L1-regularized Problem Research Articles

Barzilai–Borwein-like rules in proximal gradient schemes for ℓ1-regularized problems

We propose a novel steplength selection rule in proximal gradient methods for minimizing the sum of a differentiable function plus an ℓ 1 -norm penalty term. The proposed rule modifies one of the classical Barzilai–Borwein steplength, extending analogous results obtained in the context of gradient projection methods for constrained optimization. We analyse the spectral properties of the Barzilai–Borwein-like steplength when the differentiable part is quadratic, showing that its reciprocal lies in the spectrum of the submatrix of the Hessian that depends on both the nonzero and the nonoptimal zero components of the current iterate, allowing for acceleration effects when the optimal zero components start to be identified. Furthermore, we insert the modified rule into a proximal gradient method with a nonmonotone line search, for which we prove global convergence towards a stationary point. Numerical experiments show the ability of the proposed rule to sweep the spectrum of the reduced Hessian on a series of quadratic ℓ 1 -regularized problems, as well as its effectiveness in recovering the ground truth in a least squares regularized problem arising in image restoration.

Comparing solution paths of sparse quadratic minimization with a Stieltjes matrix

This paper studies several solution paths of sparse quadratic minimization problems as a function of the weighing parameter of the bi-objective of estimation loss versus solution sparsity. Three such paths are considered: the “ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path” where the discontinuous ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-function provides the exact sparsity count; the “ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path” where the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-function provides a convex surrogate of sparsity count; and the “capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path” where the nonconvex nondifferentiable capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-function aims to enhance the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-approximation. Serving different purposes, each of these three formulations is different from each other, both analytically and computationally. Our results deepen the understanding of (old and new) properties of the associated paths, highlight the pros, cons, and tradeoffs of these sparse optimization models, and provide numerical evidence to support the practical superiority of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path. Our study of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path is interesting in its own right as the path pertains to computable directionally stationary (= strongly locally minimizing in this context, as opposed to globally optimal) solutions of a parametric nonconvex nondifferentiable optimization problem. Motivated by classical parametric quadratic programming theory and reinforced by modern statistical learning studies, both casting an exponential perspective in fully describing such solution paths, we also aim to address the question of whether some of them can be fully traced in strongly polynomial time in the problem dimensions. A major conclusion of this paper is that a path of directional stationary solutions of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-regularized problem offers interesting theoretical properties and practical compromise between the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path and the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path. Indeed, while the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path is computationally prohibitive and greatly handicapped by the repeated solution of mixed-integer nonlinear programs, the quality of ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path, in terms of the two criteria—loss and sparsity—in the estimation objective, is inferior to the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path; the latter can be obtained efficiently by a combination of a parametric pivoting-like scheme supplemented by an algorithm that takes advantage of the Z-matrix structure of the loss function.

Image restoration and reconstruction by non-convex total variation and shearlet regularizations

The total variation (TV) model preserves edges well but causes staircase effects and fails to protect textures. To avoid these limitations, an innovative hybrid regularization model that combines minmax-concave TV (and the shearlet sparsity is proposed for simultaneous image deblurring and image reconstruction. Although the proposed cost function is a non-convex L1-regularized optimization problem, it can maintain the convexity of the cost function by giving the proper nonconvexity parameter to minimize it. Then, an alternating iterative scheme using variable splitting and the alternating direction method of multipliers is introduced to optimize the proposed model. The extensive experiments demonstrate the efficiency and viability of the proposed method in terms of both subjective vision and objective measures.

A Sparse Denoising-Based Super-Resolution Method for Scanning Radar Imaging

Scanning radar enables wide-range imaging through antenna scanning and is widely used for radar warning. The Rayleigh criterion indicates that narrow beams of radar are required to improve the azimuth resolution. However, a narrower beam means a larger antenna aperture. In practical applications, due to platform limitations, the antenna aperture is limited, resulting in a low azimuth resolution. The conventional sparse super-resolution method (SSM) has been proposed for improving the azimuth resolution of scanning radar imaging and achieving superior performance. This method uses the L1 norm to represent the sparse prior of the target and solves the L1 regularization problem to achieve super-resolution imaging under the regularization framework. The resolution of strong-point targets is improved efficiently. However, for some targets with typical shapes, the strong sparsity of the L1 norm treats them as strong-point targets, resulting in the loss of shape characteristics. Thus, we can only see the strong points in its processing results. However, in some applications that need to identify targets in detail, SSM can lead to false judgments. In this paper, a sparse denoising-based super-resolution method (SDBSM) is proposed to compensate for the deficiency of traditional SSM. The proposed SDBSM uses a sparse minimization scheme for denoising, which helps to reduce the influence of noise. Then, the super-resolution imaging is achieved by alternating iterative denoising and deconvolution. As the proposed SDBSM uses the L1 norm for denoising rather than deconvolution, the strong sparsity constraint of the L1 norm is reduced. Therefore, it can effectively preserve the shape of the target while improving the azimuth resolution. The performance of the proposed SDBSM was demonstrated via simulation and real data processing results.

A Superfast Super-Resolution Method for Radar Forward-Looking Imaging

The super-resolution method has been widely used for improving azimuth resolution for radar forward-looking imaging. Typically, it can be achieved by solving an undifferentiable regularization problem. The split Bregman algorithm (SBA) is a great tool for solving this undifferentiable problem. However, its real-time imaging ability is limited to matrix inversion and iterations. Although previous studies have used the special structure of the coefficient matrix to reduce the computational complexity of each iteration, the real-time performance is still limited due to the need for hundreds of iterations. In this paper, a superfast SBA (SFSBA) is proposed to overcome this shortcoming. Firstly, the super-resolution problem is transmitted into an regularization problem in the framework of regularization. Then, the proposed SFSBA is used to solve the nondifferentiable regularization problem. Different from the traditional SBA, the proposed SFSBA utilizes the low displacement rank features of Toplitz matrix, along with the Gohberg-Semencul (GS) representation to realize fast inversion of the coefficient matrix, reducing the computational complexity of each iteration from to . It uses a two-order vector extrapolation strategy to reduce the number of iterations. The convergence speed is increased by about 8 times. Finally, the simulation and real data processing results demonstrate that the proposed SFSBA can effectively improve the azimuth resolution of radar forward-looking imaging, and its performance is only slightly lower compared to traditional SBA. The hardware test shows that the computational efficiency of the proposed SFSBA is much higher than that of other traditional super-resolution methods, which would meet the real-time requirements in practice.

Multi-atlas segmentation and correction model with level set formulation for 3D brain MR images

We present an efficient multi-atlas segmentation and correction model with level set formulation for 3D brain MR images in this paper. We define a new energy functional by combining a weighted label fusion term, a bias field based image information fitting term and a regularization term together. More image information is taken into consideration in the new image data term to substantially improve the segmentation accuracy, especially when serious inhomogeneity and bias field exist in regions of interest in MR images. We introduce a spatially weight function and incorporate it into the label fusion term to increase the robustness of our segmentation algorithm to atlases with different registration accuracy. The new energy functional is in the form of L1 regularization problems, and we minimize it with the split Bregman method to ensure the segmentation efficiency. We apply the proposed model to segment six tissues in 3D brain MR images, including the amygdala, caudate, hippocampus, pallidum, putamen and thalamus. Experimental results have shown that our model can segment regions of interest accurately and eliminate bias field simultaneously. Quantitative comparisons with related methods have demonstrated the superiority of our model in terms of accuracy, efficiency and robustness.

Nonblind Image Deblurring Based on Bi-Composition Decomposition by Local Smoothness and Nonlocal Self-Similarity Priors

Image deblurring is a classical inverse problem in image processing and computer vision. The vital task is to construct the proper image prior model to obtain the high-quality restored image with salient edges and rich details. A new nonblind image deblurring method by combining local smoothness and nonlocal self-similarity of natural images in the regularization framework is proposed. First, the observed image is decomposed into two components: structure component and detail component by a global gradient extraction scheme. Second, the four-directional anisotropic total variation regularization satisfying the local smoothness property is adopted for the structure component, and a new nonlocal statistical modeling for self-similarity is used for the detail component, respectively. At last, the split Bregman-based iteration algorithm and four-directional fast gradient projection algorithm are introduced to optimize the proposed L 1 -regularized problem. The extensive experiments demonstrate the efficiency and viability of the proposed method for preserving salient edges and texture details while alleviating the artifacts.

Structural damage detection based on iteratively reweighted l1 regularization algorithm

Structural damage usually appears in a few sections or members only, which is sparse compared with the total elements of the entire structure. According to the sparse recovery theory, the recently developed damage detection methods employ the l1 regularization technique to exploit the sparsity condition of structural damage. However, in practice, the solution obtained by the l1 regularization is typically suboptimal. The l0 regularization technique outperforms the l1 regularization in various aspects for sparse recovery, whereas the associated nonconvex optimization problem is NP-hard and computationally infeasible. In this study, a damage detection method based on the iteratively reweighted l1 regularization algorithm is proposed. An iterative procedure is employed such that the nonconvex optimization problem of the l0 regularization can be efficiently solved through transforming it into a series of weighted l1 regularization problems. Experimental example demonstrates that the proposed damage detection method can accurately locate the sparse damage over a large number of elements. The advantage of the iteratively reweighted l1 regularization algorithm over the l1 regularization in damage detection is also demonstrated.

Automatic velocity analysis using high-resolution hyperbolic Radon transform

Velocity analysis is crucial in reflection seismic data processing and imaging. Velocity picking is widely used in the industry for building the initial velocity model. When the size of the seismic data becomes extremely large, we cannot afford the corresponding human endeavor that is required by the velocity picking. In such situations, an automatic velocity-picking algorithm is highly demanded. We have developed a novel automatic velocity-analysis algorithm that is based on the high-resolution hyperbolic Radon transform. We formulate the automatic velocity-analysis problem as a constrained optimization problem. To solve the optimization problem with a hard constraint on the sparsity and distribution of the velocity spectrum, we relax it to a more familiar L1-regularized optimization problem in two steps. We use the iterative preconditioned least-squares method to solve the L1-regularized problem, and then we apply the hard constraint of the target optimization during the iterative inversion. Using synthetic and field-data examples, we determine the successful performance of our algorithm.

Selection of regularization parameter for l1-regularized damage detection

The l1 regularization technique has been developed for structural health monitoring and damage detection through employing the sparsity condition of structural damage. The regularization parameter, which controls the trade-off between data fidelity and solution size of the regularization problem, exerts a crucial effect on the solution. However, the l1 regularization problem has no closed-form solution, and the regularization parameter is usually selected by experience. This study proposes two strategies of selecting the regularization parameter for the l1-regularized damage detection problem. The first method utilizes the residual and solution norms of the optimization problem and ensures that they are both small. The other method is based on the discrepancy principle, which requires that the variance of the discrepancy between the calculated and measured responses is close to the variance of the measurement noise. The two methods are applied to a cantilever beam and a three-story frame. A range of the regularization parameter, rather than one single value, can be determined. When the regularization parameter in this range is selected, the damage can be accurately identified even for multiple damage scenarios. This range also indicates the sensitivity degree of the damage identification problem to the regularization parameter.

Structural damage detection based onl1regularization using natural frequencies and mode shapes

Conventional vibration-based damage detection methods employ the Tikhonov regularization in model updating to deal with the problems of underdeterminacy and measurement noise. However, the Tikhonov regularization technique tends to provide over smooth solutions that the identified damage is distributed to many structural elements. This result does not match the sparsity property of the actual damage scenario, in which structural damage typically occurs at a small number of locations only in comparison with the total elements of the entire structure. In this study, an l1 regularization-based model updating technique is developed by utilizing the sparsity of the structural damage. Both natural frequencies and mode shapes are employed during the model updating. A strategy of selecting the regularization parameter for the l1 regularization problem is also developed. A numerical and an experimental examples are utilized to demonstrate the effectiveness of the proposed damage detection method. The results showed that the proposed l1 regularization-based method is able to locate and quantify the sparse damage correctly over a large number of elements. The effects of the mode number on the damage detection results are also investigated. The advantage of the present l1 regularization over the traditional l2 regularization method in damage detection is also demonstrated.

Obtaining sparse distributions in 2D inverse problems

The mathematics of inverse problems has relevance across numerous estimation problems in science and engineering. L1 regularization has attracted recent attention in reconstructing the system properties in the case of sparse inverse problems; i.e., when the true property sought is not adequately described by a continuous distribution, in particular in Compressed Sensing image reconstruction. In this work, we focus on the application of L1 regularization to a class of inverse problems; relaxation-relaxation, T1–T2, and diffusion-relaxation, D–T2, correlation experiments in NMR, which have found widespread applications in a number of areas including probing surface interactions in catalysis and characterizing fluid composition and pore structures in rocks. We introduce a robust algorithm for solving the L1 regularization problem and provide a guide to implementing it, including the choice of the amount of regularization used and the assignment of error estimates. We then show experimentally that L1 regularization has significant advantages over both the Non-Negative Least Squares (NNLS) algorithm and Tikhonov regularization. It is shown that the L1 regularization algorithm stably recovers a distribution at a signal to noise ratio<20 and that it resolves relaxation time constants and diffusion coefficients differing by as little as 10%. The enhanced resolving capability is used to measure the inter and intra particle concentrations of a mixture of hexane and dodecane present within porous silica beads immersed within a bulk liquid phase; neither NNLS nor Tikhonov regularization are able to provide this resolution. This experimental study shows that the approach enables discrimination between different chemical species when direct spectroscopic discrimination is impossible, and hence measurement of chemical composition within porous media, such as catalysts or rocks, is possible while still being stable to high levels of noise.

ℓ1-Regularized full-waveform inversion with prior model information based on orthant-wise limited memory quasi-Newton method

Full-waveform inversion (FWI) is an ill-posed optimization problem which is sensitive to noise and initial model. To alleviate the ill-posedness of the problem, regularization techniques are usually adopted. The ℓ1-norm penalty is a robust regularization method that preserves contrasts and edges. The Orthant-Wise Limited-Memory Quasi-Newton (OWL-QN) method extends the widely-used limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method to the ℓ1-regularized optimization problems and inherits the efficiency of L-BFGS. To take advantage of the ℓ1-regularized method and the prior model information obtained from sonic logs and geological information, we implement OWL-QN algorithm in ℓ1-regularized FWI with prior model information in this paper. Numerical experiments show that this method not only improve the inversion results but also has a strong anti-noise ability.

The inversion of 2D NMR relaxometry data using L1 regularization

NMR relaxometry has been used as a powerful tool to study molecular dynamics. Many algorithms have been developed for the inversion of 2D NMR relaxometry data. Unlike traditional algorithms implementing L2 regularization, high order Tikhonov regularization or iterative regularization, L1 penalty term is involved to constrain the sparsity of resultant spectra in this paper. Then fast iterative shrinkage-thresholding algorithm (FISTA) is proposed to solve the L1 regularization problem. The effectiveness, noise vulnerability and practical utility of the proposed algorithm are analyzed by simulations and experiments. The results demonstrate that the proposed algorithm has a more excellent capability to reveal narrow peaks than traditional inversion algorithms. The L1 regularization implemented by our algorithm can be a useful complementary to the existing algorithms.

A new QR decomposition-based RLS algorithm using the split Bregman method for L1-regularized problems

The split Bregman (SB) method can solve a broad class of L1-regularized optimization problems and has been widely used for sparse signal processing in a variety of applications. To achieve lower complexity and to cope with time-varying environments, we develop a new adaptive version of the SB method for finding online sparse solutions. This algorithm is derived from the recursive least squares (RLS) optimization problem, where the SB method is used to separate the regularization term from the constrained optimization. This algorithm is numerically more stable and easily amenable to multivariate implementation due to the use of a QR decomposition (QRD) structure. An efficient method is further developed for selecting the thresholding rule, which controls the sparsity level of the estimator. Moreover, the SB-QRRLS algorithm is extended to a multivariate version to solve the sparse principal component analysis (SPCA) problem. Simulation results are presented to illustrate the effectiveness of the proposed algorithms in sparse system estimation and SPCA. We show that the convergence and tracking performance of the proposed algorithms compares favorably with conventional algorithms.

A centroid-based gene selection method for microarray data classification

For classification problems based on microarray data, the data typically contains a large number of irrelevant and redundant features. In this paper, a new gene selection method is proposed to choose the best subset of features for microarray data with the irrelevant and redundant features removed. We formulate the selection problem as a L1-regularized optimization problem, based on a newly defined linear discriminant analysis criterion. Instead of calculating the mean of the samples, a kernel-based approach is used to estimate the class centroid to define both the between-class separability and the within-class compactness for the criterion. Theoretical analysis indicates that the global optimal solution of the L1-regularized criterion can be reached with a general condition, on which an efficient algorithm is derived to the feature selection problem in a linear time complexity with respect to the number of features and the number of samples. The experimental results on ten publicly available microarray datasets demonstrate that the proposed method performs effectively and competitively compared with state-of-the-art methods.

Single image super resolution using local smoothness and nonlocal self-similarity priors

Single image super resolution (SISR) is an inverse problem, so an effective image prior is necessary to reconstruct a high resolution (HR) image from a single low resolution (LR) image. On the one hand, natural images satisfy the property of local smoothness; on the other hand, the patches could find some similar patches in different locations within the same image, and this property is known as nonlocal self-similarity. In this paper, we propose a SISR method by incorporating the local smoothness and nonlocal self-similarity priors in the reconstruction-based SISR framework simultaneously, and the Split Bregman Iteration (SBI) optimization algorithm is imitated to solve the L1-regularized problem. Experimental results show that, in most case, the proposed method quantitatively and qualitatively outperforms the state-of-the-art SISR algorithms.

Robust destriping of MODIS and hyperspectral data using a hybrid unidirectional total variation model

Imaging from a degenerated push broom scanner usually leads to an undesired stripe noise which seriously affected the image quality. To eliminate this kind of artifact, a robust hybrid unidirectional total variation model is presented. The traditional unidirectional total variation model produces an excellent performance only on weak and moderate-amplitude stripe images while does a poor job on heavy ones. By introducing a simple weighted matrix, a hybrid unidirectional total variation model with two combined ℓ1 data-fidelity terms is launched to handle various stripe noises with different intensity. An efficient numerical algorithm based on the split Bregman iteration is developed to solve the hybrid ℓ1-regularized optimization problem. Comparative results on simulated and real striped images taken with MODIS and hyperspectral imaging systems demonstrated that the proposed method not only can effectively remove the stripe noise but also preserve the edge and detail information.

A second-order method for strongly convex $$\ell _1$$ ℓ 1 -regularization problems

In this paper a robust second-order method is developed for the solution of strongly convex $$\ell _1$$l1-regularized problems. The main aim is to make the proposed method as inexpensive as possible, while even difficult problems can be efficiently solved. The proposed approach is a primal-dual Newton conjugate gradients (pdNCG) method. Convergence properties of pdNCG are studied and worst-case iteration complexity is established. Numerical results are presented on synthetic sparse least-squares problems and real world machine learning problems.

Image super-resolution using sparse coding over redundant dictionary based on effective image representations

Recent years have shown a growing research interest in the sparse-representation of signals. Signals are described through sparse linear combinations of signal-atoms over a redundant-dictionary. Therefore, we propose a novel super-resolution framework using an overcomplete-dictionary based on effective image-representations such as edges, contours and high-order structures. This scheme recovers the vector of common sparse-representations between low-resolution and corresponding high-resolution image-patches by solving the ℓ1-regularized least-squared problem; subsequently, it reconstructs the HR output by multiplying it with the learned dictionary. The dictionary used in the proposed-technique contains more effective image-representations than those in previous approaches because it contains feature-descriptors such as edges, contours and motion-selective features. Therefore, the proposed-technique is more robust to various types of distortion. A saliency-map quickens this technique by confining the optimization-process to visually salient regions. Experimental analyses confirm the effectiveness of the proposed-scheme, and its quantitative and qualitative performance as compared with other state-of-the-art super-resolution algorithms.