Augmented Coupled Dictionary Learning for Image Super-Resolution
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1 Augmented Coupled Dictionary Learning for Image Super-Resolution Muhammad Rushdi and Jeffrey Ho Computer and Information Science and Engineering University of Florida Gainesville, Florida, U.S.A. Abstract Recent approaches in image super-resolution suggest learning dictionary pairs to model the relationship between low-resolution and high-resolution image patches with sparsity constraints on the patch representation. Most of the previous approaches in this direction assume for simplicity that the sparse codes for a low-resolution patch are equal to those of the corresponding high-resolution patch. However, this invariance assumption is not quite accurate especially for large scaling factors where the optimal weights and indices of representative features are not fixed across the scaling transformation. In this paper, we propose an augmented coupled dictionary learning scheme that compensates for the inaccuracy of the invariance assumption. First, we learn a dictionary for the low-resolution image space. Then, we compute an augmented dictionary in the high-resolution image space where novel augmented dictionary atoms are inferred from the training error of the low-resolution dictionary. For a low-resolution test image, the sparse codes of the low-resolution patches and the lowresolution dictionary training error are combined with the trained high-resolution dictionary to produce a high-resolution image. Our experimental results compare favourably with the non-augmented scheme. Keywords-dictionary learning; coupled features; superresolution; I. INTRODUCTION The availability of high-resolution (HR) degradation-free images is highly important for the success of many applications in medicine ( [10], [11]), remote sensing [6], biometric identification [15], and other fields. However, high-resolution imaging devices are still quite expensive. As well, the storage and transmission of high-resolution images are limited by space and bandwidth considerations. On the other hand, while low-resolution (LR) images are cheap to capture, store, and transmit, they still suffer from loss of details, blurring, noise, and interference. Superresolution (SR) techniques aim at creating high-resolution images from low-resolution ones while overcoming the inherent limitations of low-resolution imaging [8]. However, this is an ill-posed problem since one low-resolution patch can correspond to many high-resolution patches. Moreover, the low-resolution observations are blurred, noisy, and misaligned. So, interpolation, reconstruction and example-based techniques have been proposed to regularize the solution [2]. Example-based techniques learn the mapping between the low- and high-resolution patches from a training dataset and apply this mapping to low-resolution patches of test images [3], [4]. Dictionary learning schemes improve these techniques by relating the low- and high-resolution features through sparse representations with respect to coupled overcomplete dictionaries [13], [14], [16], [12], [7]. A simplifying assumption of these schemes is the invariance of the sparse representation: the sparse codes of a lowresolution patch with respect to the low-resolution dictionary are identical to the sparse codes of the corresponding highresolution patch with respect to the high-resolution dictionary. Nevertheless, this assumption is inaccurate and does not hold in particular for large magnification factors. Jia et al [5] relax this assumption by allowing the sparse codes of low- and high-resolution patch pairs to have different values while they still share the same support. Wang et al [9] learn a linear map between the sparse codes of the low- and high-resolution patches. In this paper, we compensate for the inaccuracy of the invariance assumption of sparse representations by explicitly incorporating the lowresolution dictionary learning error in the high-resolution image reconstruction model. In particular, we augment the high-resolution dictionary with additional atoms that relate the residual error of the low-resolution dictionary to the high-resolution patches. Our experiments show that this augmented dictionary models better the LR-HR coupling and outperforms the basic coupled dictionary learning. As well, we show how the dimensionality of the low-resolution features affects the augmented dictionary scheme. In Section II, we review the coupled dictionary learning () approach then introduce our augmented coupled dictionary learning () and synthesis approach. We compare the superresolution performance of our method with the baseline approach in Section III. Conclusions and future work are given in Section IV. II. DICTIONARY AUGMENTATION FOR COUPLED SPACES A. Problem Formulation Suppose we are given two coupled feature spaces, the high-resolution patch spacex R nx and the low-resolution feature space Y R ny, tied by a certain mapping function F that may be non-linear and unknown. The goal of dictionary learning approaches is to learn two dictionaries D x R nx K and D y R ny K for X and Y such that
2 the sparse representation of x i X in terms of D x should be related by a map W (chosen typically as the identity map for simplicity) to that of y i Y in terms of D y, where y i = F(x i ). Yang et al [14] addressed this problem by generalizing the basic sparse coding scheme to minimize D x,d y,γ subject to X D x Γ 2 F + Y D y Γ 2 F +λ Γ 1 { k Dx (:,k) 2 1 D y (:,k) 2 1. where X R nx N and Y R ny N are data matrices from the high-resolution patch space X and the lowresolution feature space Y, respectively. Γ R K N are the data sparse codes (common to both dictionaries) and λ is a regularization parameter. The identity-map assumption between the spare representations of the features in the low- and high-resolution spaces is rather restrictive and inaccurate for large super-resolution factors. We can alleviate this problem and get a more truthful dictionary model by augmenting the high-resolution space dictionary D x with an augmenting dictionary D a R nx ny. The augmenting dictionary should compensate for the modelling error of the non-augmented coupled dictionary scheme. This modelling error can be defined as the lowresolution-space dictionary learning residual (1) R = Y D y Γ. (2) The augmenting dictionary D a atoms may be selected to minimize this residual R. So, the Augmented Coupled Dictionary Learning () objective function may be formulated as minimize X D xγ D a R 2 F + Y D y Γ 2 F +λ Γ 1 D x,d a,d y,γ D y, compute the low-resolution residual R t = Y t D y Γ t, then compute the high-resolution image patches as subject to B. Dictionary Training k D x (:,k) 2 1 D y (:,k) 2 1 j D a (:,j) 2 1. The energy minimization in Equation 3 is tackled by separating the objective function into three sub-problems, namely updating the sparse codes Γ of the low-resolutionspace training samples Y, updating the low-resolution-space dictionary D y, and hence jointly constructing the highresolution-space and the augmenting dictionaries D x, D a. C. Optimization for the Low-Resolution Dictionary The first two sub-problems are solved through the K- SVD dictionary training procedure [1]. In this procedure, the sparse coding and dictionary update are alternated to solve the decoupled problem (3) minimize D y,γ subject to Y D y Γ 2 F +λ Γ 1 { k Dy (:,k) 2 1. The outputs of the K-SVD procedure are the lowresolution-space dictionary D y and the sparse codes Γ of the the low-resolution training data Y with respect to this dictionary. D. Computing the High-Resolution and Augmenting Dictionaries Once the low-resolution dictionary D y and the sparse codesγof the training data are obtained, the high-resolution and augmenting dictionaries could be easily obtained from Let and (4) minimize D x,d a X D x Γ D a R 2 F (5) Φ = ( D x D a ) (6) Λ = ( Γ). (7) R Then the solution of Equation 5 is the pseudo-inverse expression E. Synthesis Φ = XΛ = XΛ T (ΛΛ T ) 1. (8) Given a test image, we extract the low-resolution features Y t from the image patches, find the sparse codes Γ t of the features with respect to trained low-resolution dictionary X t = D x Γ t +D a R t. (9) The high-resolution image patches are then put back in their respective locations and averaged in overlapping areas to produce the output image. A. Implementation Details III. EXPERIMENTAL RESULTS For dictionary training, we used the natural image dataset provided by [13]. The dataset consists of 91 images of flowers, faces, vehicles, and other natural scenes. Since the human visual system is more sensitive to the luminance information, we convert all training images to gray-level ones and discard the color information. Regrading the images as the high-resolution examples, we blur and downsample all images with a factor of 3. The downsampled images are then upscaled back to their original sizes to simplify the
3 algorithmic details. The upscaled images have lost highresolution details during the downsampling process and this why these images are considered to be of low resolution. Following [13], [14], four horizontal and vertical derivative filters (namely,f 1 = [ 1,0,1] = f T 2,f 3 = [1,0, 2,0,1] = f T 4 ) are applied to the LR images to extract localized highfrequency content. Patches of size 9 9 are extracted from the filtered images. The patches are stacked together forming 324-dimensional feature vectors. Then PCA is applied to ignore components that contribute no more than 0.1% of the average feature energy. This reduces the LR dimensionality to 30. For the high-resolution images, low frequencies are removed by subtracting each low-resolution (upscaled) image from its high-resolution counterpart. About 130,000 LR-HR pairs are extracted (20% of which are used for validation). We compare our method against bicubic interpolation as well as the basic coupled dictionary learning () variant of Zeyde et al [16]. For both dictionary learning schemes, we use a total high-resolution dictionary size of 2048 atoms, 40 iterations of the K-SVD algorithm, and a maximal sparsity of 3 for the OMP coding scheme. B. Quantitative and Visual Comparison We tested the super-resolution performance on 14 standard test images and computed the average quality metrics. For each image, the dictionary-based synthesis was applied to the luminance channel while bicubic interpolation was applied to the chrominance channels. Table I shows the Structural Similarity Index (SSIM) and the Peak Signal-to- Noise Ratio (PSNR) measures in the first and second row for each image respectively. Our augmented dictionary learning scheme surpasses the baseline coupled dictionary learning on all 14 images. On average, our scheme and the baseline one show an improvement over bicubic interpolation of db and db, respectively. Figure 4 shows samples of the ground-truth and reconstructed test images. Our results look sharper than those of the bicubic interpolation and are also visually better or similar to their counterparts in the baseline scheme. Figure 3 shows the trained dictionaries of the augmented scheme. The augmenting dictionary shows clear structures and improves the overall super-resolution quality although it is small in size. C. Effect of the High-Resolution Dictionary Size We explore the effect of the HR dictionary size in Figure 1. Dictionary sizes of 128, 256, 512, 1024, and 2048 are used. The average SSIM and PSNR measures (Figure 1a,b) of our augmented scheme are consistently better across all dictionary size values. This comes at no cost in synthesis time. Indeed, our average synthesis time is slightly less than that of the baseline scheme (Figure 1c) since for the same total number of HR dictionary atoms, our scheme allocates fewer atoms to the over-complete dictionary and hence reduces the sparse coding complexity (which is known to be the bottleneck of the K-SVD algorithm). D. Effect of the Low-Resolution Feature Dimensionality From Equation 3, we see that the number of augmenting dictionary D a atoms depends on the dimensionality of the LR feature vectors which in turn can be controlled through the PCA dimensionality reduction step. Figure 2 shows the dependence of the average performance metrics on the amount of discarded feature energy. On one hand, for a large percentage of discarded energy, there is a clear drop in the average performance metrics. Indeed, the performance gap between the augmented and baseline schemes gets smaller since the augmenting dictionary will have very few atoms in this case. On the other hand, for small percentages of the discarded energy, there is no noticeable improvement over the mid-range values but there is a big increase in average synthesis time. So, a discarded energy percentage of 0.1% is a reasonable trade-off here between accuracy and time complexity. IV. CONCLUSIONS AND FUTURE WORK In this work, we have introduced a novel coupled dictionary learning scheme for image super-resolution. Our scheme models more accurately the relation between lowresolution and high-resolution patches by incorporating an augmenting dictionary that links the low-resolution reconstruction error to the high-resolution dictionary model. We have shown that the new model produces improved superresolution results in comparison with the standard coupled dictionary learning scheme. For future work, we plan to put reasonable structural constraints on the augmenting dictionary to better capture relevant features in the residual data and to bound the time and space complexity in the case of large dictionary sizes. Moreover, we would like to explore more generalized coupled dictionary models for image super-resolution and similar applications. REFERENCES [1] M. Aharon, M. Elad, and A. Bruckstein. K-svd: An algorithm for designing overcomplete dictionaries for sparse representation. Signal Processing, IEEE Transactions on, 54(11): , nov. 6. [2] G. Cristóbal, E. Gil, F. Šroubek, J. Flusser, C. Miravet, and F. B. Rodríguez. Superresolution imaging: a survey of current techniques. In Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, volume 7074 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Aug. 8. [3] W. Freeman, T. Jones, and E. Pasztor. Example-based superresolution. Computer Graphics and Applications, IEEE, 22(2):56 65, mar/apr 2. [4] D. Glasner, S. Bagon, and M. Irani. Super-resolution from a single image. In ICCV, 9. [5] K. Jia, X. Tang, and X. Wang. Image transformation based on learning dictionaries across image spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on, PP(99):1, 2012.
4 (a) Original: foreman (b) : ( ) (e) Original: monarch (f) : ( ) (d) : ( ) (g) :0.9802(29.535) 3 (j) : (22.439) (k) : ( ) 5 (n) : ( ) (o) : ( ) Figure 4. 3 (v) : (25.912) (s) : ( ) (r) : (24.365) (u) Original: flowers (p) : ( ) (l) : ( ) (q) Original: barbara (h) :0.9809( ) (m) Original: zebra (i) Original: ppt3 (c) : (31.918) 3 (w) : ( ) (t) :0.8963( ) 4 3 (x) : (27.306) Visual results for sample test images. Quantitative metrics are shown as SSIM(PSNR). 4
5 Average PSNR Average PSNR HR Dictionary Size Discarded LR feature energy (a) Average PSNR (a) Average PSNR Average SSIM Average SSIM HR Dictionary Size Discarded LR feature energy (b) Average SSIM (b) Average SSIM Average Time 6 Average Time HR Dictionary Size Discarded LR feature energy (c) Average Synthesis Time (c) Average Synthesis Time Figure 1. Effect of the HR dictionary size on the average super-resolution performance metrics of 14 test images. Figure 2. Effect of the dimensionality of the LR features on the average SR performance metrics of 14 test images.
6 (a) Low-Resolution Dictionary (b) High-Resolution Dictionary bicubic [16] baboon barbara bridge coastguard comic face flowers foreman lenna man monarch pepper ppt zebra Average Table I SUPER-RESOLUTION QUANTITATIVE RESULTS. THE SSIM AND PSNR MEASURES ARE SHOWN FOR EACH IMAGE IN THE 1ST AND 2ND ROWS, RESPECTIVELY. (c) Augmenting High-Resolution Dictionary Figure 3. The trained dictionaries of the augmented coupled dictionary learning () scheme. [6] F. Li, X. Jia, D. Fraser, and A. Lambert. Super resolution for remote sensing images based on a universal hidden markov tree model. Geoscience and Remote Sensing, IEEE Transactions on, 48(3): , march [7] D. Lin and X. Tang. Coupled space learning of image style transformation. In Computer Vision, 5. ICCV 5. Tenth IEEE International Conference on, volume 2, pages Vol. 2, oct. 5. [8] S. C. Park, M. K. Park, and M. G. Kang. Super-resolution image reconstruction: a technical overview. Signal Processing Magazine, IEEE, 20(3):21 36, may 3. [9] Y. L. S. Wang, L. Zhang and Q. Pan. Semi-coupled dictionary learning with applications to image super-resolution and photo-sketch image synthesis. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), [10] B. Scherrer, A. Gholipour, and S. K. Warfield. Superresolution reconstruction of diffusion-weighted images from distortion compensated orthogonal anisotropic acquisitions. In Mathematical Methods in Biomedical Image Analysis (MMBIA), 2012 IEEE Workshop on, pages , jan [11] D. Wallach, F. Lamare, G. Kontaxakis, and D. Visvikis. Super-resolution in respiratory synchronized positron emission tomography. Medical Imaging, IEEE Transactions on, 31(2): , feb [12] J. Yang, Z. Wang, Z. Lin, S. Cohen, and T. Huang. Couple dictionary training for image super-resolution. Image Processing, IEEE Transactions on, PP(99):1, [13] J. Yang, J. Wright, T. Huang, and Y. Ma. Image superresolution as sparse representation of raw image patches. In Computer Vision and Pattern Recognition, 8. CVPR 8. IEEE Conference on, pages 1 8, june 8. [14] J. Yang, J. Wright, T. Huang, and Y. Ma. Image superresolution via sparse representation. Image Processing, IEEE Transactions on, 19(11): , nov [15] X. Zeng and H. Huang. Super-resolution method for multiview face recognition from a single image per person using nonlinear mappings on coherent features. Signal Processing Letters, IEEE, 19(4): , april [16] R. Zeyde, M. Elad, and M. Protter. On single image scale-up using sparse-representations. In Curves and Surfaces, pages , 2010.
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