Group Sparsity Residual Constraint for Image Denoising

Group-based sparse representation has shown great potential in image denoising. However, most existing methods only consider the nonlocal self-similarity (NSS) prior of noisy input image. That is, the similar patches are collected only from degraded input, which makes the quality of image denoising largely depend on the input itself. However, such methods often suffer from a common drawback that the denoising performance may degrade quickly with increasing noise levels. In this paper we propose a new prior model, called group sparsity residual constraint (GSRC). Unlike the conventional group-based sparse representation denoising methods, two kinds of prior, namely, the NSS priors of noisy and pre-filtered images, are used in GSRC. In particular, we integrate these two NSS priors through the mechanism of sparsity residual, and thus, the task of image denoising is converted to the problem of reducing the group sparsity residual. To this end, we first obtain a good estimation of the group sparse coefficients of the original image by pre-filtering, and then the group sparse coefficients of the noisy image are used to approximate this estimation. To improve the accuracy of the nonlocal similar patch selection, an adaptive patch search scheme is designed. Furthermore, to fuse these two NSS prior better, an effective iterative shrinkage algorithm is developed to solve the proposed GSRC model. Experimental results demonstrate that the proposed GSRC modeling outperforms many state-of-the-art denoising methods in terms of the objective and the perceptual metrics

From Rank Estimation to Rank Approximation: Rank Residual Constraint for Image Restoration

In this paper, we propose a novel approach to the rank minimization problem, termed rank residual constraint (RRC) model. 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 observations, we progressively approximate 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. Towards 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 are updated gradually and jointly in each iteration. 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 quality

Convolutional Sparse Coding with Overlapping Group Norms

The most widely used form of convolutional sparse coding uses an $\ell_1$ regularization term. While this approach has been successful in a variety of applications, a limitation of the $\ell_1$ penalty is that it is homogeneous across the spatial and filter index dimensions of the sparse representation array, so that sparsity cannot be separately controlled across these dimensions. The present paper considers the consequences of replacing the $\ell_1$ penalty with a mixed group norm, motivated by recent theoretical results for convolutional sparse representations. Algorithms are developed for solving the resulting problems, which are quite challenging, and the impact on the performance of the denoising problem is evaluated. The mixed group norms are found to perform very poorly in this application. While their performance is greatly improved by introducing a weighting strategy, such a strategy also improves the performance obtained from the much simpler and computationally cheaper $\ell_1$ norm

Image Reconstruction Using Deep Learning

This paper proposes a deep learning architecture that attains statistically significant improvements over traditional algorithms in Poisson image denoising espically when the noise is strong. Poisson noise commonly occurs in low-light and photon- limited settings, where the noise can be most accurately modeled by the Poission distribution. Poisson noise traditionally prevails only in specific fields such as astronomical imaging. However, with the booming market of surveillance cameras, which commonly operate in low-light environments, or mobile phones, which produce noisy night scene pictures due to lower-grade sensors, the necessity for an advanced Poisson image denoising algorithm has increased. Deep learning has achieved amazing breakthroughs in other imaging problems, such image segmentation and recognition, and this paper proposes a deep learning denoising network that outperforms traditional algorithms in Poisson denoising especially when the noise is strong. The architecture incorporates a hybrid of convolutional and deconvolutional layers along with symmetric connections. The denoising network achieved statistically significant 0.38dB, 0.68dB, and 1.04dB average PSNR gains over benchmark traditional algorithms in experiments with image peak values 4, 2, and 1. The denoising network can also operate with shorter computational time while still outperforming the benchmark algorithm by tuning the reconstruction stride sizes

A survey of sparse representation: algorithms and applications

Sparse representation has attracted much attention from researchers in fields of signal processing, image processing, computer vision and pattern recognition. Sparse representation also has a good reputation in both theoretical research and practical applications. Many different algorithms have been proposed for sparse representation. The main purpose of this article is to provide a comprehensive study and an updated review on sparse representation and to supply a guidance for researchers. The taxonomy of sparse representation methods can be studied from various viewpoints. For example, in terms of different norm minimizations used in sparsity constraints, the methods can be roughly categorized into five groups: sparse representation with $l_0$-norm minimization, sparse representation with $l_p$-norm (0$<$p$<$1) minimization, sparse representation with $l_1$-norm minimization and sparse representation with $l_{2,1}$-norm minimization. In this paper, a comprehensive overview of sparse representation is provided. The available sparse representation algorithms can also be empirically categorized into four groups: greedy strategy approximation, constrained optimization, proximity algorithm-based optimization, and homotopy algorithm-based sparse representation. The rationales of different algorithms in each category are analyzed and a wide range of sparse representation applications are summarized, which could sufficiently reveal the potential nature of the sparse representation theory. Specifically, an experimentally comparative study of these sparse representation algorithms was presented. The Matlab code used in this paper can be available at: http://www.yongxu.org/lunwen.html.Comment: Published on IEEE Access, Vol. 3, pp. 490-530, 201

Deep Residual Auto-Encoders for Expectation Maximization-inspired Dictionary Learning

We introduce a neural-network architecture, termed the constrained recurrent sparse auto-encoder (CRsAE), that solves convolutional dictionary learning problems, thus establishing a link between dictionary learning and neural networks. Specifically, we leverage the interpretation of the alternating-minimization algorithm for dictionary learning as an approximate Expectation-Maximization algorithm to develop auto-encoders that enable the simultaneous training of the dictionary and regularization parameter (ReLU bias). The forward pass of the encoder approximates the sufficient statistics of the E-step as the solution to a sparse coding problem, using an iterative proximal gradient algorithm called FISTA. The encoder can be interpreted either as a recurrent neural network or as a deep residual network, with two-sided ReLU non-linearities in both cases. The M-step is implemented via a two-stage back-propagation. The first stage relies on a linear decoder applied to the encoder and a norm-squared loss. It parallels the dictionary update step in dictionary learning. The second stage updates the regularization parameter by applying a loss function to the encoder that includes a prior on the parameter motivated by Bayesian statistics. We demonstrate in an image-denoising task that CRsAE learns Gabor-like filters, and that the EM-inspired approach for learning biases is superior to the conventional approach. In an application to recordings of electrical activity from the brain, we demonstrate that CRsAE learns realistic spike templates and speeds up the process of identifying spike times by 900x compared to algorithms based on convex optimization

Robust Non-linear Regression: A Greedy Approach Employing Kernels with Application to Image Denoising

We consider the task of robust non-linear regression in the presence of both inlier noise and outliers. Assuming that the unknown non-linear function belongs to a Reproducing Kernel Hilbert Space (RKHS), our goal is to estimate the set of the associated unknown parameters. Due to the presence of outliers, common techniques such as the Kernel Ridge Regression (KRR) or the Support Vector Regression (SVR) turn out to be inadequate. Instead, we employ sparse modeling arguments to explicitly model and estimate the outliers, adopting a greedy approach. The proposed robust scheme, i.e., Kernel Greedy Algorithm for Robust Denoising (KGARD), is inspired by the classical Orthogonal Matching Pursuit (OMP) algorithm. Specifically, the proposed method alternates between a KRR task and an OMP-like selection step. Theoretical results concerning the identification of the outliers are provided. Moreover, KGARD is compared against other cutting edge methods, where its performance is evaluated via a set of experiments with various types of noise. Finally, the proposed robust estimation framework is applied to the task of image denoising, and its enhanced performance in the presence of outliers is demonstrated

Group-Sparse Signal Denoising: Non-Convex Regularization, Convex Optimization

Convex optimization with sparsity-promoting convex regularization is a standard approach for estimating sparse signals in noise. In order to promote sparsity more strongly than convex regularization, it is also standard practice to employ non-convex optimization. In this paper, we take a third approach. We utilize a non-convex regularization term chosen such that the total cost function (consisting of data consistency and regularization terms) is convex. Therefore, sparsity is more strongly promoted than in the standard convex formulation, but without sacrificing the attractive aspects of convex optimization (unique minimum, robust algorithms, etc.). We use this idea to improve the recently developed 'overlapping group shrinkage' (OGS) algorithm for the denoising of group-sparse signals. The algorithm is applied to the problem of speech enhancement with favorable results in terms of both SNR and perceptual quality.Comment: 14 pages, 11 figure

A Greedy Approach to ℓ0,∞\ell_{0,\infty} Based Convolutional Sparse Coding

Sparse coding techniques for image processing traditionally rely on a processing of small overlapping patches separately followed by averaging. This has the disadvantage that the reconstructed image no longer obeys the sparsity prior used in the processing. For this purpose convolutional sparse coding has been introduced, where a shift-invariant dictionary is used and the sparsity of the recovered image is maintained. Most such strategies target the $\ell_0$ "norm" or the $\ell_1$ norm of the whole image, which may create an imbalanced sparsity across various regions in the image. In order to face this challenge, the $\ell_{0,\infty}$ "norm" has been proposed as an alternative that "operates locally while thinking globally". The approaches taken for tackling the non-convexity of these optimization problems have been either using a convex relaxation or local pursuit algorithms. In this paper, we present an efficient greedy method for sparse coding and dictionary learning, which is specifically tailored to $\ell_{0,\infty}$, and is based on matching pursuit. We demonstrate the usage of our approach in salt-and-pepper noise removal and image inpainting. A code package which reproduces the experiments presented in this work is available at https://web.eng.tau.ac.il/~rajaComment: Accepted for publication in SIAM Journal on Imaging Sciences (SIIMS

Compression, Restoration, Re-sampling, Compressive Sensing: Fast Transforms in Digital Imaging

Transform image processing methods are methods that work in domains of image transforms, such as Discrete Fourier, Discrete Cosine, Wavelet and alike. They are the basic tool in image compression, in image restoration, in image re-sampling and geometrical transformations and can be traced back to early 1970-ths. The paper presents a review of these methods with emphasis on their comparison and relationships, from the very first steps of transform image compression methods to adaptive and local adaptive transform domain filters for image restoration, to methods of precise image re-sampling and image reconstruction from sparse samples and up to "compressive sensing" approach that has gained popularity in last few years. The review has a tutorial character and purpose.Comment: 41 pages, 16 figure