Image restoration with a high-order total variation minimization method

In this paper, we propose a fast and efficient way to restore blurred and noisy images with a high-order total variation minimization technique. The proposed method is based on an alternating technique for image deblurring and denoising. It starts by finding an approximate image using a Tikhonov regularization method. This corresponds to a deblurring process with possible artifacts and noise remaining. In the denoising step, a high-order total variation algorithm is used to remove noise in the deblurred image. We see that the edges in the restored image can be preserved quite well and the staircase effect is reduced effectively in the proposed algorithm. We also discuss the convergence of the proposed regularization method. Some numerical results show that the proposed method gives restored images of higher quality than some existing total variation restoration methods such as the fast TV method and the modified TV method with the lagged diffusivity fixed-point iteration.     

Anisotropic Total Variation Filtering

Total variation regularization and anisotropic filtering have been established as standard methods for image denoising because of their ability to detect and keep prominent edges in the data. Both methods, however, introduce artifacts: In the case of anisotropic filtering, the preservation of edges comes at the cost of the creation of additional structures out of noise; total variation regularization, on the other hand, suffers from the stair-casing effect, which leads to gradual contrast changes in homogeneous objects, especially near curved edges and corners. In order to circumvent these drawbacks, we propose to combine the two regularization techniques. To that end we replace the isotropic TV semi-norm by an anisotropic term that mirrors the directional structure of either the noisy original data or the smoothed image. We provide a detailed existence theory for our regularization method by using the concept of relaxation. The numerical examples concluding the paper show that the proposed introduction of an anisotropy to TV regularization indeed leads to improved denoising: the stair-casing effect is reduced while at the same time the creation of artifacts is suppressed.     

基于分数阶微积分正则化的图像处理

全变分正则化方法已被广泛地应用于图像处理,利用此方法可以较好地去除噪声,并保持图像的边缘特征,但得到的优化解会产生"阶梯"效应.为了克服这一缺点,本文通过分数阶微积分正则化方法,建立了一个新的图像处理模型.为了克服此模型中非光滑项对求解带来的困难,本文研究了基于不动点方程的迫近梯度算法.最后,本文利用提出的模型与算法进行了图像去噪、图像去模糊与图像超分辨率实验,实验结果表明分数阶微积分正则化方法能较好的保留图像纹理等细节信息.     

Speckle noise removal in ultrasound images by first- and second-order total variation

Speckle noise contamination is a common issue in ultrasound imaging system. Due to the edge-preserving feature, total variation (TV) regularization-based techniques have been extensively utilized for speckle noise removal. However, TV regularization sometimes causes staircase artifacts as it favors solutions that are piecewise constant. In this paper, we propose a new model to overcome this deficiency. In this model, the regularization term is represented by a combination of total variation and high-order total variation, while the data fidelity term is depicted by a generalized Kullback-Leibler divergence. The proposed model can be efficiently solved by alternating direction method with multipliers (ADMM). Compared with some state-of-the-art methods, the proposed method achieves higher quality in terms of the peak signal to noise ratio (PSNR) and the structural similarity index (SSIM). Numerical experiments demonstrate that our method can remove speckle noise efficiently while suppress staircase effects on both synthetic images and real ultrasound images.     

Non-convex and non-smooth variational decomposition for image restoration

The variational image decomposition model decomposes an image into a structural and an oscillatory component by regularization technique and functional minimization. It is an important task in various image processing methods, such as image restoration, image segmentation, and object recognition. In this paper, we propose a non-convex and non-smooth variational decomposition model for image restoration that uses non-convex and non-smooth total variation (TV) to measure the structure component and the negative Sobolev space $H^{-1}$ to model the oscillatory component. The new model combines the advantages of non-convex regularization and weaker-norm texture modeling, and it can well remove the noises while preserving the valuable edges and contours of the image. The iteratively reweighted l¹ (IRL1) algorithm is employed to solve the proposed non-convex minimization problem. For each subproblem, we use the alternating direction method of multipliers (ADMM) algorithm to solve it. Numerical results validate the effectiveness of the proposed model for both synthetic and real images in terms of peak signal-to-noise ratio (PSNR) and mean structural similarity index (MSSIM).     

Image reconstruction model for the exterior problem of computed tomography based on weighted directional total variation

The exterior problem of computed tomography (CT) is a special imaging modality and it is an important research issue in Non-Destructive Testing (NDT). In the exterior problem, the missing projection data vary with the position of the pixels. Furthermore, amounts of theoretical and experimental results showed that the edges that are tangent to the radial direction are much harder to be reconstructed than edges that are tangent to the angular direction. In this paper, a weighted directional total variation (WDTV) based regularization model was proposed to better deal with the exterior problem. By introducing the WDTV regularization term, our model can reconstruct high quality images. First, WDTV of an image also describes the sparsity of image gradient magnitude which can enforce the reconstructed result to be a nearly flat image. Second, our model can preserve edges and reduce artifacts near edges that are tangent to the radial direction. Third, a numerical implementation called SART+WDTV algorithm was developed to solve our model. Simulated and real experiments demonstrated that our model was more capable of suppressing artifacts and preserving edges.     

IMAGE RESTORATION UNDER CAUCHY NOISE WITH SPARSE REPRESENTATION PRIOR AND TOTAL GENERALIZED VARIATION

This article introduces a novel variational model for restoring images degraded by Cauchy noise and/or blurring.The model integrates a nonconvex data-fidelity term with two regularization terms,a sparse representation prior over dictionary learning and total generalized variation(TGV)regularization.The sparse representation prior exploiting patch information enables the preservation of fine features and textural patterns,while adequately denoising in homogeneous regions and contributing natural visual quality.TGV regularization further assists in effectively denoising in smooth regions while retaining edges.By adopting the penalty method and an alternating minimization approach,we present an efficient iterative algorithm to solve the proposed model.Numerical results establish the superiority of the proposed model over other existing models in regard to visual quality and certain image quality assessments.     

Regularization parameter selection and an efficient algorithm for total variation-regularized positron emission tomography

In positron emission tomography, image data corresponds to measurements of emitted photons from a radioactive tracer in the subject. Such count data is typically modeled using a Poisson random variable, leading to the use of the negative-log Poisson likelihood fit-to-data function. Regularization is needed, however, in order to guarantee reconstructions with minimal artifacts. Given that tracer densities are primarily smoothly varying, but also contain sharp jumps (or edges), total variation regularization is a natural choice. However, the resulting computational problem is quite challenging. In this paper, we present an efficient computational method for this problem. Convergence of the method has been shown for quadratic regularization functions and here convergence is shown for total variation regularization. We also present three regularization parameter choice methods for use on total variation-regularized negative-log Poisson likelihood problems. We test the computational and regularization parameter selection methods on two synthetic data sets.     

Homotopy method for a mean curvature-based denoising model

Variational image denoising models based on regularization of gradients have been extensively studied. The total variation model by Rudin, Osher, and Fatemi (1992) [38] can preserve edges well but for images without edges (jumps), the solution to this model has the undesirable staircasing effect. To overcome this, mean curvature-based energy minimization models offer one approach for restoring both smooth (no edges) and nonsmooth (with edges) images. As such models lead to fourth order (instead of the usual second order) nonlinear partial differential equations, development of fast solvers is a challenging task. Previously stabilized fixed point methods and their associated multigrid methods were developed but the underlying operators must be regularized by a relatively large parameter. In this paper, we first present a fixed point curvature method for solving such equations and then propose a homotopy approach for varying the regularized parameter so that the Newton type method becomes applicable in a predictor-corrector framework. Numerical experiments show that both of our methods are able to maintain all important information in the image, and at the same time to filter out noise.     

A hybrid convex variational model for image restoration

We propose a new hybrid model for variational image restoration using an alternative diffusion switching non-quadratic function with a parameter. The parameter is chosen adaptively so as to minimize the smoothing near the edges and allow the diffusion to smooth away from the edges. This model belongs to a class of edge-preserving regularization methods proposed in the past, the ?-function formulation. This involves a minimizer to the associated energy functional. We study the existence and uniqueness of the energy functional of the model. Using real and synthetic images we show that the model is effective in image restoration.     

On weak solutions for an image denoising-deblurring model

A new denoising-deblurring model in image restoration is proposed,in which the regularization term carries out anisotropic diffusion on the edges and isotropic diffusion on the regular regions.The existence and uniqueness of weak solutions for this model are proved,and the numerical model is also testified.Compared with the TV diffusion,this model preferably reduces the staircase appearing in the restored images.     

Fraction-order total variation blind image restoration based on L1-norm

A fraction-order total variation blind image restoration algorithm based on L1-norm was proposed for restoring the images blurred by unknown point spread function (PSF) during imaging. According to the form of total variation, this paper introduced an arithmetic operator of fraction-order total variation and generated a mathematical model of cost. Semi-quadratic regularization was used to solve the model iteratively so that the solution of this algorithm became easier. This paper also analyzed the convergence of this algorithm and then testified its feasibility in theory. The experimental results showed the proposed algorithm can increase the PSNR of the restored image by 1 dB in relation to the first order total variation blind restoration method and Bayesian blind restoration method. The details in real blurred image were also pretty well restored. The effectiveness of the proposed algorithm revealed that it was practical in the blind image restoration.     

An ADMM algorithm for second-order TV-based MR image reconstruction

In this paper, we propose a new model for MR image reconstruction based on second order total variation ( \(\text {TV}^{2}\) ) regularization and wavelet, which can be considered as requiring the image to be sparse in both the spatial finite differences and wavelet transforms. Furthermore, by applying the variable splitting technique twice, augmented Lagrangian method and the Barzilai-Borwein step size selection scheme, an ADMM algorithm is designed to solve the proposed model. It reduces the reconstruction problem to several unconstrained minimization subproblems, which can be solved by shrinking operators and alternating minimization algorithms. The proposed algorithm needs not to solve a fourth-order PDE but to solve several second-order PDEs so as to improve calculation efficiency. Numerical results demonstrate the effectiveness of the presented algorithm and illustrate that the proposed model outperforms some reconstruction models in the quality of reconstructed images.     

An automatic regularization parameter selection algorithm in the total variation model for image deblurring

Image restoration is an inverse problem that has been widely studied in recent years. The total variation based model by Rudin-Osher-Fatemi (1992) is one of the most effective and well known due to its ability to preserve sharp features in restoration. This paper addresses an important and yet outstanding issue for this model in selection of an optimal regularization parameter, for the case of image deblurring. We propose to compute the optimal regularization parameter along with the restored image in the same variational setting, by considering a Karush Kuhn Tucker (KKT) system. Through establishing analytically the monotonicity result, we can compute this parameter by an iterative algorithm for the KKT system. Such an approach corresponds to solving an equation using discrepancy principle, rather than using discrepancy principle only as a stopping criterion. Numerical experiments show that the algorithm is efficient and effective for image deblurring problems and yet is competitive.     

Algorithms and software for total variation image reconstruction via first-order methods

This paper describes new algorithms and related software for total variation (TV) image reconstruction, more specifically: denoising, inpainting, and deblurring. The algorithms are based on one of Nesterov’s first-order methods, tailored to the image processing applications in such a way that, except for the mandatory regularization parameter, the user needs not specify any parameters in the algorithms. The software is written in C with interface to Matlab (version 7.5 or later), and we demonstrate its performance and use with examples.     

An edge driven wavelet frame model for image restoration

Wavelet frame systems are known to be effective in capturing singularities from noisy and degraded images. In this paper, we introduce a new edge driven wavelet frame model for image restoration by approximating images as piecewise smooth functions. With an implicit representation of image singularities sets, the proposed model inflicts different strength of regularization on smooth and singular image regions and edges. The proposed edge driven model is robust to both image approximation and singularity estimation. The implicit formulation also enables an asymptotic analysis of the proposed models and a rigorous connection between the discrete model and a general continuous variational model. Finally, numerical results on image inpainting and deblurring show that the proposed model is compared favorably against several popular image restoration models.     

求解分段常数图象分割模型的一个快速算法

针对Xue-ChengTai等提出的分段常数图象分割模型,我们提出了一个新的快速求解算法。通过引进一个函数来选择模型中的正则化参数β的值,并判断在迭代过程中何时求解不含惩罚项的泛函F。此函数的引入有效地加速了算法的收敛速度。结合原始-对偶Newton方法来求解总变差最小化问题。数值试验表明新算法具有很快的收敛速度与良好的分割效果,且算法对初始值的要求不高。     

基于Huber正则化的红外与可见光图像融合

图像融合通常是指从多源信道采集同一目标图像,将互补的多焦点、多模态、多时相和/或多视点图像集成在一起,形成新图像的过程.在本文中,我们采用基于Huber正则化的红外与可见光图像的融合模型.该模型通过约束融合图像与红外图像相似的像素强度保持热辐射信息,以及约束融合图像与可见光图像相似的灰度梯度和像素强度保持图像的边缘和纹理等外观信息,同时能够改善图像灰度梯度相对较小区域的阶梯效应.为了最小化这种变分模型,我们结合增广拉格朗日方法(ALM)和量身定做有限点方法(TFPM)的思想设计数值算法,并给出了算法的收敛性分析.最后,我们将所提模型和算法与其他七种图像融合方法进行定性和定量的比较,分析了本文所提模型的特点和所提数值算法的有效性.     

A “Nonconvex+Nonconvex” approach for image restoration with impulse noise removal

In this paper, we propose a new method for image restoration problems, which are degraded by impulsive noise, with nonconvex data fitting term and nonconvex regularizer.The proposed method possesses the advantages of nonconvex data fitting and nonconvex regularizer simultaneously, namely, robustness for impulsive noise and efficiency for restoring neat edge images.Further, we propose an efficient algorithm to solve the “Nonconvex+Nonconvex” structure problem via using the alternating direction minimization, and prove that the algorithm is globally convergent when the regularization parameter is known. However, the regularization parameter is unavailable in general. Thereby, we combine the algorithm with the continuation technique and modified Morozov’s discrepancy principle to get an improved algorithm in which a suitable regularization parameter can be chosen automatically. The experiments reveal the superior performances of the proposed algorithm in comparison with some existing methods.