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.     

High-order total variation-based Poissonian image deconvolution with spatially adapted regularization parameter

Images captured by image acquisition systems using photon-counting devices such as astronomical imaging, positron emission tomography and confocal microscopy imaging, are often contaminated by Poisson noise. Total variation (TV) regularization, which is a classic regularization technique in image restoration, is well-known for recovering sharp edges of an image. Since the regularization parameter is important for a good recovery, Chen and Cheng (2012) proposed an effective TV-based Poissonian image deblurring model with a spatially adapted regularization parameter. However, it has drawbacks since the TV regularization produces staircase artifacts. In this paper, in order to remedy the shortcoming of TV of their model, we introduce an extra high-order total variation (HTV) regularization term. Furthermore, to balance the trade-off between edges and the smooth regions in the images, we also incorporate a weighting parameter to discriminate the TV and the HTV penalty. The proposed model is solved by an iterative algorithm under the framework of the well-known alternating direction method of multipliers. Our numerical results demonstrate the effectiveness and efficiency of the proposed method, in terms of signal-to-noise ratio (SNR) and relative error (RelRrr).     

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.     

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.     

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.     

IMAGE DENOISING VIA TIME-DELAY REGULARIZATION COUPLED NONLINEAR DIFFUSION EQUATIONS

A novel nonlinear anisotropic diffusion model is proposed for image denoising which can be viewed as a novel regularized model that preserves the cherished features of Perona-Malik to some extent. It is characterized by a local dependence in the diffusivity which manifests itself through the presence of $p(x)$-Laplacian and time-delay regularization. The proposed model offers a new nonlinear anisotropic diffusion which makes it possible to effectively enhance the denoising capability and preserve the details while avoiding artifacts. Accordingly, the restored image is very clear and becomes more distinguishable. By Galerkin's method, we establish the well-posedness in the weak setting. Numerical experiments illustrate that the proposed model appears to be overwhelmingly competitive in restoring the images corrupted by Gaussian noise.     

New time dependent pretreat models based on total variational image restoration

In this paper,we propose new pretreat models for total variation (TV) minimization problems in image deblurring and denoising.Specially,blur operator is considered as useful information in restoration.New models in form is equivalent to pretreat the initial value by image blur operator.We successfully get a new (L.Rudin,S.Osher,and E.Fatemi) ROF model,a new level set motion model and a new anisotropic diffusion model respectively.Numerical experiments demonstrate that,under the same stopping rule,the proposed methods significantly accelerate the convergence of the mothed,save computation time and get the same restored effect.     

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.     

Edge-preserving wavelet thresholding for image denoising

In this paper we consider a general setting for wavelet based image denoising methods. In fact, in both deterministic regularization methods and stochastic maximum a posteriori estimations, the denoised image is obtained by minimizing a functional, which is the sum of a data fidelity term and a regularization term that enforces a roughness penalty on the solution. The latter is usually defined as a sum of potentials, which are functions of a derivative of the image. By considering particular families of dyadic wavelets, we propose the use of new potential functions, which allows us to preserve and restore important image features, such as edges and smooth regions, during the wavelet denoising process. Numerical results are presented, showing the optimal performance of the denoising algorithm obtained.     

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.     

Compression and denoising using <Emphasis Type="Italic">l</Emphasis><Subscript>0</Subscript>-norm

In this paper, we deal with l ₀-norm data fitting and total variation regularization for image compression and denoising. The l ₀-norm data fitting is used for measuring the number of non-zero wavelet coefficients to be employed to represent an image. The regularization term given by the total variation is to recover image edges. Due to intensive numerical computation of using l ₀-norm, it is usually approximated by other functions such as the l ₁-norm in many image processing applications. The main goal of this paper is to develop a fast and effective algorithm to solve the l ₀-norm data fitting and total variation minimization problem. Our idea is to apply an alternating minimization technique to solve this problem, and employ a graph-cuts algorithm to solve the subproblem related to the total variation minimization. Numerical examples in image compression and denoising are given to demonstrate the effectiveness of the proposed algorithm.     

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.     

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.     

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.     

Image reconstruction model for limited-angle CT based on prior image induced relative total variation

Limited-angle computed tomography (CT) reconstruction has a great potential to reduce X-ray radiation dose or scanning time. Suppressing shading artifacts is challenging, but of great practical significance in limited-angle CT. Traditional methods based on total variation (TV) cannot effectively remove the shading artifacts, prior image constrained compressed sensing (PICCS) is a promising method, but is sensitive to the quality of the prior image. In micro-CT, a prior image reconstructed by filtered back-projection (FBP) may contain high-level noise. An image reconstructed by PICCS tends to inherit both structures and noise of the prior image. In this study, to suppress noise and shading artifacts, we propose a new limited-angle CT reconstruction model called prior image induced relative total variation (piiRTV), that uses the structure information of a prior image to guide limited-angle CT reconstruction. The proposed piiRTV is compared to TV and PICCS. Numerical simulations and experiments on real CT projections demonstrate the effectiveness of piiRTV in suppression of noise and shading artifacts. In addition, the proposed piiRTV is more robust to the prior image quality than PICCS.     

L1 regularization method in electrical impedance tomography by using the L1-curve (Pareto frontier curve)

Electrical impedance tomography (EIT), as an inverse problem, aims to calculate the internal conductivity distribution at the interior of an object from current-voltage measurements on its boundary. Many inverse problems are ill-posed, since the measurement data are limited and imperfect. To overcome ill-posedness in EIT, two main types of regularization techniques are widely used. One is categorized as the projection methods, such as truncated singular value decomposition (SVD or TSVD). The other categorized as penalty methods, such as Tikhonov regularization, and total variation methods. For both of these methods, a good regularization parameter should yield a fair balance between the perturbation error and regularized solution. In this paper a new method combining the least absolute shrinkage and selection operator (LASSO) and the basis pursuit denoising (BPDN) is introduced for EIT. For choosing the optimum regularization we use the L1-curve (Pareto frontier curve) which is similar to the L-curve used in optimising L2-norm problems. In the L1-curve we use the L1-norm of the solution instead of the L2 norm. The results are compared with the TSVD regularization method where the best regularization parameters are selected by observing the Picard condition and minimizing generalized cross validation (GCV) function. We show that this method yields a good regularization parameter corresponding to a regularized solution. Also, in situations where little is known about the noise level σ, it is also useful to visualize the L1-curve in order to understand the trade-offs between the norms of the residual and the solution. This method gives us a means to control the sparsity and filtering of the ill-posed EIT problem. Tracing this curve for the optimum solution can decrease the number of iterations by three times in comparison with using LASSO or BPDN separately.     

A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising

The total variation model proposed by Rudin, Osher and Fatemi performs very well for removing noise while preserving edges. However, it favors a piecewise constant solution in BV space which often leads to the staircase effect, and small details such as textures are often filtered out with noise in the process of denoising. To preserve the textures and eliminate the staircase effect, we improve the total variation model in this paper. This is accomplished by the following steps: (1) we define a new space of functions of fractional-order bounded variation called the BV_α space by using the Grünwald–Letnikov definition of fractional-order derivative; (2) we model the structure of the image as a function belonging to the BV_α space, and the textures in different scales as functions belonging to different negative Sobolev spaces. Thus, we propose a class of fractional-order multi-scale variational models for image denoising. (3) We analyze some properties of the fraction-order total variation operator and its conjugate operator. By using these properties, we develop an alternation projection algorithm for the new model and propose an efficient condition of the convergence of the algorithm. The numerical results show that the fractional-order multi-scale variational model can improve the peak signal to noise ratio of image, preserve textures and eliminate the staircase effect efficiently in the process of denoising.     

Image denoising using the Gaussian curvature of the image surface

A number of high‐order variational models for image denoising have been proposed within the last few years. The main motivation behind these models is to fix problems such as the staircase effect and the loss of image contrast that the classical Rudin–Osher–Fatemi model [Leonid I. Rudin, Stanley Osher and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), pp. 259–268] and others also based on the gradient of the image do have. In this work, we propose a new variational model for image denoising based on the Gaussian curvature of the image surface of a given image. We analytically study the proposed model to show why it preserves image contrast, recovers sharp edges, does not transform piecewise smooth functions into piecewise constant functions and is also able to preserve corners. In addition, we also provide two fast solvers for its numerical realization. Numerical experiments are shown to illustrate the good performance of the algorithms and test results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1066–1089, 2016     

Restoration of images with rotated shapes

Methods for image restoration which respect edges and other important features are of fundamental importance in digital image processing. In this paper, we present a novel technique for the restoration of images containing rotated (linearly transformed) rectangular shapes which avoids the round-off effects at vertices produced by known edge-preserving denoising techniques. Following an idea of Berkels et al. our approach is also based on two steps: the determination of the angles related to the rotated shapes and a subsequent restoration step which incorporates the knowledge of the angles. However, in contrast to Berkels et al., we find the smoothed rotation angles of the shapes by minimizing a simple quadratic functional without constraints which involves only first order derivatives so that we finally have to solve only a linear system of equations. Moreover, we propose to perform the restoration step either by quadratic programming or by solving an anisotropic diffusion equation. We focus on a discrete approach which approximates derivatives by finite differences. Particular attention is paid to the choice of the difference filters. We prove some relations concerning the preservation of rectangular shapes for our discrete setting. Finally, we present numerical examples for the denoising of artificial images with rotated rectangles and parallelograms and for the denoising of a real-world image.     

A multiscale fourth‐order model for the image inpainting and low‐dimensional sets recovery

We consider a fourth‐order variational model, to solve the image inpainting problem, with the emphasis on the recovery of low‐dimensional sets (edges and corners) and the curvature of the edges. The model permits also to perform simultaneously the restoration (filtering) of the initial image where this one is available. The multiscale character of the model follows from an adaptive selection of the diffusion parameters that allows us to optimize the regularization effects in the neighborhoods of the small features that we aim to preserve. In addition, because the model is based on the high‐order derivatives, it favors naturally the accurate capture of the curvature of the edges, hence to balance the tasks of obtaining long curved edges and the obtention of short edges, tip points, and corners. We analyze the method in the framework of the calculus of variations and the Γ‐convergence to show that it leads to a convergent algorithm. In particular, we obtain a simple discrete numerical method based on a standard mixed‐finite elements with well‐established approximation properties. We compare the method to the Cahn–Hilliard model for the inpainting, and we present several numerical examples to show its performances. Copyright © 2016 John Wiley & Sons, Ltd.