Truncated Fractional-Order Total Variation Model for Image Restoration

In recent years, the Total Generalized Variation (TGV) model has received lots of attention in image processing community. Though this model can restore image with natural intensity transitions, its spatial identical parameter setting limits its performance. In this paper, we propose a novel Adaptive Weighted Total Generalized Variation model for image restoration. We analyze the TGV model from Bayesian Probability view and derive a novel adaptive parameter calculation scheme for it, exploiting the image's self-similarity. Experiment results on image deblurring and reconstruction show that by adapting the parameters in TGV model to image contents, the proposed model can restore image's edges and details well and achieve significant improvement over state of the art variational based models.

Nonconvex and nonsmooth total generalized variation model for image restoration

The total generalized variation (TGV) denoising model is the extension of the total variation (TV) model and is capable of avoiding the staircase artifacts of the TV model. However, the TGV model independently disposes the pixels, ignoring the structural similarity prior of the processed image. Thus, the TGV model is not robust to high-amplitude noise. The motivation of this paper is to employ the structural similarity and improve the TGV denoising effect. By introducing the overlapping group sparsity to the TGV model, a new improved TGV model is then proposed, exploring both the first-order and second-order neighborhood differential gradient information to improve the robustness of the TGV to heavy noise pollution. To solve the proposed model, we adopted the accelerated alternating direction method of multipliers (ADMM), in which the multi-constrained problem is divided into several sub-problems. To avoid large-scale matrix computations in the spatial domain, we regard the differential operators as the convolution form, and thus the fast Fourier transform and the convolution theorem are employed to solve the proposed model efficiently. Finally, experiments were conducted on several seismic signals under different types of noise to verify the proposed method. The findings are listed as follows (1). The proposed model is particularly good at removing the heavy noise in smooth areas (2). The accelerated ADMM with a restart process is capable of solving the proposed model and is much faster than the traditional ADMM (3). The group size should be chosen properly to arrive the best performance of the proposed method.

We consider the image denoising problem under Poisson noise. The total generalized variation (TGV)-based method is high efficient in eliminating the staircase effect which often occurs in the total variation-based methods. However, the TGV-based method may produces over-smoothing edges. In order to avoid over-smoothing edges, a weighted total generalized variation (WTGV) model is proposed. Computationally, the WTGV minimization problem is transformed into a saddle-point problem by using the dual technique of optimization. Then, the Chambolle–Pock’s first-order primal–dual algorithm is used to solve the transformed saddle-point problem. At last, compared with several existing state-of-the-art methods, experimental results demonstrate the performance of our proposed method in edges preservation and staircase effect elimination.

Electrical impedance tomography (EIT) is a potential imaging technique for reconstructing the interior conductivity distribution within an object. The reconstruction is realized by processing the boundary voltage measured from the EIT. Mathematically, the reconstruction of conductivity distribution is a highly nonlinear ill-posed inverse problem as the solution is not unique and is sensitive to the noise, which largely hinders the practical application of the EIT technique. In this paper, a novel modified non-convex ${L}_{1}$ -norm penalty based total generalized variation (NCP-TGV) model is proposed to solve the inverse problem and reconstruct the conductivity distribution. An iteratively reweighted ${L}_{1}$ algorithm is developed to convert the non-convex model to convex function and the Chambolle-Pock primal-dual algorithm is then applied to solve the minimization problem. To demonstrate the performance of the proposed NCP-TGV model in reconstructing the conductivity distribution, extensive numerical simulation and experimental work have been carried out. Additionally, the proposed approach has been validated by comparing its performance with the convex ${L}_{1}$ -norm penalty based total variation (TV) and TGV inverse models. The results indicate that the images reconstructed by the proposed NCP-TGV model show better quality with the advantages of simultaneous reduction of staircase effect and preservation of edge in comparison to the TV and TGV approaches.

In medical applications, computed tomography (CT) is widely used to evaluate various sample characteristics. However, image quality of CT reconstruction can be degraded due to artifacts. To propose and test a truncated total variation (truncation TV) model to solve the problem of large penalties for the total variation (TV) model. In this study, a truncated TV image denoising model in the fractional B-spline wavelet domain is developed to obtain the best solution. The method is validated by the analysis of CT reconstructed images of actual biological Pigeons samples. For this purpose, several indices including the peak signal-to-noise ratio (PSNR), structural similarity index (SSIM) and mean square error (MSE) are used to evaluate the quality of images. Comparing to the conventional truncated TV model that yields 22.55, 0.688 and 361.17 in PSNR, SSIM and MSE, respectively, using the proposed fractional B-spline-truncated TV model, the computed values of these evaluation indices change to 24.24, 0.898 and 244.98, respectively, indicating substantial reduction of image noise with higher PSNR and SSIM, and lower MSE. Study results demonstrate that compared with many classic image denoising methods, the new denoising algorithm proposed in this study can more effectively suppresses the reconstructed CT image artifacts while maintaining the detailed image structure.

When the first order variational models are used for multiplicative noise removal, there always some staircase effect, contract reduction, and corner smearing. In this paper, we will design a new second order variational model based on the total generalized variation (TGV) regularizer to solve these problems. The second order variation model is proposed originally for additive noise removal and we revise it in this paper for multiplicative noise removal. For the sake of computational efficiency, we transform this proposed model into a Split Bregman iterative scheme by introducing some auxiliary variables and iterative parameters, and then solve it via alternating optimization strategy. In order to speed up the computational efficiency, we also apply the fast Fourier transform (FFT), generalized soft threshold formulas and gradient descent method to the related sub-problems in each step. The experimental results show that in comparison with the first order total variation (TV) model, the proposed TGV model can effectively overcome the staircase effect; Also in comparison with the second order bounded Hessian regularization, the TGV model shows the advantage of preserving corners and edges in images.

Imaging through turbid medium has many potential applications such as looking through clouds, seeing into seawater and observing through biological tissues. The transmission matrix (TM) method is one of the main imaging technologies that has potential in imaging of large targets. With aid of pre-measured TM, several optimization models are proposed to recover targets from speckle patterns, including ℓ₂ norm optimization model, sparse representation (SR) framework and total variation (TV) model. However, the solution of ℓ₂ norm optimization model contains large reconstruction noise, while the SR framework and TV model are two kinds of compressive sensing strategies, which require that the targets are sparse. In this paper, in order to image non-sparse targets and suppress the reconstruction noise, we apply the maximum entropy method (MEM) model to recover the target images from speckle patterns. Simulation results show that, for non-sparse target, the MEM model has better reconstruction performance under different noise levels compared with the TV model. For example, peak signal-to-noise ratio (PSNR) and correlation coefficient (CC) of images reconstructed by MEM model at SNR=15 dB are comparable with those by TV model at SNR=35 dB.

The problem for image restoration is usually reduced to a constraint optimization problem. Different choice of optimization operator leads to various restoration models, e.g. least squares model and original total variation (TV) model. The TV model and its modified version can efficiently preserve the edge of the restored image well, but there exist obvious staircases in smooth area of the restored image. To reduce those staircases, we propose a new modified TV model, by adding a constraint term for smooth area protection as a penalty function. The numerical experiment shows our model can not only preserve the edge as well as TV model, but also efficiently reduce the staircase appearing in the smooth areas. Furthermore, It is shown that the restored image by our model has higher signal-to-noise ratio, less mean square error and better visual effect than those by the least squares model and by the TV models.

The total variation model performs very well for removing noise while preserving edges. However, it gives a piecewise constant solution which often leads to the staircase effect, consequently small details such as textures are filtered out in the denoising process. Fractional‐order total variation method is one of the major approaches to overcome such drawbacks. Unlike their good quality of fractional order, all these methods use a fixed fractional order for the whole of the image. In this paper, a novel variable‐order total fractional variation model is proposed for image denoising, in which the order of fractional derivative will be allocated automatically for each pixel based on the context of the image. This kind of selection is able to capture the edges and texture of the image simultaneously. In this regard, we prove the existence and uniqueness of the presented model. The split Bregman method is adapted to solve the model. Finally, the results illustrate the efficiency of the proposed model that yielded good visual effects and a better signal‐to‐noise ratio.

To overcome the weakness of a total variation based model for image restoration, various high order (typically second order) regularization models have been proposed and studied recently. In this paper we analyze and test a fractional-order derivative based total $\alpha$-order variation model which can outperform the currently popular high order regularization models. There exist several previous works using total $\alpha$-order variations for image restoration; however, first, no analysis has been done yet, and second, all tested formulations, differing from each other, utilize the zero Dirichlet boundary conditions which are not realistic (while nonzero boundary conditions violate definitions of fractional-order derivatives). This paper first reviews some results of fractional-order derivatives and then analyzes the theoretical properties of the proposed total $\alpha$-order variational model rigorously. It then develops four algorithms for solving the variational problem---one based on the variational...

Several methods based on Total Variation (TV) have been proposed for Hyperspectral Image (HSI) denoising. However, the TV terms of these methods just use various l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norms and penalize image gradient magnitudes, having a negative influence on the preprocessing of HSI denoising and further HSI classification task. In this paper, a novel l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> Total Variation (l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> TV) is first introduced and analyzed for the HSI noise removal framework to preserve more information for classification. We propose a novel Tensor low-rank constraint and l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> Total Variation (TLR-l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> TV) model in this paper. l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> TV directly controls the number of non-zero gradients and focuses on recovering the sharp image edges. The spectral-spatial information among all bands is exploited uniformly for removing mixed noise, which facilitates the subsequent classification after denoising. Including the Weighted Sum of Weighted Nuclear Norm (WSWNN) and the Weighted Sum of Weighted Tensor Nuclear Norm (WSWTNN), we propose two TLR-l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> TV-based algorithms, namely WSWNN-l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> TV and WSWTNN-l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> TV. The Alternating Direction Method of Multipliers (ADMM) and the Augmented Lagrange Multiplier (ALM) are employed to solve the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> TV model and TLR-l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> TV model, respectively. In both simulated and real data, the proposed models achieve superior performances in mixed noise removal of HSI. Especially, HSI classification accuracy is improved more effectively after denoising by the proposed TLR-l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> TV method.

Following the traditional total variational denoising model in removing medical image noise with blurred image texture details, among other problems, an adaptive medical image fractional-order total variational denoising model with an improved sparrow search algorithm is proposed in this study. This algorithm combines the characteristics of fractional-order differential operators and total variational models. The model preserves the weak texture region of the image improvement based on the unique amplitude-frequency characteristics of the fractional-order differential operator. The order of the fractional-order differential operator is adaptively determined by the improved sparrow search algorithm using both the sine search strategy and the diversity variation processing strategy, which can greatly improve the denoising ability of the fractional-order differential operator. The experimental results reveal that the model not only achieves the adaptivity of fractional-order total variable differential order, but also can effectively remove noise, preserve the texture structure of the image to the maximum extent, and improve the peak signal-to-noise ratio of the image; it also displays favorable prospects for applications in medical image denoising.

In this paper, we introduce two novel total variation models to deal with speckle noise in ultrasound image in order to retain the fine details more effectively and to improve the speed of energy diffusion during the process. Firstly, two new convex functions are introduced as regularization term in the adaptive total variation model, and then, the diffusion performances of Hypersurface Total Variation (HYPTV) model and Logarithmic Total Variation (LOGTV) model are analyzed mathematically through the physical characteristics of local coordinates. We have shown that the larger positive parameter in the model is set, the greater energy diffusion speed appears to be, but it will cause the image to be too smooth that required adequate attention. Numerical experimental results show that our proposed LOGTV model for speckle noise removal is superior to traditional models, not only in visual effect but also in quantitative measures.

New discretization of total variation functional for image processing tasks