Single image super‐resolution based on sparse representation using dictionaries trained with input image patches

IET Image ProcessingVolume 14, Issue 8 p. 1587-1593 Research ArticleFree Access Single image super-resolution based on sparse representation using dictionaries trained with input image patches Rasoul Asgarian Dehkordi, Rasoul Asgarian Dehkordi Electrical and Robotics Engineering Department, Shahrood University of Technology, Shahrood, IranSearch for more papers by this authorHossein Khosravi, Corresponding Author Hossein Khosravi hosseinkhosravi@shahroodut.ac.ir orcid.org/0000-0003-4798-0109 Electrical and Robotics Engineering Department, Shahrood University of Technology, Shahrood, IranSearch for more papers by this authorAlireza Ahmadyfard, Alireza Ahmadyfard orcid.org/0000-0002-5084-8910 Electrical and Robotics Engineering Department, Shahrood University of Technology, Shahrood, IranSearch for more papers by this author Rasoul Asgarian Dehkordi, Rasoul Asgarian Dehkordi Electrical and Robotics Engineering Department, Shahrood University of Technology, Shahrood, IranSearch for more papers by this authorHossein Khosravi, Corresponding Author Hossein Khosravi hosseinkhosravi@shahroodut.ac.ir orcid.org/0000-0003-4798-0109 Electrical and Robotics Engineering Department, Shahrood University of Technology, Shahrood, IranSearch for more papers by this authorAlireza Ahmadyfard, Alireza Ahmadyfard orcid.org/0000-0002-5084-8910 Electrical and Robotics Engineering Department, Shahrood University of Technology, Shahrood, IranSearch for more papers by this author First published: 14 May 2020 https://doi.org/10.1049/iet-ipr.2019.0129Citations: 3AboutSectionsPDF 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 In this study, an efficient self-learning method for image super-resolution (SR) is presented. In the proposed algorithm, the input image is divided into equal size patches. Using these patches, a dictionary is learned based on K-SVD, referred to as high resolution (HR) dictionary. Then, by down-sampling, the columns of the dictionary, called atoms, a low resolution (LR) version of the dictionary is obtained. An initial estimate of the SR image is constructed using the bicubic interpolation on the input image. Then in an iterative algorithm, the difference between the down-sampled version of the estimated SR image and the input image is obtained. This difference image, which includes reconstructed details is enlarged using sparse representation and LR/HR dictionaries. The enlarged detail is added to the latest reconstructed SR image. This process gradually improves the quality of the initial SR image. After several iterations, the reconstructed image is an SR version of the input image. Experimental results confirm that the proposed method performance is promising. 1 Introduction One of the most important concepts in the field of image processing is super-resolution (SR). Resolution is a characteristic of an image that indicates the quality of the image for presenting details. In most applications, including remote sensing, medical imaging, and surveillance cameras, high-resolution (HR) images are required. The higher the resolution, the more detail of the scene is provided in the captured image. To capture a HR image, expensive imaging hardware is needed. Due to the high cost and also the big storage required, it is practically impossible to use such imaging devices in typical applications such as video surveillance and remote sensing. Hence, another solution is considered to increase the resolution without changing the hardware. Methods supporting this request are based on signal processing techniques. These techniques can be applied to a low-resolution (LR) image and reconstruct a HR image. These methods, called super-resolution, do not require additional hardware [1-5]. SR techniques can be divided into three general categories: interpolation-based and learning-based methods, which both work on a single image, and multi-frame methods that use the information of several LR frames to produce a single HR frame. The most common methods for image enlargement are interpolation techniques. These methods have low complexity, but cannot reconstruct scene details properly and usually soften the high-frequency segments like the edges of the image. In multi-frame reconstruction methods, multiple LR frames from the same scene are used to create an SR image. These methods require a considerable number of frames with low displacements in the range of sub-pixels. This is a major restriction for these methods [6-9]. A learning-based method tries to model the relation between SR and LR image patches by learning this relation from the training dataset. In these methods, images are usually divided into patches of the same size. The training set can be obtained from an image dataset (example-based methods) or extracted from the input image itself (self-learning methods) [10]. Methods that use a training dataset to learn the relationship between HR and LR patches are based on the idea of Freeman's article [11]. These methods require a large number of pairs of LR and HR image patches for training. Yang et al. [12] used the dictionary learning and sparse representation on LR and HR patches to reconstruct SR patches. In this method, the HR images of the database are down-sampled to provide corresponding LR images. Then, two dictionaries of the LR and HR patches are constructed using the training images. The main assumption in this method is that each patch of the input image is similar to at least one of the patches in the training image dataset. Given a large dataset of diverse images, this assumption is acceptable. Using sparse representation, sparse coefficients are obtained for the patches of the input image. These coefficients are then applied to the dictionary related to the HR patches and SR patches are constructed. In [13], after constructing HR/LR equivalent dictionaries from patches of the training images, the input image is divided into patches and the closest atoms are identified in the LR dictionary for each patch of the input image. Given the LR atoms, the equivalent atoms in the HR dictionaries are used to make SR patches after a light recovery process. This is the first estimate of the SR image. In the next phase, to eliminate the effect of blocking and to improve the brightness of the patches, another estimate of the SR image is constructed using the bicubic interpolation method. Finally, by averaging the two estimated images, the final SR image is obtained. Zeyde et al. [14] provided an LR dataset by down-sampling HR images. Subsequently, by removing the low-frequency elements from the dataset and employing the PCA algorithm, they created the dictionaries needed to construct the SR image. Methods in [15, 16] utilise deep neural networks. In [15], the authors proposed an attention-based back projection network (ABPN) for SR. Since the shrinkage of the SR image should be close to that of the LR image, they proposed enhanced back-projection blocks to iteratively update the LR and HR feature residues. In [16] to model the down-sampling process, an iterative kernel correction (IKC) method for blur kernel estimation in blind SR problem, is suggested. This method tries to obtain a better estimate of the shrink image by modifying the kernel in the iteration loop. The authors proposed an SR network using the spatial feature transform (SFT) layers to handle multiple blur kernels, named SFTMD. Given that example-based methods have some limitations, such as computational complexity and the need for a large image dataset, recently, self-learning methods have attracted the attention of many researchers in this field. These methods learn the relations between LR and HR patches from the input image itself. These methods rely on the fact that many small patches within a natural image are very similar. Glasner et al. [17] investigated this issue statistically. As the authors reported in an image, about 90% of the patches have nine or more similar patches. Therefore, they proposed to reconstruct the SR version of an LR image without the need for an external database. Tsai et al. [18] used segmentation to extract similar patches in an image and for each section in the LR and HR images, they constructed HR/LR dictionaries and eventually used these dictionaries to construct whole SR image. The main problem with this method is its high computational complexity. Chen and Fowler [19], assumed that the neighbouring patches of the image, are almost similar, so each LR patch can be expressed as a linear combination of the adjacent SR patches. The coefficients of this linear combination are obtained by Tikhonov regularisation. Zhu et al. [20] presented a new methodfor SR. In this method, some image patches are randomly extracted. Then smoothpatches (low-frequency patches) are left out and high-frequency patches are used tomake dictionaries. In this algorithm, a fast method based on K-singular valuedecomposition (K-SVD) is used to construct the dictionary of image patches. In [21], which is the previous work of one of the authors, a fast method is proposed for SR based on self-learning. In this method, the input image is divided into patches on a regular grid. For each patch, a dictionary is created using the patch itself and its eight neighbouring patches. Also, a down-sampled version of the input image is obtained using the bicubic interpolation method. Using the same strategy from the down-sampled version, image patches are extracted. For each patch of this image, a dictionary is constructed in the same manner as patches for the original input image. Then, the relationship between the corresponding patches in these images is learned using their dictionaries. Using this model, the SR version of image patches is constructed. The proposed method in this paper is a self-learning method that only uses the information of the input image. In this algorithm, the input image is divided into equal patches. Using the KSVD method, a dictionary for the HR image is obtained. Another dictionary for the LR image is constructed by down-sampling the atoms of the HR dictionary using the bicubic interpolation. Then, an initial SR estimate of the input image is obtained using the bicubic interpolation. To improve the quality of this image, an iterative method is suggested. In each step, the details of the SR image are estimated by the orthogonal matching pursuit (OMP) method [22] using the LR/HR dictionaries and the obtained details at each step are added to the currently constructed SR image. After a few iterations, the algorithm converges and the SR version of the input image is constructed. The rest of this paper is organised as follows: in Section 2 the proposed algorithm is described. Experimental results and a comparison with the previous works are given in Section 3. Finally, the conclusion is presented in Section 4. 2 Proposed method Fig. 1 shows the flowchart of the proposed algorithm. As outlined in the figure, the proposed method consists of two phases: in the first phase, the top red box, the input image is divided into patches of the same dimensions on a regular grid. These patches are reshaped into the column vectors and the sparse representation dictionary of the input image is obtained by the KSVD method [22]. This dictionary is referred to as the HR dictionary, and its columns are called dictionary atoms. Each atom of the HR dictionary is converted to the corresponding atom in the LR dictionary by down-sampling and consequently, the LR dictionary is constructed. Subsequently, using the bicubic interpolation, the input image is up-sampled and an initial estimate of the SR image is obtained. Fig. 1Open in figure viewerPowerPoint Flowchart of the proposed algorithm In the second phase of the proposed method, the bottom green box in Fig. 1, the estimation of the SR image is updated in an iterative algorithm. In each iteration, new details are computed using sparse representation coefficients. These details are then added to the current SR image and produce a better estimate of the SR image. The details of the algorithm are described in the following sections. 2.1 Phase 1: learning and initialisation phase In the first phase, the input image (ILR) is divided into patches of size m × m on regular bases. All of these patches are used to train the HR dictionary (DH). For this purpose, we first reshape each patch into an m2 × 1 column vector by concatenating its columns. A matrix of image patches is formed using all image patches where each patch is a column of this matrix. We call this matrix of HR patches as DataH. For an input image with k patches, the dimension of the patch matrix is m2 × k. According to the sparse representation theory, a properly learned dictionary of HR patches is able to represent each image patch while only a few number of atoms are involved in this representation. The DH dictionary has Z atoms and each atom is an m2 × 1 vector. Therefore, the dimensions of this dictionary are m2 × Z, where Z << k. One of the most common methods for learning dictionary is the KSVD method [20-22]. This method is based on minimising the mean square error of patch representation subject to the sparsity of vector, known as sparse coefficients: (1)in (1) is the matrix of image patches, is the HR dictionary and α is the sparse coefficient matrix. As mentioned, the number of dictionary columns (atoms) is less than the number of columns, but the number of rows of both matrices is equal to the number of pixels in each input image patch. Fig. 2 depicts the image of Barbara and the atoms in the learned dictionary. Each atom of the dictionary is reshaped as a square image patch for better visualisation. The number of elements per atom is 400. Fig. 2Open in figure viewerPowerPoint The Barbara image and its atoms in the learned dictionary (a) Atoms of the learned dictionary (enlarged for better visualization), (b) Barbara image After obtaining the DH dictionary, we must find the LR dictionary, DL. To this end, we used the method proposed by Zhu et al. [20]. In this way, we first resize each m × m patches into smaller patches of size n × n (n < m) using the bicubic interpolation. Then, we transform the n × n matrix into an n2 × 1 vector; this vector is embedded as a LR atom into the DL matrix. The size of the DL dictionary will be n2 × Z, in other words, the ith atom of the DL is the LR version of the ith atom in the DH dictionary. After creating the dictionaries, to create an initial estimate of the SR image, the input image is up-sampled using the bicubic interpolation. Although interpolation methods are very fast, they smooth the image details like edges. However, it can be used as the initial estimate for the SR image. In the next section, we explain how the proposed method iteratively obtains image details to reconstruct the SR image. 2.2 Phase 2: iterative method to reconstruct the SR image In the second phase of the proposed algorithm, we improve the quality of the initial SR image using details learned from and dictionaries. This process is shown in the green box of Fig. 1. In this stage, we try to restore the missing details to the SR image, gradually in an iterative manner. For this purpose, in each iteration of the algorithm, the last version of the SR image is compared with the input image. Since these two images have different dimensions, the SR image is down-sampled to the size of the input image dimension. We denote the down-sampled version of the SR image in each iteration of the algorithm by . The difference between the ILR input image and the is considered as : (2)If the current SR image is well reconstructed, the difference between the two images denoted by is negligible. Therefore, the next steps in the algorithm must be organised in such a way that the norm of this matrix converges to zero. So we use (3) as the stopping criterion: (3)here M and N represent the number of rows and columns of the matrix, respectively. The smaller values of mean better , so, the algorithm continues while is greater than a predefined value. Fig. 3 shows the graph for the two test images: Gold-Hill and Girl. The goal is to increase the dimensions of these images from 256 × 256 to 512 × 512. As can be seen, after about 15 to 20 iterations the MSE error becomes <10−5 so the algorithm stops. Fig. 3Open in figure viewerPowerPoint for two test images in 30 iterations It must be mentioned that consists of the details which currently do not exist in the SR image. These details will be added to the SR image using the concept of sparse representation. For this purpose, in the next section, the up-sampled version of the difL is obtained using learned dictionaries and added to the latest reconstructed image (SR). Given that the difL image has the same size as the input image, ILR, it is divided into patches of size n × n with a 50% overlap. Each patch is converted to a column vector and sparse coefficients for each vector are determined from the dictionary using the OMP method by minimising (4). The OMP algorithm tries to combine the columns of the dictionary to reconstruct each patch accurately. Details of the OMP method can be found in [22] (4)In (4), X is one of the input patches and is the vector of sparse coefficients, which should be obtained separately for each patch. The ℓ0-norm of λ must be smaller than Z, which means that the number of atoms used to reconstruct the patches must be <Z. As stated in Section 2.1, atoms of the DL and DH dictionaries are corresponding to each other. Therefore, these dictionaries can be used to convert the small patches into the large patches and vice versa. For example, if we have an n × n patch, we can use the DL dictionary to calculate the sparse coefficients for it and using the DH dictionary, an m × m patch can be constructed which is an enlarged approximation of the n × n patch. Image difS is the enlarged version of image using LR/HR dictionaries. So to obtain this image at any stage it is enough that the sparse coefficients for each LR patch, calculated in the previous step, be used to reconstruct the SR patch. Equation (5) shows how to create a patch in the image using the sparse coefficients (5)So, in brief, to compute the we must perform the following steps: (i) The image is divided into n × n patches. For each patch, the sparse vector is obtained separately using the dictionary (4). (ii) The sparse vector obtained for each image patch is applied to the dictionary to obtain the equivalent patch in the image (5). After obtaining for updating we have: (6)The process of difS and difS construction and update of the ISR continues until the stopping criterion is satisfied. Once the algorithm stopped, the ISR image can be considered as the final SR image of the input image. 3 Experimental results To evaluate the proposed method and compare its performance against a number of existing methods, we used the USC-SIPI image dataset. As the output images for other algorithms and their codes are not available, a fair visual comparison between the proposed method and other algorithms is not possible, so we compare the performance of methods using peak signal to noise ratio (PSNR), similarity structural index measure (SSIM), root mean square error (RMSE) and the computation time as criteria. In qualitative comparison, only the proposed method and the bicubic interpolation method are compared. The size of the patches in the input and output image is very important. In [22], the effect of choosing the different size of patches in the construction of the dictionary and reconstruction of the SR images has been investigated and it has been proven that if the patches are smaller, we can get a more accurate reconstruction of the image and output patches for a certain number of atoms. Of course, in the proposed method, according to the experiments conducted, the same behaviour was approved. Table 1 shows the results of some experiments conducted on the test set to reconstruct the output image. The size of the input image was 256 × 256 and the output image was 512 × 512. The number of atomic dictionaries for all experiments is 100. More atoms have almost no effect on the results. As it is clear from the table, if we consider non-overlapping patches, (first and third modes in Table 1), the computational time reduces, but the reconstruction RMS error increases. More RMS error is due to the blocking effect that comes from the brightness changes at the border pixels of the patches. Increasing the overlap to 3, as in the fifth mode, reduces the blocking effect, but, increases the run time significantly. Table 1. Results of the proposed method for different patch size Experiment Mode1 Mode2 Mode3 Mode4 Mode5 input patch dimension 2 × 2 2 × 2 3 × 3 3 × 3 3 × 3 output patch dimension 4 × 4 3 × 3 5 × 5 5 × 5 5 × 5 overlapping rows/cols (input image) 0 1 0 1 2 overlapping rows/cols (output image) 0 1 0 1 3 no. of patches 16,384 64,516 7225 16,129 64,009 no. of dictionary atoms 100 100 100 100 100 time, s 15 52 7 17 70 RMSE 7.43 6.42 7.71 7.20 6.51 The second and fourth modes show the results for the case that the adjacent patches overlap in only one row/column. In the fourth mode that the size of the patches increased to 5 × 5, the total number of patches reduced and as a result, we have better speed but worse accuracy. To have good accuracy and speed, patches of size 2 × 2 and 3 × 3 are selected for input and output, respectively. Fig. 4 compares the proposed method with the bicubic interpolation for doubling the dimension of four images. Fig. 4Open in figure viewerPowerPoint Results of the bicubic interpolation (left) and the proposed method (right) when the dimensions of the images increase from 256 × 256 to 512 × 512 As illustrated in Fig. 4, the proposed method has better behaviour in the reconstruction of the details and the edge areas. The main reason for this is the use of an iterative method to reduce errors and increase the accuracy of the algorithm. Fig. 5 shows and images for Barbara's image at the first iteration. These two images contain some important details and the proposed method attempts to gradually add these details to the image. Fig. 5Open in figure viewerPowerPoint and for the Barbara image (a) , (b) Tables 2–4 show the PSNR, RMSE, and SSIM of the proposed method and several other methods for X2 SR (doubling the image size from 256 × 256 to 512 × 512). The values reported in these tables are taken from the corresponding references. The size of the patches in all the methods has been chosen for the best fit. Among the methods compared, Yang et al. method [12] require a database of images for creating the dictionary. In this method, the sizes of the input and output patches are 5 × 5 and 10 × 10, respectively. According to their paper, a large image database has been used to train dictionaries, which includes hundreds of images. Table 2. PSNR criterion for doubling the image size from 256 × 256 to 512 × 512 G.Hill Peppers Girl Crowd Barbara Man Boat Clown Couple Cameraman Bicubic 31.45 31.76 36.64 32.66 25.35 31.05 29.95 27.78 32.65 29.55 35.72 Yang et al. [12] 32.54 32.89 38.68 35.14 25.92 32.79 31.5 28.96 34.82 30.96 38.69 ABPN [15] 32.39 34.11 38.79 35.10 24.27 32.45 31.00 27.58 34.22 31.32 35.58 IKC [16] 31.22 30.90 33.53 31.26 20.94 30.02 28.95 24.51 30.53 29.66 30.96 Chen and Fowler [19] 32.23 32.35 37.96 34.08 25.78 32.16 30.1 28.67 33.81 30.4 37.9 Zhu et al. [20] 32.34 32.66 38.05 34.27 25.87 32.4 31.15 28.8 34.35 30.52 38.5 Mokari and Ahmadyfard [21] 32.45 32.38 37.57 34.23 25.81 32.46 31.11 28.71 34.45 30.48 38.38 proposed method 32.39 32.62 38.55 34.32 25.88 32.56 31.21 28.76 34.59 30.53 38.47 Table 3. RMSE criterion for doubling the image size from 256 × 256 to 512 × 512 G.Hill Peppers Girl Crowd Barbara Man Boat Clown Couple Cameraman Bicubic 6.82 6.59 3.76 5.93 13.77 7.15 8.11 10.41 5.95 8.49 4.17 Yang 6.02 5.78 2.97 4.46 12.89 5.85 6.78 9 4.63 7.22 2.96 ABPN 6.12 5.02 2.93 4.48 15.59 6.08 7.18 10.65 4.96 6.92 4.24 IKC 7.00 7.27 5.37 6.97 22.87 8.04 9.09 15.16 7.58 8.38 7.22 Chen 6.24 6.15 3.22 5.12 13.1 6.29 7.21 9.4 5.1 7.69 3.22 Zhu 6.13 5.94 3.02 4.93 12.97 6.03 7.06 9.29 4.79 7.59 3.03 Mokari 6.08 6.13 3.37 4.95 13.05 6.07 7.09 9.35 4.83 7.63 3.07 proposed method 6.12 5.96 3.01 4.90 12.95 6.00 7.01 9.30 4.75 7.57 3.05 Table 4. SSIM criterion for doubling the image size from 256 × 256 to 512 × 512 G.Hill Peppers Girl Crowd Barbara Man Boat Clown Couple Cameraman Bicubic 0.9926 0.9917 0.9918 0.9919 0.9461 0.9948 0.9834 0.9684 0.9980 0.9808 0.9939 Yang 0.9947 0.9965 0.9976 0.9983 0.9809 0. 996 0.9950 0.9941 0.9952 0.9948 0.9978 ABPN 0.9968 0.9980 0.9981 0.9979 0.9936 0.9964 0.9954 0.9910 0.9958 0.9952 0.9982 IKC 0.9878 0.9844 0.9794 0.9887 0.9454 0.9849 0.9846 0.9559 0.9874 0.9872 0.9879 Chen 0.9951 0.9974 0.9982 0.9974 0.9815 0.9965 0.9955 0.9946 0.9914 0.9949 0.9993 Zhu 0.9953 0.9975 0.9984 0.9987 0.9919 0.9967 0.9956 0.9947 0.9961 0.9953 0.9995 Mokari 0.9951 0.9971 0.9979 0.9978 0.9901 0.9963 0.9954 0.9945 0.9925 0.9948 0.9991 proposed method 0.9956 0.9978 0.9980 0.9989 0.9917 0.9965 0.9959 0.9946 0.9962 0.9951 0.9992 In the ABPN method [15], the size of the LR image patches is 32 × 32 and the size of the SR image patches is 64 × 64. In the IKC method [16], the kernel size is 21 × 21. In the Chen and Fowler method [19], the input patches are 4 × 4 and the output image patches are 8 × 8 pixels. Considering that this method requires the use of neighbour patches to create a dictionary of each patch, an 8 × 8 area is formed at the centre of each patch, and all patches in this area are used to build the dictionary. In the Zhu et al. method [20], the size of the input image patches is 4 × 4 and the size of the output image patches is 8 × 8. In this method, after extracting image patches, 3000 patches are selected to construct the dictionary. In Mokari and Ahmadyfard method [21], the size of the input image patches is 2 × 2 and the size of the output image patches are 4 × 4. In the proposed method, as we mentioned the size of the input image patches is 2 × 2 and the size of the output image patches is 3 × 3. Table 5 shows the average PSNR, RMSE, SSIM, and run-time criteria for different algorithms. The execution time of Yang, Bicubic, Chen, and Zhu methods in [20] are computed on a computer with Core i7 CPU at 3.4 GHz and 8 GB RAM. To compare the run-time of the proposed method against other methods, a PC with similar specification was used. The ABPN and IKC are implemented on NVIDIA GeForce GTX 960M GPU. Table 5. Quantitative comparison of algorithms Time, s PSNR SSIM RMSE Bicubic 0.096 31.32 0.9778 7.37 Yang 502 33.00 0.9946 6.05 ABPN 10.60(GPU) 32.43 0.9960 6.74 IKC 3.14(GPU) 29.31 0.9794 9.54 Chen 349 32.31 0.9947 6.61 Zhu 7.05 32.62 0.9963 6.43 Mokari 344 32.54 0.9955 6.52 proposed method 52 32.71 0.9963 6.42 4 Conclusion In this paper, a method is presented for the single-image SR using sparse representation. In this method, the input image is firstly divided into equal size patches and from these patches, a pair of HR/LR dictionaries are constructed. Then, using the bicubic interpolation method, an initial estimate of the SR image is obtained. 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