Through exploiting the image nonlocal self-similarity (NSS) prior by clustering similar patches to construct patch groups, recent studies have revealed that structural sparse representation (SSR) models can achieve promising performance in various image restoration tasks. However, most existing SSR methods only exploit the NSS prior from the input degraded (internal) image, and few methods utilize the NSS prior from external clean image corpus; how to jointly exploit the NSS priors of internal image and external clean image corpus is still an open problem. In this article, we propose a novel approach for image restoration by simultaneously considering internal and external nonlocal self-similarity (SNSS) priors that offer mutually complementary information. Specifically, we first group nonlocal similar patches from images of a training corpus. Then a group-based Gaussian mixture model (GMM) learning algorithm is applied to learn an external NSS prior. We exploit the SSR model by integrating the NSS priors of both internal and external image data. An alternating minimization with an adaptive parameter adjusting strategy is developed to solve the proposed SNSS-based image restoration problems, which makes the entire algorithm more stable and practical. Experimental results on three image restoration applications, namely image denoising, deblocking and deblurring, demonstrate that the proposed SNSS produces superior results compared to many popular or state-of-the-art methods in both objective and perceptual quality measurements.

Sparse coding has achieved a great success in various image processing tasks. However, a benchmark to measure the sparsity of image patch/group is missing since sparse coding is essentially an NP-hard problem. This work attempts to fill the gap from the perspective of rank minimization. We firstly design an adaptive dictionary to bridge the gap between group-based sparse coding (GSC) and rank minimization. Then, we show that under the designed dictionary, GSC and the rank minimization problems are equivalent, and therefore the sparse coefficients of each patch group can be measured by estimating the singular values of each patch group. We thus earn a benchmark to measure the sparsity of each patch group because the singular values of the original image patch groups can be easily computed by the singular value decomposition (SVD). This benchmark can be used to evaluate performance of any kind of norm minimization methods in sparse coding through analyzing their corresponding rank minimization counterparts. Towards this end, we exploit four well-known rank minimization methods to study the sparsity of each patch group and the weighted Schatten p-norm minimization (WSNM) is found to be the closest one to the real singular values of each patch group. Inspired by the aforementioned equivalence regime of rank minimization and GSC, WSNM can be translated into a non-convex weighted ℓp-norm minimization problem in GSC. By using the earned benchmark in sparse coding, the weighted ℓp-norm minimization is expected to obtain better performance than the three other norm minimization methods, i.e., ℓ1-norm, ℓp-norm and weighted ℓ1-norm. To verify the feasibility of the proposed benchmark, we compare the weighted ℓp-norm minimization against the three aforementioned norm minimization methods in sparse coding. Experimental results on image restoration applications, namely image inpainting and image compressive sensing recovery, demonstrate that the proposed scheme is feasible and outperforms many state-of-the-art methods.

In this paper, we propose a novel approach for the rank minimization problem, termed rank residual constraint (RRC). Different from existing low-rank based approaches, such as the well-known nuclear norm minimization (NNM) and the weighted nuclear norm minimization (WNNM), which estimate the underlying low-rank matrix directly from the corrupted observation, we progressively approximate (approach) the underlying low-rank matrix via minimizing the rank residual. Through integrating the image nonlocal self-similarity (NSS) prior with the proposed RRC model, we apply it to image restoration tasks, including image denoising and image compression artifacts reduction. Toward this end, we first obtain a good reference of the original image groups by using the image NSS prior, and then the rank residual of the image groups between this reference and the degraded image is minimized to achieve a better estimate to the desired image. In this manner, both the reference and the estimated image in each iteration are improved gradually and jointly. Based on the group-based sparse representation model, we further provide a theoretical analysis on the feasibility of the proposed RRC model. Experimental results demonstrate that the proposed RRC model outperforms many state-of-the-art schemes in both the objective and perceptual qualities.

Group sparse representation (GSR) has made great strides in image restoration producing superior performance, realized through employing a powerful mechanism to integrate the local sparsity and nonlocal self-similarity of images. However, due to some form of degradation (e.g., noise, down-sampling or pixels missing), traditional GSR models may fail to faithfully estimate sparsity of each group in an image, thus resulting in a distorted reconstruction of the original image. This motivates us to design a simple yet effective model that aims to address the above mentioned problem. Specifically, we propose group sparsity residual constraint with nonlocal priors (GSRC-NLP) for image restoration. Through introducing the group sparsity residual constraint, the problem of image restoration is further defined and simplified through attempts at reducing the group sparsity residual. Towards this end, we first obtain a good estimation of the group sparse coefficient of each original image group by exploiting the image nonlocal self-similarity (NSS) prior along with self-supervised learning scheme, and then the group sparse coefficient of the corresponding degraded image group is enforced to approximate the estimation. To make the proposed scheme tractable and robust, two algorithms, i.e., iterative shrinkage/thresholding (IST) and alternating direction method of multipliers (ADMM), are employed to solve the proposed optimization problems for different image restoration tasks. Experimental results on image denoising, image inpainting and image compressive sensing (CS) recovery, demonstrate that the proposed GSRC-NLP based image restoration algorithm is comparable to state-of-the-art denoising methods and outperforms several state-of-the-art image inpainting and image CS recovery methods in terms of both objective and perceptual quality metrics.

Image nonlocal self-similarity (NSS) property has been widely exploited via various sparsity models such as joint sparsity (JS) and group sparse coding (GSC). However, the existing NSS-based sparsity models are either too restrictive, e.g., JS enforces the sparse codes to share the same support, or too general, e.g., GSC imposes only plain sparsity on the group coefficients, which limit their effectiveness for modeling real images. In this paper, we propose a novel NSS-based sparsity model, namely, low-rank regularized group sparse coding (LR-GSC), to bridge the gap between the popular GSC and JS. The proposed LR-GSC model simultaneously exploits the sparsity and low-rankness of the dictionary-domain coefficients for each group of similar patches. An alternating minimization with an adaptive adjusted parameter strategy is developed to solve the proposed optimization problem for different image restoration tasks, including image denoising, image deblocking, image inpainting, and image compressive sensing. Extensive experimental results demonstrate that the proposed LR-GSC algorithm outperforms many popular or state-of-the-art methods in terms of objective and perceptual metrics.

Sparse representation has achieved great success in various image processing and computer vision tasks. For image processing, typical patch-based sparse representation (PSR) models usually tend to generate undesirable visual artifacts, while group-based sparse representation (GSR) models lean to produce over-smooth effects. In this paper, we propose a new sparse representation model, termed joint patch-group based sparse representation (JPG-SR). Compared with existing sparse representation models, the proposed JPG-SR provides an effective mechanism to integrate the local sparsity and nonlocal self-similarity of images. We then apply the proposed JPG-SR to image restoration tasks, including image inpainting and image deblocking. An iterative algorithm based on the alternating direction method of multipliers (ADMM) framework is developed to solve the proposed JPG-SR based image restoration problems. Experimental results demonstrate that the proposed JPG-SR is effective and outperforms many state-of-the-art methods in both objective and perceptual quality.