa_CPRN and CPRN_S are two CPRN variants which will be discussed in Ablation study in the ensuing subsection.
Iv-D Ablation study
Our proposed CPRN consists of a deep sub-network and a shallow sub-network as well as a step-wise connection mechanism for better sup-resolution performance. We evaluated three network architectures beyond the standard CPRN to investigate how these designs contribute to the overall performance: 1) A monotonous version that executes either the coupled-projection blocks (CP_SD) or the residual blocks (RN_SD) alone; 2) A parallel version Pa_CPRN that executes shallow and deep sub-networks in parallel and averages their outputs as the final output; 3) A step-wise version CPRN_S that improves CPRN with our proposed step-wise connection mechanism. Table II shows experimental results. As Table II shows, CP_SD and RN_SD obtain lower PSNR and SSIM as they only process either coupled-projection blocks or residual blocks alone without fusing features of different types. Pa_CPRN combines the two sub-network in parallel which does not show clear performance improvement, showing that running the two sub-networks in parallel does not capture their merits. CPRN_S and CPRN obtain the best performance on Brats and ATLAS_standardized datasets, of which CPRN_S performs better than CPRN especially on the scenario. Additionally, CPRN_S uses much less parameters than CPRN, more details to be discussed in the following Discussion part.
Fig. 4 illustrates the super-resolution by the CPRN, Pa_CPRN, and CPRN_S, respectively. As Fig. 4 shows, CPRN and CPRN_S generate clearer texture patterns than Pa_CPRN, and the reconstructed images are subjectively closer to the ground truth. This shows that the proposed two-stage network structure in CPRN can retain more image details and reconstruct clearer images. CPRN_S propagates features progressively by step-wise connecting the output of each down projection in the shallow sub-network with the corresponding output of the residual block in the deep sub-network. Such step-wise connection mechanism is similar to the human cognitive process which helps the network to achieve great super-resolution performance.
To further verify the effectiveness of our proposed network structure, we experimented using different and and Fig. 5 shows the obtained PSNR and SSIM. Specifically, we first fixed to measure the effect of different . The results show that PSNR and SSIM perform the best when . In addition, we fixed to measure the effect of different , and found that the network obtains the best performance when . Further, the two parameters and are quite consistent which achieve the optimal performance when they are set around 6 and 16 across the three studied datasets.
The numbers of channels in the deep sub-network () and shallow sub-network () also affect the image super-resolution performance. Table III shows experimental results. As Table III shows, the proposed CPRN achieve the optimal performance when and are set around 32 and 64, respectively, consistently across the three studied datasets. Specifically, the PSNR and SSIM drop sharply when becomes larger. In addition, increasing in the shallow sub-network empowers the network to better learn the relationship between and during the coupled-projection, and the reconstructed features were consistent with that of . On the other hand, the relationship between and features was not learned well when was too small.
Previous works show difference influence of BN layer while introduced for image super-resolution [57, 46, 35]. We also studied the effect of introducing BN layers in our shallow and deep sub-networks. Table IV shows experimental results. As Table IV shows, our proposed CPRN achieves the highest PSNR and SSIM without using the BN layers. Fig. 6 further illustrates that removing BN layers in the shallow and deep sub-networks can better reconstruct texture details. The three cases on the right tend to produce stripe patterns or blurred details that degrade the image super-resolution performance. This study show that it is good to exclude BN layers which helps to get better super-resolution images and also reduce the network parameters and complexity.
We also studied the computational cost of different super-resolution networks, and explored how different network structures influence the speed of our proposed models (CPRN, Pa_CPRN, CPRN_S). Table V shows the experimental results where the execution time of each network is evaluated over 30 dataset images. It can be seen that SRCNN takes the shortest time as it has only three fully connected layers. VDSR takes more time than EDSR and DDBPN as VDSR has a much deep structure by cascading 20 residual blocks with BN layers. Although the depth of Pa_CPRN is reduced, the increase of the network width leads to an increase in the number of channels. CPRN_S use step-wise connection to progressively and hierarchically spread and reuse features and this design reduces the network depth and meanwhile avoids increasing the number of channels. Its computational cost is therefore lower than that of Pa_CPRN and CPRN. Specifically, the parameters of CPRN in shallow network are similar to the deep network and CPRN_S saves nearly 60% of the parameters compared to CPRN, CPRN_S therefore saves around 30% of network parameters as compared with CPRN, leading to the lower computational costs.
This paper presented a deep network for MRI super-resolution. Unlike the previous methods whose network structures are monotonous, our proposed network consists of two main parts: a shallow network and a deep one. In the shallow network, a coupled-projection mechanism helps to build interconnected up- and down-projection stages, making the reconstructed closer to the in content. In the deep network, we cascade multiple residual blocks to attain the high-frequency residuals of the and . Furthermore, to be consistent with human cognitive processes (from simple to complex), we develop an enhanced version CPRN_S which stepwisely connects each down projection in the shallow network with the corresponded residual blocks. The results in terms of PSNR and SSIM, show that our network significantly outperforms current state-of-the-art methods in reconstructing clear images from degraded ones.
This work was supported in part by the Natural Science Foundation of China under Grant 61573248, Grant 61802267 and Grant 61732011, in part by the Natural Sci-ence Foundation of Guangdong Province (Grant 2017A030313367), and in part by the Shenzhen Municipal Science and Technology Innovation Council under Grant JCYJ20180305124834854 and JCYJ20160429182058044.
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