Super-resolution: a comprehensive survey <s> Wavelet transform <s> In this paper we propose an image super-resolution algorithm using wavelet-domain hidden Markov tree (HMT) model. Wavelet-domain HMT models the dependencies of multiscale wavelet coefficients through the state probabilities of wavelet coefficients, whose distribution densities can be approximated by the Gaussian mixture. Because wavelet-domain HMT accurately characterizes the statistics of real-world images, we reasonably specify it as the prior distribution and then formulate the image super-resolution problem as a constrained optimization problem. And the cycle-spinning technique is used to suppress the artifacts that may exist in the reconstructed high-resolution images. Quantitative error analyses are provided and several experimental images are shown for subjective assessment. <s> BIB001 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> We consider the estimation of the unknown hyperparameters for the problem of reconstructing a high-resolution image from multiple undersampled, shifted, degraded frames with subpixel displacement errors. We derive mathematical expressions for the iterative calculation of the maximum likelihood estimate (MLE) of the unknown hyperparameters given the low resolution observed images. Experimental results are presented for evaluating the accuracy of the proposed method. <s> BIB002 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> We propose a single-frame, learning-based super-resolution restoration technique by using the wavelet domain to define a constraint on the solution. Wavelet coefficients at finer scales of the unknown high-resolution image are learned from a set of high-resolution training images and the learned image in the wavelet domain is used for further regularization while super-resolving the picture. We use an appropriate smoothness prior with discontinuity preservation in addition to the wavelet-based constraint to estimate the super-resolved image. The smoothness term ensures the spatial correlation among the pixels, whereas the learning term chooses the best edges from the training set. Because this amounts to extrapolating the high-frequency components, the proposed method does not suffer from oversmoothing effects. The results demonstrate the effectiveness of the proposed approach. © 2004 Wiley Periodicals, Inc. Int J Imaging Syst Technol 14, 105–112, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ima.20013 <s> BIB003 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> Wavelet coefficient thresholding is effective in reducing spatial domain noise in wavelet-based super-resolution algorithms. Here, the effect of the threshold level on reconstructed image quality in second-generation wavelet super-resolution is investigated. The choice of optimal threshold involves a tradeoff between noise filtering and blurring introduced by thresholding. A measure based on the singular values of the image matrix is employed as a reliable gauge of generated high-resolution image quality. <s> BIB004 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> We are concerned with analysis of linear reconstruction-based super-resolution (SR) techniques. We propose a novel Fourier interpretation of an "archetype" linear SR algorithm and derive practical results on the expected performances of such techniques. <s> BIB005 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> The super-resolution problem is considered using a mean-squared error minimizing solution of a generalized undersampling model in this non-iterative frequency domain approach. While previous frequency domain approaches have been based on a bandlimited image model, this approach uses a non-bandlimited stationary spectral model. This allows improved reconstruction of certain image features. The model and algorithm are presented along with an example <s> BIB006 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> In this paper, we study the problem of reconstructing a high-resolution image from several blurred low-resolution image frames. The image frames consist of decimated, blurred and noisy versions of the high-resolution image. The high-resolution image is modeled as a Markov random field (MRF), and a maximum a posteriori (MAP) estimation technique is used for the restoration. We show that with the periodic boundary condition, the high-resolution image can be restored efficiently by using fast Fourier transforms. We also apply the preconditioned conjugate gradient method to restore the high-resolution image. Computer simulations are given to illustrate the effectiveness of the proposed method. <s> BIB007 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> Hyperspectral images are used for aerial and space imagery applications, including target detection, tracking, agricultural, and natural resource exploration. Unfortunately, atmospheric scattering, secondary illumination, changing viewing angles, and sensor noise degrade the quality of these images. Improving their resolution has a high payoff, but applying super-resolution techniques separately to every spectral band is problematic for two main reasons. First, the number of spectral bands can be in the hundreds, which increases the computational load excessively. Second, considering the bands separately does not make use of the information that is present across them. Furthermore, separate band super resolution does not make use of the inherent low dimensionality of the spectral data, which can effectively be used to improve the robustness against noise. In this paper, we introduce a novel super-resolution method for hyperspectral images. An integral part of our work is to model the hyperspectral image acquisition process. We propose a model that enables us to represent the hyperspectral observations from different wavelengths as weighted linear combinations of a small number of basis image planes. Then, a method for applying super resolution to hyperspectral images using this model is presented. The method fuses information from multiple observations and spectral bands to improve spatial resolution and reconstruct the spectrum of the observed scene as a combination of a small number of spectral basis functions. <s> BIB008 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> In this paper we propose a computationally efficient method for image super-resolution reconstruction. We concentrate on pure translational motion and shift-invariant blur. This is the case of interlaced sampling where each low resolution image is uniformly sampled but the overall sampling is non-uniform with respect to the high resolution grid. The reconstruction problem is considered in a multi-resolution framework using separable B-spline wavelets as basis. This allows performing the reconstruction on a uniform higher resolution grid by digital filtering. A computationally efficient structure ? the transposed modified Farrow structure is used to achieve a near least squares solution without using any matrix inversions or iterations. The reconstruction can also be performed at non-dyadic scales. Our method minimizes the aliasing distortions and gives results comparable with other least squares techniques. <s> BIB009 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> We study the problem of reconstruction of a high-resolution image from several blurred low-resolution image frames. The image frames consist of blurred, decimated, and noisy versions of a high-resolution image. The high-resolution image is modeled as a Markov random field (MRF), and a maximum a posteriori (MAP) estimation technique is used for the restoration. We show that with the periodic boundary condition, a high-resolution image can be restored efficiently by using fast Fourier transforms. We also apply the preconditioned conjugate gradient method to restore high-resolution images in the aperiodic boundary condition. Computer simulations are given to illustrate the effectiveness of the proposed approach. <s> BIB010 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> Super-resolution algorithms reconstruct a high-resolution image from a set of low-resolution images of a scene. Precise alignment of the input images is an essential part of such algorithms. If the low-resolution images are undersampled and have aliasing artifacts, the performance of standard registration algorithms decreases. We propose a frequency domain technique to precisely register a set of aliased images, based on their low-frequency, aliasing-free part. A high-resolution image is then reconstructed using cubic interpolation. Our algorithm is compared to other algorithms in simulations and practical experiments using real aliased images. Both show very good visual results and prove the attractivity of our approach in the case of aliased input images. A possible application is to digital cameras where a set of rapidly acquired images can be used to recover a higher-resolution final image. <s> BIB011 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> A learning-based super-resolution system consisting of training and synthesis processes is presented. In the proposed system, a multiresolution wavelet approach is applied to carry out the robust synthesis of both the global geometric structure and the local high-frequency detailed features of a facial image. In the training process, the input image is transformed into a series of images of increasingly lower resolution using the Haar discrete wavelet transform (DWT). The images at each resolution level are divided into patches, which are then projected onto an eigenspace to derive the corresponding projection weight vectors. In the synthesis process, a low-resolution input image is divided into patches, which are then projected onto the same eigenspace as that used in the training process. Modeling the resulting projection weight vectors as a Markov network, the maximum a posteriori (MAP) estimation approach is then applied to identity the best-matching patches with which to reconstruct the image at a higher level of resolution. The experimental results demonstrate that the proposed reconstruction system yields better results than the bi-cubic spline interpolation method. <s> BIB012 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> High resolution image reconstruction is an image process to reconstruct a high resolution image from a set of blurred, degraded and shifted low resolution images. In this paper, the reconstruction problem is treated as a function approximation. We use linear interpolation to build up an algorithm to obtain the relationship between the detail coefficients in wavelet subbands and the set of low resolution images. We use Haar wavelet as an example and establish the connection between the Haar wavelet subband and the low resolution images. Experiments show that we can use just 3 low resolution images to obtain a high resolution image which has better quality than Tikhonov least-squares approach and Chan et al. Algorithm 3 in low noise cases. We also propose an error correction extension for our method which can lead to very good results even in noisy cases. Moreover, our approach is very simple to implement and very efficient. <s> BIB013 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> In many applications, the sampling frequency is limited by the physical characteristics of the components: the pixel pitch, the rate of the analog-to-digital (AID) converter, etc. A low- pass filter is usually applied before the sampling operation to avoid aliasing. However, when multiple copies are available, it is possible to use the information that is inherently present in the aliasing to reconstruct a higher resolution signal. If the different copies have unknown relative offsets, this is a nonlinear problem in the offsets and the signal coefficients. They are not easily separable in the set of equations describing the super-resolution problem. Thus, we perform joint registration and reconstruction from multiple unregistered sets of samples. We give a mathematical formulation for the problem when there are M sets of N samples of a signal that is described by L expansion coefficients. We prove that the solution of the registration and reconstruction problem is generically unique if MN ges L + M - 1. We describe two subspace-based methods to compute this solution. Their complexity is analyzed, and some heuristic methods are proposed. Finally, some numerical simulation results on one- and two-dimensional signals are given to show the performance of these methods. <s> BIB014 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> Super resolution imaging refers to inferring the missing high resolution image from low resolution image(s). Super resolution methods are generally classified into reconstruction based and learning based methods. The learning based methods fully learn the intensity prior between all bands of images. For most of these, some fake information can be inevitably introduced into synthesized high resolution image though the image may offer visually good effects. For medical image analysis, it might deteriorate the image analysis and diagnosis performance. In this paper we propose a better learning using real image training set that enhances the high frequency information. The method exploits richness of real-world images. The training set is preprocessed images so as to extract the structural correlation. The technique learns the fine details that correspond to different image structures seen at a low-resolution and then uses those learned relationships to predict fine details in other images. <s> BIB015 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> In this paper, we propose a new super resolution method Maximum a Posteriori based on a universal hidden Markov tree model (MAP-uHMT) for remote sensing images. The hidden Markov tree theory in the wavelet domain is used to set up a prior model for reconstructing super resolution images from a sequence of warped, blurred, sub-sampled and contaminated low resolution images. Both the simulation results with a Landsat7 panchromatic image and actual results with four Landsat7 panchromatic images which were captured on different dates show that our method achieves better super resolution images both visually and quantitatively than other methods, based on PSNR in the simulation and derived PSF with actual data. <s> BIB016 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> Super-resolution algorithms combine multiple low resolution images into a single high resolution image. They have received a lot of attention recently in various application domains such as HDTV, satellite imaging, and video surveillance. These techniques take advantage of the aliasing present in the input images to reconstruct high frequency information of the resulting image. One of the major challenges in such algorithms is a good alignment of the input images: subpixel precision is required to enable accurate reconstruction. In this paper, we give an overview of some subspace techniques that address this problem. We first formulate super-resolution in a multichannel sampling framework with unknown offsets. Then, we present three registration methods: one approach using ideas from variable projections, one using a Fourier description of the aliased signals, and one using a spline description of the sampling kernel. The performance of the algorithms is evaluated in numerical simulations. <s> BIB017 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> We present an analysis and algorithm for the problem of super-resolution imaging, that is the reconstruction of HR (high-resolution) images from a sequence of LR (low-resolution) images. Super-resolution reconstruction entails solutions to two problems. One is the alignment of image frames. The other is the reconstruction of a HR image from multiple aligned LR images. Both are important for the performance of super-resolution imaging. Image alignment is addressed with a new batch algorithm, which simultaneously estimates the homographies between multiple image frames by enforcing the surface normal vectors to be the same. This approach can handle longer video sequences quite well. Reconstruction is addressed with a wavelet-based iterative reconstruction algorithm with an efficient de-noising scheme. The technique is based on a new analysis of video formation. At a high level our method could be described as a better-conditioned iterative back projection scheme with an efficient regularization criteria in each iteration step. Experiments with both simulated and real data demonstrate that our approach has better performance than existing super-resolution methods. It can remove even large amounts of mixed noise without creating artifacts. <s> BIB018 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> We proposed a compressed sensing Super Resolution algorithm based on wavelet. The proposed algorithm performs well with a smaller quantity of training image patches and outputs images with satisfactory subjective quality. It is tested on classical images commonly adopted by Super Resolution researchers with both generic and specialized training sets for comparison with other popular commercial software and state-of-the-art methods. Experiments demonstrate that, the proposed algorithm is competitive among contemporary Super Resolution methods. <s> BIB019 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> This paper proposes a new algorithm to generate a super-resolution image from a single, low-resolution input without the use of a training data set. We do this by exploiting the fact that the image is highly compressible in the wavelet domain and leverage recent results of compressed sensing (CS) theory to make an accurate estimate of the original high-resolution image. Unfortunately, traditional CS approaches do not allow direct use of a wavelet compression basis because of the coherency between the point-samples from the downsampling process and the wavelet basis. To overcome this problem, we incorporate the downsampling low-pass filter into our measurement matrix, which decreases coherency between the bases. To invert the downsampling process, we use the appropriate inverse filter and solve for the high-resolution image using a greedy, matching-pursuit algorithm. The result is a simple and efficient algorithm that can generate high quality, high-resolution images without the use of training data. We present results that show the improved performance of our method over existing super-resolution approaches. <s> BIB020 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> In this paper, we propose a novel face hallucination method which mainly focuses on recovering the lost high-frequency information of low resolution face images. Both the global information and local detailed features are taken into account in the proposed method. Firstly, the resolution of the low-resolution face image is enhanced by an edge-protecting interpolation algorithm. Then, the structural information of the interpolated face is extracted by principal component analysis. Utilizing the training database, we synthesize the local high-frequency details according to its corresponding structural information. At last, we can get a final high-resolution result by integrating the interpolated face with its corresponding local high-frequency details. Experiments demonstrate that this method can achieve face hallucination with good visual quality. <s> BIB021 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> This paper presents an artifact-free super resolution texture mapping from multiple-view images. The multiple-view images are upscaled with a learning-based super resolution technique and are mapped onto a 3D mesh model. However, mapping multiple-view images onto a 3D model is not an easy task, because artifacts may appear when different upscaled images are mapped onto neighboring meshes. We define a cost function that becomes large when artifacts appear on neighboring meshes, and our method seeks the image-and mesh assignment that minimizes the cost function. Experimental results with real images demonstrate the effectiveness of our method. <s> BIB022 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> We present a novel approach for online shrinkage functions learning in single image super-resolution. The proposed approach leverages the classical Wavelet Shrinkage denoising technique where a set of scalar shrinkage functions is applied to the wavelet coefficients of a noisy image. In the proposed approach, a unique set of learned shrinkage functions is applied to the overcomplete representation coefficients of the interpolated input image. The super-resolution image is reconstructed from the post-shrinkage coefficients. During the learning stage, the lowresolution input image is treated as a reference high-resolution image and a super-resolution reconstruction process is applied to a scaled-down version of it. The shapes of all shrinkage functions are jointly learned by solving a Least Squares optimization problem that minimizes the sum of squared errors between the reference image and its super-resolution approximation. Computer simulations demonstrate superior performance compared to state-of-the-art results. <s> BIB023 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> In this paper, we propose a new learning-based approach for super-resolving an image captured at low spatial resolution. Given the low spatial resolution test image and a database consisting of low and high spatial resolution images, we obtain super-resolution for the test image. We first obtain an initial high-resolution (HR) estimate by learning the high-frequency details from the available database. A new discrete wavelet transform (DWT) based approach is proposed for learning that uses a set of low-resolution (LR) images and their corresponding HR versions. Since the super-resolution is an ill-posed problem, we obtain the final solution using a regularization framework. The LR image is modeled as the aliased and noisy version of the corresponding HR image, and the aliasing matrix entries are estimated using the test image and the initial HR estimate. The prior model for the super-resolved image is chosen as an Inhomogeneous Gaussian Markov random field (IGMRF) and the model parameters are estimated using the same initial HR estimate. A maximum a posteriori (MAP) estimation is used to arrive at the cost function which is minimized using a simple gradient descent approach. We demonstrate the effectiveness of the proposed approach by conducting the experiments on gray scale as well as on color images. The method is compared with the standard interpolation technique and also with existing learning-based approaches. The proposed approach can be used in applications such as wildlife sensor networks, remote surveillance where the memory, the transmission bandwidth, and the camera cost are the main constraints. <s> BIB024 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> We introduce a class of inverse problem estimators computed by mixing adaptively a family of linear estimators corresponding to different priors. Sparse mixing weights are calculated over blocks of coefficients in a frame providing a sparse signal representation. They minimize an l1 norm taking into account the signal regularity in each block. Adaptive directional image interpolations are computed over a wavelet frame with an O(N logN) algorithm, providing state-of-the-art numerical results. <s> BIB025 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> From the perspective of linear algebra, the performance of super-resolution reconstruction (SR) depends on the conditioning of the linear system characterizing the degradation model. This is analyzed in the Fourier domain using the perturbation theory. By proposing a new SR error bound in terms of the point spread function (PSF), we reveal that the blur function dominates the condition number (CN) of degradation matrix, and the advantage of non-integer magnification factors (MFs) over the integer ones comes from sampling zero crossings of the DFT of the PSF. We also explore the effect of regularization by integrating it into the SR model, and investigate the influence of the optimal regularization parameter. A tighter error bound is derived given the optimal regularization parameter. Two curves of error bounds vs. MFs are presented, and verified by processing real images. It explains that with proper regularization, SR at the integer MFs is still valid. <s> BIB026 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> In this correspondence, the authors propose an image resolution enhancement technique based on interpolation of the high frequency subband images obtained by discrete wavelet transform (DWT) and the input image. The edges are enhanced by introducing an intermediate stage by using stationary wavelet transform (SWT). DWT is applied in order to decompose an input image into different subbands. Then the high frequency subbands as well as the input image are interpolated. The estimated high frequency subbands are being modified by using high frequency subband obtained through SWT. Then all these subbands are combined to generate a new high resolution image by using inverse DWT (IDWT). The quantitative and visual results are showing the superiority of the proposed technique over the conventional and state-of-art image resolution enhancement techniques. <s> BIB027 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> In this paper, we propose a novel single image super-resolution method based on MAP statistical reconstruction. Our approach takes the wavelet domain Implicit Markov Random Field (IMRF) model as the prior constraint. Specifically, we apply the IMRF model to show the probability distribution of natural images in wavelet domain, and utilize the MAP theory of Bayesian estimation to construct the objective function with this model. Furthermore, we employ steepest descent method to optimize this objective function. In the experiments, we use both Peak Signal to Noise Ratio (PSNR) and visual effect to evaluate our method. The experimental results demonstrate that our method obtains the superior performance in comparison with traditional single image super-resolution approaches. <s> BIB028 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> This paper addresses the reconstruction of high-resolution omnidirectional images from multiple low-resolution images with inexact registration. When omnidirectional images from low-resolution vision sensors can be uniquely mapped on the 2-sphere, such a reconstruction can be described as a transform-domain super-resolution problem in a spherical imaging framework. We describe how several spherical images with arbitrary rotations in the SO(3) rotation group contribute to the reconstruction of a high-resolution image with help of the spherical Fourier transform (SFT). As low-resolution images might not be perfectly registered in practice, the impact of inaccurate alignment on the transform coefficients is analyzed. We then cast the joint registration and super-resolution problem as a total least-squares norm minimization problem in the SFT domain. A l1-regularized total least-squares problem is considered and solved efficiently by interior point methods. Experiments with synthetic and natural images show that the proposed methods lead to effective reconstruction of high-resolution images even when large registration errors exist in the low-resolution images. The quality of the reconstructed images also increases rapidly with the number of low-resolution images, which demonstrates the benefits of the proposed solution in super-resolution schemes. Finally, we highlight the benefit of the additional regularization constraint that clearly leads to reduced noise and improved reconstruction quality. <s> BIB029 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> In multi-slice magnetic resonance imaging (MRI) the resolution in the slice direction is usually reduced to allow faster acquisition times and to reduce the amount of noise in each 2-D slice. In this paper, a novel image super resolution (SR) algorithm is presented that is used to improve the resolution of the 3D MRI volumes in the slice direction. The proposed SR algorithm uses a complex wavelet-based de-blurring approach with a Gaussian scale mixture model sparseness constraint. The algorithm takes several multi-slice volumes of the same anatomical region captured at different angles and combines these low-resolution images together to form a single 3D volume with much higher resolution in the slice direction. Our results show that the 3D volumes reconstructed using this approach have higher quality than volumes produced by the best previously proposed approaches. <s> BIB030 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> In multi-slice magnetic resonance imaging (MRI) the resolution in the slice direction is usually reduced to allow faster acquisition times and to reduce the amount of noise in each 2-D slice. To address this issue a number of super resolution (SR) methods have been proposed to improve the resolution of 3D MRI volumes. These methods typically involve the use of prior models of the MRI data as regularization terms in an ill-conditioned inverse problem. However; an inappropriate choice of these models may reduce the overall performance of the algorithm. In this paper, we propose a novel SR algorithm which utilizes a complex wavelet-based de-noising approach. The proposed algorithm uses a bivariate Laplacian mixture model in a sparseness constraint to regularize the SR inverse problem. Our results show that the 3D MRI volumes reconstructed using this approach have quality superior to volumes produced by the best previously proposed approaches. <s> BIB031 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> We consider the problem of super-resolution from unregistered aliased images with unknown spatial scaling factors and shifts. Due to the limitation of pixel size in the image sensor, the sampling rate for each image is lower than the Nyquist rate of the scene. Thus, we have aliasing in captured images, which makes it hard to register the low-resolution images and then generate a high-resolution image. To work out this problem, we formulate it as a multichannel sampling and reconstruction problem with unknown parameters, spatial scaling factors and shifts. We can estimate the unknown parameters and then reconstruct the high-resolution image by solving a nonlinear least square problem using the variable projection method. Experiments with synthesized 1-D signals and 2-D images show the effectiveness of the proposed algorithm. <s> BIB032 </s> Super-resolution: a comprehensive survey <s> Wavelet transform <s> We propose a new technique of non-iterative super-resolution image reconstruction. A closed form solution to reconstruction is derived in the discrete cosine transform (DCT) domain from a MAP-based cost functional. An average image is used in order to avoid iterative operations. Symmetric convolution with appropriate types of DCT suppresses boundary distortion. Experimental results demonstrate the effectiveness of the proposed technique. <s> BIB033
The wavelet transform as an alternative to the Fourier transform has been widely used in frequency domain-based SR algorithms. Usually it is used to decompose the input image into structurally correlated sub-images. This allows exploiting the self-similarities between local neighboring regions BIB015 BIB027 . For example, in BIB027 the input image is first decomposed into subbands. Then, the input image and the high-frequency subbands are both interpolated. Then, the results of a stationary wavelet transform of the highfrequency subbands are used to improve the interpolated subbands. Then, the super-resolved HR output is generated by combining all of these subbands using an inverse discrete wavelet transform (DWT). Similar methods based on the DWT have been developed for SR in BIB001  BIB004 BIB009 BIB022 . In BIB003 BIB012 BIB013 BIB016 BIB018 BIB023 BIB024 BIB025 BIB028 BIB030 BIB031 , but the results obtained by DWT are used as a regularization term in maximum a posteriori (MAP) formulation of the problem (Sect. 5.1.6). In BIB019 BIB020 they have been used with compressive sensing (CS) methods (Sect. 5.2.1) and in BIB021 within a PCA-based face hallucination algorithm (Sect. 5.2). Wavelet-based methods may have difficulties in efficient implementation of degraded convolution filters, while they can be done efficiently using the Fourier transform. Therefore, these two transforms have sometimes been combined together into the Fourier-wavelet regularized deconvolution BIB019 . In addition to the above-mentioned methods in the frequency domain, some other SR algorithms of this domain have borrowed the methods that have been usually used in the spatial domain; among them are: BIB002  BIB014 BIB017 BIB032 which have used a maximum likelihood (ML) method (Sect. 5.1.5),  BIB005 BIB006 which have used a regularized ML method, BIB007 BIB010 BIB011 BIB026 BIB029 BIB033 which have used a MAP method (Sect. 5.1.6), and  BIB008 which have implemented a projection onto convex set (POCS) method (Sect. 5.1.4). These will all be explained in the next section.