On Single Image Scale-Up using Sparse-Representation Roman Zeyde, Matan Protter and Michael Elad The Computer Science Department Technion – Israel Institute of Technology – Haifa 32000, Israel {romanz,matanpr,elad}@cs.technion.ac.il February 27, 2010 Abstract In this paper we use sparse-representation modeling for the single image scale-up problem. The goal is to recover an original image from its blurred and down-scaled noisy version. Since this problem is highly ill-posed, a prior is needed in order to solve it in a robust fashion. The literature offers various ways to address this problem, ranging from simple linear space-invariant interpolation schemes (e.g., bicubic interpolation), to spatially adaptive and non-linear filters of various sorts. In this paper, we embark from a recently-proposed algorithm by Yang et. al. [1, 2], and similarly assume a local Sparse-Land model on image patches, thus stabilizing the problem. We introduce several important modifications to the above-mentioned solution, and show that these lead to improved results. These modifications include a major simplification of the overall process both in terms of the computational complexity and the algorithm architecture, using a different training approach for the dictionary-pair, and operating without a training-set by boot-strapping the scale-up task from the given low-resolution image. We demonstrate the results on true images, showing both visual and PSNR improvements. Keywords: Sparse representations, Single Image Scale-Up, Super-Resolution, K-SVD, OMP. 1 Introduction Single image scale-up1 problem can be formulated as follows: denote the original high-resolution image as yh ∈ RNh , represented as a vector of length Nh pixels. In addition, denote the blur and decimation operators as H:RNh → RNh and S:RNh → RNl (where Nl < Nh) respectively. We shall assume hereafter that H applies known low-pass filter to the image, and S performs a decimation by an integer factor s, by discarding rows/columns from the input image. We define zl to be the low-resolution noisy version of the original image as zl = SHyh + v (1) for an additive Gaussian noise, denoted as v ∼ N ¡ 0, σ2I ¢ . Given zl, our problem is to find ˆy ∈ RNh such that ˆy ≈ yh. Due to the Gaussianity of v, the maximum-likelihood estimation is obtained by the minimization kSHˆy − zlk2. However, since SH 1This problem is often referred to in the literature as super-resolution. The term super-resolution may be confusing, as it is also used in the context of fusing several low-resolution images into one high-resolution result. 1 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 is rectangular with more columns than rows, it cannot be inverted stably, and there are infinitely many perfect solutions that lead to a zero value in the above-mentioned least-squares term. Existing single-image scale-up algorithms use various priors on the image in order to stabilize this inverse problem. In this work we shall use the Sparse-Land local model, as introduced in [3, 4] in the context of denoising, for the scale-up problem. This model assumes that each patch from the images considered can be represented well as a linear combination of a few atoms from a dictionary using a sparse coefficient vector. This assumption will help us in developing an algorithm for image scale-up. We note that this is also the path taken by [1, 2], and our work differs from their solution is four important aspects: • The proposed algorithm is much simplified by removing redundant steps; • Numerical shortcuts bring the proposed algorithm to be highly efficient and much faster; • A different training approach is used for the dictionary-pair2; and • Our algorithm operates without a training-set, by boot-strapping the scale-up task from the given low-resolution image. This idea is similar in spirit to the concept posed in [9], but our solution is simpler and yet very effective. This paper is organized as follows: The combination of the Sparse-Land model into the scale-up problem is shown in Section 2. The third section describes the actual implementation details of the algorithm. Experiments and comparative results are brought in Section 4, and the conclusions are drawn in Section 5. 2 Incorporating the Sparse-Land Prior In order to avoid complexities caused by the different resolutions of zl and yh, and in order to simplify the overall recovery algorithm, we shall assume hereafter that the image zl is scaled-up by a simple interpolation operator Q:RNl → RNh that fills-in the missing rows/columns, returning to the size of yh. We emphasize that this decision will not influence badly the computational complexity of the algorithm, and in fact, the eventual scale-up algorithm we build here is much faster than the one proposed in [1, 2, 8]. The scaled-up image shall be denoted by yl, and it satisfies the relation yl = Qzl = Q(SHyh + v) = QSHyh + Qv = Lallyh + ˜v.(2) 2We have recently found that a parallel work reported in [8] uses a similar (but different!!) training method to the one we pose here. 2 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 Our goal is to process yl ∈ RNh and produce a result ˆyh ∈ RNh , which will get as close as possible to the original high-resolution image, yh ∈ RNh . The processing we are about to present operates on patches extracted from yl, aiming to estimate the corresponding patch from yh. Let ph k = Rkyh ∈ Rn be a high-resolution image patch of size √n×√n, extracted by the operator Rk :RNh → Rn from the image yh in location k. We shall assume that the locations to consider, k, are only those centered around true pixels in the low-resolution image yl (as opposed to filled-in pixels due to the interpolation). We shall refer hereafter to this set of samples as the set Ω. It is now time to invoke the Sparse-Land model: We shall further assume that ph k can be represented sparsely by qk ∈ Rm over the dictionary Ah ∈ Rn×m, namely, ph k = Ahqk, where kqkk0 ¿ n. The `0-pseudo-norm counts the number of non-zeros in the vector qk. Consider the corresponding low-resolution patch pl k = Rkyl, extracted from yl in the same lo- cation3 k, such that its size is √n × √n. Since the operator Lall = QSH transforms the complete high-resolution image yh to the low-resolution one, yl, we can assume that pl k = Lph k + ˜vk, where L is a local operator being a portion of Lall, and ˜vk is the additive noise in this patch. Note that L is a spatially independent operator, as we consider only locations k ∈ Ω from yl. Since we have assumed that ph k = Ahqk, multiplication of this equation by L gives Lph k = LAhqk.(3) Exploiting the relation between the low-resolution and the high-resolution patches, pl k = Lph k + ˜vk, we thus get LAhqk = Lph k = pl k − ˜vk (4) implying that kpl k − LAhqk2 ≤ ² (5) where ² is related to the noise power σ of ˜v. The key observation from the above derivations is that the low-resolution patch should be repre- sented by the same sparse vector qk over the effective dictionary Al = LAh, with a controlled error ²l. This implies that for a given low-resolution patch pl k, we should find its sparse representation vector, qk, and then we can recover ph k by simply multiplying this representation by the dictionary Ah. This is the core idea behind the image scale-up algorithm as developed by Yang et. al, [1, 2], and we follow it as well, with important modifications. 3The patches pl k and ph k are centered around the same pixel. 3 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 3 The Single-Image Scale-Up Algorithm 3.1 Overall Structure The scale-up algorithm we are about to present consists of a training phase (that can be done off-line) and a testing phase, performing the scale-up on the test image, using the trained model from the previous phase. We start the discussion with the training phase, which includes the following steps: 1. Patch-Pairs Construction: For a set of training images that are given in high and low-resolution pairs {yj h, yj l }j, we extract pairs of matching patches that form the training database, P = {ph k, pl k}j. Each of these patch-pairs undergoes a pre-processing stage that removes the low- frequencies from ph k and extracts features from pl k. 2. Dictionary Training: A dictionary Al is trained for the low-resolution patches, such that they can be represented sparsely. A corresponding dictionary Ah is constructed for the high- resolution patches, such that it matches the low-resolution one. The above training phase is done off-line, producing the two dictionaries, Al and Ah, to be used in the super-resolution reconstruction. Given a test low-resolution image yl to be scaled-up (recall that yl is already interpolated to the destination size, and all we need to perform is a spatial non-linear filtering that sharpens this image), we extract pre-processed patches pl k from each location k ∈ Ω, and then sparse-code them using the trained dictionary Al. The found representation qk is then used to recover the high-resolution patch by multiplying it with Ah. We now turn to describe all the above process is more details. 3.2 Training Set Construction The training phase starts by collecting several images {yj h}j, which are the high-resolution examples. Each of these images is blurred and down-scaled by a factor s. This leads to the formation of the corresponding low-resolution images {zj l }j, which are then scaled up back to the original size, resulting with the set {yj l }j. Thus, yj l = Lallyj h + ˜vj.(6) It is important to note that the same operator Lall should be used both in the training and the testing phases. 4 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 3.3 Preprocessing and Feature Extraction The next step to perform is pre-processing. Rather than extracting small image-patches and applying this step on them, we employ the desired pre-processing directly on the full images, and only then extract the patches. This avoids boundary problems due to the small patch size4. The pre-processing applied on the high-resolution images consists of a removal of their low- frequencies. This is done by computing the difference images ej h = yj h − yj l . The reason for this step is the desire to focus the training on characterizing the relation between the low-resolution patches, and the edges and texture content within the corresponding high-resolution ones. As for the pre-processing of the low-resolution images, these are filtered using K high-pass filters, in order to extract local features that correspond to their high-frequency content. Thus, each low- resolution image yj l leads to a set of K filtered images, fk ∗ yj l for i = 1, 2,...,K(where ∗ stands for a convolution). After the two pre-processing steps described above, we are ready to extract the local patches, and form the data-set P = {ph k, pl k}k. Considering only locations k ∈ Ω, we extract patches of size √n × √n pixels from the high-resolution images ej h. The corresponding low-resolution patches are extracted from the same locations in the filtered images fk ∗ yj l and using the same size (√n × √n pixels). Thus, every corresponding K such low-resolution patches are merged into one vector ˜pl k of length nK. Note that the high-resolution patch size should be at least of size s×s so as to cover the high-resolution image. If the patch size is larger, we get overlaps that improve the reconstruction result. 3.4 Dimensionality Reduction The last step before turning to the dictionary learning stage is to reduce the dimension of the input low-resolution patches, {˜pl k}k. We apply the Principal Component Analysis (PCA) algorithm on these vectors, seeking a subspace on which we could project these patches while preserving 99.9% of their average energy. The motivation for this step is the knowledge that the low-resolution patches could not occupy the full nK space, since they are effectively built of patches of size √n/s×√n/s in the low-resolution image zj l that go through a set of K linear filters. The scaling up and the filtering are not expected to increase the dimensionality of the resulting patches. By such a dimensionality reduction we expect to save computations in the subsequent training and super-resolution algorithms. 4A patch of size √n × √n in yl should correspond to a larger patch in yh, because of the spatial extent of the blur and the scale-up operations. Nevertheless, this additional “band” of pixels can be disregarded, as if we concentrate on predicting only the center of the destination patch from yh. 5 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 We denote by B ∈ Rnl×nK the projection operator that transforms the patch ˜pl k to its reduced feature vector, pl k = B˜pl k ∈ Rnl . 3.5 Dictionary Learning Turning to the dictionary learning part in the training, we start with the low-resolution training patches {pl k}k. We apply K-SVD dictionary training procedure for these patches, resulting with the dictionary Al ∈ Rnl×m. Within this learning procedure, sparse-coding is performed using the OMP with a fixed number of atoms per example, L. As a side product of this training, we obtain the sparse representation vector qk that corresponds to the training patch pl k. After constructing Al, we proceed to the high-resolution dictionary construction. Recall that it is our intention to recover the patch ph k by approximating it as being ph k ≈ Ahqk, namely, use the found sparse representation vector for the low-resolution patch, and multiply it by the high-resolution dictionary. Thus, we should seek a dictionary Ah such that this approximation is as exact as it can be. Thus, we define this dictionary to be the solver of the problem Ah = arg min Ah X k kph k − Ahqkk2 2 (7) = arg min Ah kPh − AhQk2 F, where we have constructed the matrix Ph, such that the high-resolution training patches, {ph k}k from its columns, and similarly, Q contain {qk}k as its column. The solution of the problem is given by Ah = PhQ+ = PhQT(QQT)−1.(8) We should note that the above approach disregards the fact that the high-resolution patches overlap, and thus a better training procedure can be envisioned for computing Ah. Bearing in mind that the eventual high-resolution images (in the test stage) would be constructed by positioning these patches and averaging over their overlaps, we should optimize Ah such that these resulting images are as close as possible to their origin. Put formally, the image ˆyj h should be constructed by the following formula: ˆyj h = yj l + "X k∈Ω RT k Rk #−1 "X k∈Ω RT k Ahqk # .(9) 6 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 Thus, it is natural to define the best dictionary Ah as the solution of the optimization task Ah = arg min Ah X j °°°yj h − ˆyj h °°° 2 2 (10) = arg min Ah X j °°°°°° yj h − yj l − "X k∈Ω RT k Rk #−1 "X k∈Ω RT k Ahqk #°°°°°° 2 2 . The reason yj l appears in the errors computation is the fact that the patches in Ph are constructed from the difference images ej h = yj h − yj l , and this means that for the image ˆyj h to be constructed, we should return these low-frequencies. While this minimization task is more demanding, it is expected to lead to better output quality. In the experiments given below, we shall adopt the simpler way to derive Ah. See the discussion section for some notes about this method. This concludes the training phase of the super-resolution algorithm. 3.6 Testing Phase We are given a test low-resolution image zl to be magnified. This image is assumed to have been generated from a high-resolution image yh by the same blur and scale-down operations as used in the training. The following are the steps performed in order to complete the super-resolution algorithm: 1. Scale this image up by a factor of s using bicubic interpolation, resulting with yl. 2. Filter the image yl using the same K high-pass filters that were used for feature extraction in the training, and obtain fk ∗ yl. 3. Extract patches from these K images, each of size √n × √n from locations k ∈ Ω. Every K such patches that correspond to the same location are to be concatenated to form a patch vector ˜pl k. This leads to the set {˜pl k}k. 4. Multiply the found patches {˜pl k}k by the projection operator B to reduce their dimensionality, resulting with the set {pl k}k, each patch of length nl (≈ 30). 5. Apply the OMP algorithm on {pl k}k, allocating L atoms to their representation, and find the sparse representation vectors {qk}k. 6. Multiply the representation vectors {qk}k by the high-resolution dictionary Ah, and obtain the approximated high-resolution patches, {Ahqk}k = {ˆph k}k. 7 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 7. Construct the final super-resolved image by putting ˆph k to their proper location, averaging in overlap regions, and adding yl to the final image. Put differently, compute the final image by the formula, which performs exactly the above-described procedure: ˆyh = yl + "X k∈Ω RT k Rk #−1 X k∈Ω RT k ˆph k.(11) 4 Results and discussion We present several experiments that demonstrate the above-described scaling-up algorithm in action. 4.1 Text Scale-Up The first test is applied on images showing a printed text. The training image (screen-grabbed from a PDF file) is shown in Figure 1 – note that only one image is used for training, and adding more is expected to improve the results. The global operator Lall in this experiment is implemented by first blurring the high-resolution images yj h with a 1D filter [1, 3, 4, 3, 1]/12 both horizontally and vertically, and then down-sampling it by a factor s = 3. I.e., the scaled-down image zl is one-ninth of the original image size. The image yl is created by bicubic interpolation of zl, returning to the original size. Extraction of features from the low-resolution images is done using 4 filters that perform 1-st and 2-nd horizontal and vertical derivatives: f1 = [1, −1] = fT 2 and f3 = [1, −2, 1] = fT 4 . These filters are applied such that only sampled pixels are used in the filtering computation5 The patch size used is n = 9, and the PCA results with a reduction from 4 · 81 = 324 to nl ≈ 30. The dictionary training procedure applied 40 iterations of the K-SVD algorithm, with m = 1, 000 atoms in the dictionary, and allocating L = 3 atoms per patch-representation. The test image (a different image, grabbed from a different page, but having the same scale) is shown in Figure 2. This figure shows the original test image, and the scaled-down version that should be scaled-up. The scaling-up results are shown in Figure 3, and it is evident that the outcome is far better, compared to the bicubic interpolation, showing an improvement of 2.27dB. 4.2 Photo Scale-Up The second experiment is applied on the image Building. Starting from the original image yh of size 800 × 800 pixels, we filter this image with the separable filter [1, 2, 1]/4 (horizontally and vertically), 5This means that we either filter zl and then interpolate, or we filter yl with zero-padded filters of the form f1 = [0, 0, 1, 0, 0, −1] = f T 2 and f3 = [1, 0, 0, −2, 0, 0, 1] = f T 4 . 8 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 Figure 1: First experiment: The training image for the image scaling-up algorithm. This image is of size 717 × 717 pixels, and it provides a set of 54, 289 training patch-pairs. 9 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 Figure 2: First experiment: The test image zl to be scaled-up (left), and its original high-resolution version yh (right). The image zl is of size 120 × 120 pixels, and it provides a set of 12, 996 patches to operate on. Figure 3: First experiment: The scaling up results – the image yl that was scaled-up using a bicubic interpolation (left, RMSE= 47.06), and the algorithm’s result ˆyh (right, RMSE= 36.22). 10 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 and then down-scale it by factor s = 2 to obtain zl of size 400 × 400. In this experiment we train the dictionaries using the very same image, by further scaling it down by a factor s = 2, resulting with the image zll of size 200 × 200. The image pair {zl, zll} is used for the training, based on the expectation that the relation between these two images reflects also the relation that should be used to go from zl to yh. Extraction of features from the low-resolution images is done using the same 4 filters, and the dimensionality reduction leads this time to nl ≈ 42. The training data contains 37, 636 pairs of low- and high-resolution patches to be modeled. The parameters of the dictionary training all remain the same (40 iterations of the K-SVD algorithm, m = 1000 atoms in the dictionary, and L = 3 atoms per representation). Figure 4 shows the original image yh, the bicubic scaled-up image yl, and the result of the scaling up algorithm, ˆyh. The difference between the two images is 3.32dB, and in order to see where these differences reside, the figure also shows the the image |ˆyh −yh|. Figure 5 shows two 100×100 portions extracted from yh, yl, and ˆyh, to better show the visual gain achieved by the scaling-up algorithm. 4.3 PSNR Comparison with Yang et. al. We have trained our algorithm on the same training set used at [1], using s = 3 scale-up configuration. Each training image, was blurred and decimated using imresize MATLAB function, using a bicubic filter; feature extraction was done as before (using gradient and laplacian filters). We collected around 130,000 training patch-pairs and applied PCA to reduce feature dimension to nl ≈ 30. Low-resolution dictionary learning took approximately 12 minutes to apply 40 iterations of the K-SVD algorithm, with m = 1, 000 atoms in the dictionary, and allocating L = 3 atoms per patch-representation. Moreover, high-resolution dictionary training took just a few seconds, using Pseudo-Inverse expression Ah = PhQ+. We note that our training algorithm is much faster than the one used by Yang et. al. (taking several hours to run). Then, we tested our reconstruction algorithm on 14 test images (taking a few seconds on each image, using fully overlapping 3 × 3 patches in low-resolution scale) and compared its results versus bicubic interpolation and Yang et. al. reconstruction algorithm. The resulting images’ boundary was cropped (to ignore boundary effects of overlap-and-add method) and Peak-SNR was computed using the following formula: y, ˆy ∈ [0, 1]N ⊆ RN(12) PSNR = 10 log10 µ P i 12 P i(ˆyi − yi)2 ¶ = 10 log10 µ NP i(ˆyi − yi)2 ¶ (13) 11 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 Figure 4: Second experiment: The original Building image yh (top-left), the bicubic interpolated image yl (bottom-left, RMSE= 12.78), the algorithm’s result ˆyh (bottom-right, RMSE= 8.72), and the difference image |ˆyh − yh| magnified by 5 (top-right). 12 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 Figure 5: Second experiment: Portions from the original Building image yh (left), the bicubic interpolated image yl (middle), and the algorithm’s result ˆyh (right. Notice that the original portions show some compression artifacts, which do not appear in the scaled-up results. 13 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 The results are summarized at Figure 6. Image Name Bicubic Yang et. al Our algorithm baboon 23.2 23.5 23.5 barbara 26.2 26.4 26.8 bridge 24.4 24.8 25.0 coastguard 26.6 27.0 27.1 comic 23.1 23.9 24.0 face 32.8 33.1 33.5 flowers 27.2 28.2 28.4 foreman 31.2 32.0 33.2 lenna 31.7 32.6 33.0 man 27.0 27.8 27.9 monarch 29.4 30.7 31.1 pepper 32.4 33.3 34.1 ppt3 23.7 25.0 25.2 zebra 26.6 28.0 28.5 Figure 6: Third experiment: PSNR results. We compare our results for a few 100 × 100 representative patches from different images. The results are given at Figure 7, where our results are at the last column. We note that our algorithm performs visually much better than bicubic interpolcation, and on some images considerably better than Yang et. al. algorithm, having less visual artifacts and producing sharper results. Moreover, our implementation is much faster (by an order of magnitude) than Yang 14 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 Figure 7: Third experiment: Portions from various images (barbara, comic, face, pepper, zebra). Left to right: the original image, bicubic interpolation, Yang et. al and our algorithm. Note that our algorithm produces sharper results, preserves the small details of the image and has much less visual artifacts. 15 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 et. al. implementation, using optimized K-SVD and OMP implementations by [7]. 4.4 Discussion 4.4.1 Training phase When compared to the algorithms [1] and [8], our algorithm uses the same idea of training phase and testing phase. However, we use different algorithms for training our dictionaries: K-SVD for low-resolution dictionary, and pseudo-inverse for high-resolution dictionary. Moreover, we use OMP for sparse-coding, instead of LASSO optimization methods. It should be noted that we used L = 3 (as a sparsity constraint) which is much smaller than the sparsity achieved by [1], while our PSNR results are better. This may happen due to better generalization ability of our model, requiring much less atoms per patch. Moreover, while [1] suggests training high-resolution and low-resolution dictionaries together (by concatenating high-resolution and low-resolution patches together into one vector), we split the process as described above - thus achieving more stable reconstruction process (having less visual artifacts). We have compared the results of [8] to ours, and have found that our results have much better PSNR score 6. 4.4.2 Bootstrapping approach In case the scale-up process has no other information except the low-resolution image, we can still “bootstrap” our model from the image itself, as proposed by [9]. We may apply dictionary training to high-resolution and low-resolution patches collected from the to-be-scaled image zl. This approach is valid, given the training process has enough training data to build a valid Sparse-Land model. We have tested this approach on several images using various dictionary sizes, but the reconstruction results were inferior to separate training phase approach. We should note that [9] provide no “ground-truth” images, so we could not provide PSNR results. Moreover, the algorithm in [9] is much more complex than ours (requiring the solution of many Nearest-Neighbor problems for reconstruction phase) and relies heavily on patch recurrence property. Nevertheless, it does performs quite well on the testing set of images described in [9], probably due to “coarse-to-fine” approach: the image is scaled-up n times, each time by n√s factor. 6In fact, we found that using simple bicubic interpolation gives better results - both visually and by PSNR. 16 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 4.4.3 Alternative high-resolution dictionary training We have noted that given Ah and sparse representation coefficients {qk}, the optimal reconstruction is given by: ˆyh = yl + "X k RT k Rk #−1 "X k RT k Ahqk # (14) Denote the diagonal matrix D = P k RT k Rk ∈ RNh×Nh and Dk = RkD−1 ∈ Rn×Nh and rewrite: ˆyh = yl + X k DT k Ahqk (15) We would like to minimize kyh − ˆyhk2 2 with respect to Ah, denoting eh = yh − yl: Ah = arg min Ah kyh − yl − X k DT k Ahqkk2 2 = arg min Ah keh − X k DT k Ahqkk2 2 (16) Given X ∈ Rn×m, we define x ≡ vec(X) to be: xi+nj = Xij. Therefore: vec(BAC) = ¡ CT ⊗ B ¢ vec(A) (17) X k DT k Ahqk = ÃX k qT k ⊗ DT k ! vec(Ah) = M· vec(Ah) (18) (19) where M ∈ RNh×mn is defined as: M = ÃX k qk ⊗ Dk !T (20) Note that the gradient of f(Ah) = 1 2keh − P k DT k Ahqkk2 2 with respect to Ah, can be written as: ∇Ah f = MT(eh − M· vec(Ah)) (21) So the solution for Ah can be achieved using M†eh or using an iterative scheme, using the gradient expression above. 4.4.4 Back-projection We may use back-projection method as suggested in [1] and implemented in [9]. The image ˆyh we construct does not necessarily conforms with the requirement Lallˆyh ≈ yl. Thus, we may project the result ˆyh to this constraint, by the following derivation: ˆyh = arg min ˆyh kLallˆyh − ylk2 2 (22) 17 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 Differentiating with respect to ˆyh, we get an expression for the gradient, for a steepest descent algorithm, initialized by the solution of our algorithm: ∆ˆyh ∝ LT all (yl − Lallˆyh) (23) However, since the objective function above does not have a single minimum (due to Lall singular- ity) we may not converge to the correct image yh, but to any image from the affine set yh + ker Lall. Indeed, using back-projection in our scenario didn’t improved the PSNR, but even added some artifacts to the image. 4.5 Summary There are various ways to scale-up an image while preserving edges and small details. In this paper we introduced one such algorithm that illustrates how sparse representation modeling and dictionary learning can be used for this goal. The presented algorithm is based on the method proposed by Yang et. al., with few modifications. This method is relatively simple, and yet it produces a substantial improvement over bicubic inter- polation. The algorithm operates by training a pair of low- and high-resolution dictionaries, using either training images, or exploiting a lower-resolution version of the same image to be handled. Future work Various further improvements can be considered, and these are likely to enhance the algorithm’s output quality. • One such option is to force the overlapping patches ˆph k to better align with each other. This can be done by operating sequentially on the incoming patches pl k, and when applying the sparse coding stage (to produce qk, add a penalty on the distance between the newly constructed patch, ˆph k, and the ones already computed. • Another option is the optimization of feature extraction and dimensionality reduction opera- tors. We have tried to use various high-pass filters and various thresholds for PCA transform, however there must be more options to investigate. • We may extend our training set by adding more examples, by applying simple operators on the input images, e.g. rotation by 90◦, reflection, etc. • We should note that it was assumed that the blur operator H is known for all the experiments we have performed. We may suggest that in case it is not the case (i.e. while bootstrapping 18 Technion - Computer Science Department - Tehnical Report CS-2010-12 - 2010 from a single image) we have a significant degree of freedom in choosing H– which obviously will affect the results. • In addition, we may suggest combining the approaches of [1] and [9] by training a general dictionary pair {Al,Ah} and apply several more iterations on the low-resolution image and its down-scaled version {zl, zll} in order to “adapt” the reconstruction process to the specific image to be reconstructed. • We suggest that using joint sparsity (for similar image patches) during dictionary training, since usually regular images tend to have many reoccurring patches, thus stabilizing the training process and helping to achieve even better results. • We conclude by suggesting that using more than two scales (the “low” and the “high” ones) in a “coarse-to-fine” framework, may help improve the scale-up process by building multi-scale sparse-representation for the image. 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