Fractional-order Total Variation Research Articles

Primal-dual hybrid gradient image denoising algorithm based on overlapping group sparsity and fractional-order total variation

This study introduces a non-convex fractional-order hyper-Laplacian variational model for Gaussian noise removal. It employs first the primal-dual hybrid gradient algorithm to solve problems involving overlapping group sparse structures. Additionally, this paper designs a new algorithm leveraging the framework of the Chambolle-Pock algorithm with convergence and aims to recover high-quality images. The model, integrating the overlapping group sparse structure of the hyper-Laplacian prior with the non-convex fractional-order total variation, exhibits superior performance in reducing the staircase effect and maintaining sharp edge contours. To further improve the performance of the algorithm, a semi-adaptive p(x) non-convex penalty weight assignment mechanism is designed by introducing the structure tensor, which according to the characteristics of each region of the image and the noise level. The effectiveness and superiority of the proposed algorithm in image denoising with simulation experiments and comparative analyses are fully verified.

Boosting MR Image Impulse Noise Removal With Overlapping Group Sparse Fractional-Order Total Variation and Minimax Concave Penalty

The acquisition and transmission of magnetic resonance images are susceptible to noise, particularly impulse noise. Although the method based on the ℓ0-norm and overlapping group sparse total variation (ℓ0-OGSTV) is effective for impulse noise image restoration, it can only mitigate the staircase artifacts to a certain extent. To boost the impulse noise removal performance of ℓ0-OGSTV, we propose a new restoration model that consists of two terms. Specifically, in the first term, we keep using the ℓ0-norm as the data fidelity term to eliminate impulse noise. In the second term, we first introduce an overlapping group sparsity fractional-order total variation regularizer to eliminate staircase artifacts while preserving structural information. Then, we adopt the minimax-concave penalty to further accurately estimate the image edges. Finally, we employ an alternate direction method of multipliers to solve the proposed optimization model. Clinical experiments demonstrate its effectiveness in denoising medical images.

Nonconvex fractional order total variation based image denoising model under mixed stripe and Gaussian noise

In this paper, we propose a minimization-based image denoising model for the removal of mixed stripe and Gaussian noise. The objective function includes the prior information from both the stripe noise and image. Specifically, we adopted a unidirectional regularization term and a nonconvex group sparsity term for the stripe noise component, while we utilized a nonconvex fractional order total variation (FTV) regularization for the image component. The priors for stripes enable adequate extraction of periodic or non-periodic stripes from an image in the presence of high levels of Gaussian noise. Moreover, the nonconvex FTV facilitates image restoration with less staircase artifacts and well-preserved edges and textures. To solve the nonconvex problem, we employed an iteratively reweighted $ \ell_1 $ algorithm, and then the alternating direction method of multipliers was adopted for solving subproblems. This led to an efficient iterative algorithm, and its global convergence was proven. Numerical results show that the proposed model provides better denoising performance than existing models with respect to visual features and image quality evaluations.

Total fractional-order variation and bilateral filter for image denoising

Image denoising stands out as a primary goal in image processing. However, many existing methods encounter challenges in preserving features such as corners and edges of an image while deleting the noise. This study investigates and evaluates a fractional-order derivative based on the total α-order variation (TV) model and the bilateral total variation (BTV) model. This choice is motivated by the proven effectiveness of the TV model in noise removal and edge preservation, with the BTV model further utilized to enhance the restoration of fine and intricate details. The experimental results affirm the efficacy of the proposed model, supported by objective quantitative metrics and subjective assessments of visual appearance.

A combined first and fractional order regularization method for mixed Poisson-white spike noisy image restoration

In this paper we study the problem of restoring images affected by mixed Poisson-Square Cauchy noise. A multi-convex variational model is derived by integrating infimal convolution likelihood into a process of joint maximum a posteriori estimation, where a modified four-directional fractional-order total variation is united with the first order total variation to characterize the image prior. To solve the proposed model numerically, a block coefficient descent based algorithm is derived, in which variable splitting and alternating direction minimization of multipliers are utilized along with anisotropic diffusion and additive operator splitting to gain efficiency and quality. The obtained numerical results are compared with results obtained from two other total variation regularized models with different fidelities. The numerical discuss confirms the flexibility and validity of the proposed model.

Image denoising based on the fractional-order total variation and the minimax-concave

Image denoising based on the fractional-order total variation and the minimax-concave

On the preconditioning of the primal form of TFOV-based image deblurring model

To address the staircasing problem in deblurred images generated by a simple total variation (TV) based model, one approach is to use the total fractional-order variation (TFOV) image deblurring model. However, the discretization of the Euler–Lagrange equations for the TFOV-based model results in a nonlinear ill-conditioned system, which adversely influences the performance of computational methods like Krylov subspace algorithms (e.g., Generalized Minimal Residual, Conjugate Gradient). To address this challenge, three novel preconditioned matrices are proposed to improve the conditioning of the primal model when using the conjugate gradient method. These matrices are designed based on circulant approximations of the matrix associated with blurring kernel. Experimental evaluations demonstrate the effectiveness of the proposed preconditioned matrices in enhancing the convergence and accuracy of the conjugate gradient method for solving the primal form of the TFOV-based image deblurring model. The results highlight the importance of appropriate preconditioning strategies in achieving robust and high-quality image deblurring using the TFOV approach.

Preconditioning Technique for an Image Deblurring Problem with the Total Fractional-Order Variation Model

Total fractional-order variation (TFOV) in image deblurring problems can reduce/remove the staircase problems observed with the image deblurring technique by using the standard total variation (TV) model. However, the discretization of the Euler–Lagrange equations associated with the TFOV model generates a saddle point system of equations where the coefficient matrix of this system is dense and ill conditioned (it has a huge condition number). The ill-conditioned property leads to slowing of the convergence of any iterative method, such as Krylov subspace methods. One treatment for the slowness property is to apply the preconditioning technique. In this paper, we propose a block triangular preconditioner because we know that using the exact triangular preconditioner leads to a preconditioned matrix with exactly two distinct eigenvalues. This means that we need at most two iterations to converge to the exact solution. However, we cannot use the exact preconditioner because the Shur complement of our system is of the form S=K*K+λLα which is a huge and dense matrix. The first matrix, K*K, comes from the blurred operator, while the second one is from the TFOV regularization model. To overcome this difficulty, we propose two preconditioners based on the circulant and standard TV matrices. In our algorithm, we use the flexible preconditioned GMRES method for the outer iterations, the preconditioned conjugate gradient (PCG) method for the inner iterations, and the fixed point iteration (FPI) method to handle the nonlinearity. Fast convergence was found in the numerical results by using the proposed preconditioners.

Fractional-Order Total Variation Geiger-Mode Avalanche Photodiode Lidar Range-Image Denoising Algorithm Based on Spatial Kernel Function and Range Kernel Function

A Geiger-mode avalanche photodiode (GM-APD) laser radar range image has much noise when the signal-to-background ratios (SBRs) are low, making it difficult to recover the real target scene. In this paper, based on the GM-APD lidar denoising model of fractional-order total variation (FOTV), the spatial relationship and similarity relationship between pixels are obtained by using a spatial kernel function and range kernel function to optimize the fractional differential operator, and a new FOTV GM-APD lidar range-image denoising algorithm is designed. The lost information and range anomalous noise are suppressed while the target details and contour information are preserved. The Monte Carlo simulation and experimental results show that, under the same SBRs and statistical frame number, the proposed algorithm improves the target restoration degree by at least 5.11% and the peak signal-to-noise ratio (PSNR) by at least 24.6%. The proposed approach can accomplish the denoising of GM-APD lidar range images when SBRs are low.

Image Restoration with Fractional-Order Total Variation Regularization and Group Sparsity

In this paper, we present a novel image denoising algorithm, specifically designed to effectively restore both the edges and texture of images. This is achieved through the use of an innovative model known as the overlapping group sparse fractional-order total variation regularization model (OGS-FOTVR). The OGS-FOTVR model ingeniously combines the benefits of the fractional-order (FO) variation domain with an overlapping group sparsity measure, which acts as its regularization component. This is further enhanced by the inclusion of the well-established L2-norm, which serves as the fidelity term. To simplify the model, we employ the alternating direction method of multipliers (ADMM), which breaks down the model into a series of more manageable sub-problems. Each of these sub-problems can then be addressed individually. However, the sub-problem involving the overlapping group sparse FO regularization presents a high level of complexity. To address this, we construct an alternative function for this sub-problem, utilizing the mean inequality principle. Subsequently, we employ the majorize-minimization (MM) algorithm to solve it. Empirical results strongly support the effectiveness of the OGS-FOTVR model, demonstrating its ability to accurately recover texture and edge information in images. Notably, the model performs better than several advanced variational alternatives, as indicated by superior performance metrics across three image datasets, PSNR, and SSIM.

An Image-Denoising Framework Using ℓq Norm-Based Higher Order Variation and Fractional Variation with Overlapping Group Sparsity

As one of the most significant issues in imaging science, image denoising plays a major role in plenty of image processing applications. Due to the ill-posed nature of image denoising, total variation regularization is widely used in image denoising problems for its capability to suppress noise and preserve image edges. Nevertheless, traditional total variation will inevitably yield undesirable staircase artifacts when applied to recorded images. Inspired by the success of ℓq norm minimization and overlapping group sparsity in image denoising, and the effective staircase artifacts removal by fractional total variation, the hybrid model which combines the fractional order total variation with overlapping group sparsity and higher order total variation with ℓq norm is developed in this paper to restore images corrupted by Gaussian noise. An efficient algorithm based on the parallel linear alternating direction method of multipliers is developed for solving the corresponding model and the numerical experiments demonstrate the effectiveness of the proposed approach against several state-of-the-art methods, in terms of peak signal-to-noise ratio and structure similarity index measure values.

A Fractional-Order Total Variation Regularization-Based Method for Recovering Geiger-Mode Avalanche Photodiode Light Detection and Ranging Depth Images

High-quality image restoration is typically challenging due to low signal–to–background ratios (SBRs) and limited statistics frames. To address these challenges, this paper devised a method based on fractional-order total variation (FOTV) regularization for recovering Geiger-mode avalanche photodiode (GM-APD) light detection and ranging (lidar) depth images. First, the spatial differential peak-picking method was used to extract the target depth image from low SBR and limited frames. FOTV regularization was introduced based on the total variation regularization recovery model, which incorporates the fractional-order differential operator, in order to realize FOTV-regularization-based depth image recovery. These frameworks were used to establish an algorithm for GM-APD depth image recovery based on FOTV. The simulation and experimental results demonstrate that the devised FOTV-recovery algorithm improved the target reduction degree, peak signal–to–noise ratio, and structural similarity index measurement by 76.6%, 3.5%, and 6.9% more than the TV, respectively, in the same SBR and statistic frame conditions. Thus, the devised approach is able to effectively recover GM-APD lidar depth images in low SBR and limited statistic frame conditions.

Multiplicative Noise Removal and Contrast Enhancement for SAR Images Based on a Total Fractional-Order Variation Model

In this paper, we propose a total fractional-order variation model for multiplicative noise removal and contrast enhancement of real SAR images. Inspired by the high dynamic intensity range of SAR images, the full content of the SAR images is preserved by normalizing the original data in this model. Then, we propose a degradation model based on the nonlinear transformation to adjust the intensity of image pixel values. With MAP estimator, a corresponding fidelity term is introduced into the model, which is beneficial for contrast enhancement and bias correction in the denoising process. For the regularization term, a gray level indicator is used as a weighted matrix to make the model adaptive. We first apply the scalar auxiliary variable algorithm to solve the proposed model and prove the convergence of the algorithm. By virtue of the discrete Fourier transform (DFT), the model is solved by an iterative scheme in the frequency domain. Experimental results show that the proposed model can enhance the contrast of natural and SAR images while removing multiplicative noise.

Solving linear Bayesian inverse problems using a fractional total variation-Gaussian (FTG) prior and transport map

The Bayesian inference is widely used in many scientific and engineering problems, especially in the linear inverse problems in infinite-dimensional setting where the unknowns are functions. In such problems, choosing an appropriate prior distribution is an important task. Especially when the function to infer has much detail information, such as many sharp jumps, corners, and the discontinuous and nonsmooth oscillation, the so-called total variation-Gaussian (TG) prior is proposed in function space to address it. However, the TG prior is easy to lead the blocky (staircase) effect in numerical results. In this work, we present a fractional order-TG (FTG) hybrid prior to deal with such problems, where the fractional order total variation (FTV) term is used to capture the detail information of the unknowns and simultaneously uses the Gaussian measure to ensure that it results in a well-defined posterior measure. For the numerical implementations of linear inverse problems in function spaces, we also propose an efficient independence sampler based on a transport map, which uses a proposal distribution derived from a diagonal map, and the acceptance probability associated to the proposal is independent of discretization dimensionality. And in order to take full advantage of the transport map, the hierarchical Bayesian framework is applied to flexibly determine the regularization parameter. Finally we provide some numerical examples to demonstrate the performance of the FTG prior and the efficiency and robustness of the proposed independence sampler method.

Image restoration with impulse noise based on fractional-order total variation and framelet transform

Image restoration with impulse noise based on fractional-order total variation and framelet transform

A hybrid alternating minimization algorithm for structured convex optimization problems with application in Poissonian image processing

<p style='text-indent:20px;'>Motivated by applications in image processing, we consider a class of structured convex optimization problems in which the objective function is the sum of two component functions with favorable structures, and, furthermore, both of which are composed with linear operators. In particular, to recover images degraded by Poisson noise, the famous Kullback-Leibler divergence can be adopted in data fitting, in which case one component function is separable, strongly convex and easy to minimize. The other component function usually allows an easily computable proximal operator. We then propose, analyze and test a hybrid algorithm particularly tailored for the underlying model structures, which can be viewed as a combination of the famous alternating minimization algorithm of Tseng and the alternating direction method of multipliers. Under mild conditions, global iterate convergence as well as ergodic <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{O}(1/N) $\end{document}</tex-math></inline-formula> sublinear convergence rate results measured by the function value residual and constraint violation of a reformulated constrained optimization problem are established, where <inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula> denotes the iteration counter. The proposed algorithm is applied to fractional-order total variation regularized image denoising and deblurring problems, where the noise obeys Poisson distribution. Our experimental results demonstrate the effectiveness of the proposed algorithm.</p>

Wavelet oriented SAR image despeckling using fractional-order TV and a non-convex sparse prior

Speckle is an inborn deformity caused by the interaction of backscattered waves from the target in active coherent imaging sensors such as synthetic aperture radar (SAR). Its presence severely limits the utility of SAR imagery in terms of detecting ground targets, retrieving information and analyzing scenes. The wavelet-based methods employ a threshold value on the noisy image for sparse wavelet representation. However, the noisy image coefficients exceeding the threshold are the cause of spurious noise spikes around the discontinuities. The edge preserving total variation (TV) regularizer is effective against speckle. Nevertheless, it is associated with staircase artifact. In this work, the wavelet-based forward model is integrated with a fractional order TV (FrTV) regularizer in the presence of a non-convex sparse prior to exploit the benefits of both methods. This combination aids in avoiding staircasing artifacts and preserving discontinuities while eliminating noise. Under some constraints, the optimization problem is convex, and it is solved using the alternating direction method of multipliers (ADMM). The alternating framework is iteratively solved by employing a thresholding operation, a shrinkage function, and quadratic minimization. The proposed method’s convergence is investigated both mathematically and empirically. The experimental results on both simulated and practical SAR data show that the proposed technique outperforms the previous algorithms in terms of speckle removal.

Non-convex fractional-order TV model for impulse noise removal

For the pursuit of high-quality restoration effects, we introduce a novel non-convex extended scheme to reconstruct blurred images under impulse noise in this paper. Because of the remarkable advantages of the combination of non-convex ℓp-norm and fractional-order total variation regularization, our developed strategy possesses a brilliant ability to overcome the staircase-like aspects and preserve neat contours. To obtain the minimizer of the optimization problem, we adopt the alternating direction method of multipliers closely integrating with the iteratively reweighted ℓ1 algorithm. Theoretically, a convergence proof is provided for the developed algorithm. By considering salt-and-pepper noise and random-valued impulse noise, comprehensive simulation comparisons with some popular methods are carried out for image restoration. And the experimental results illustrate that our proposed model behaves better with respect to visual comparison and quantitative measurement.

Fast algorithm for box‐constrained fractional‐order total variation image restoration with impulse noise

In this paper, a novel variational model with box constraints is proposed to restore images corrupted by impulse noise. The proposed model is composed of fractional-order total variation regularization and Lp-fidelity term ( 0 < p < 1 ) $(0<p<1)$ . Moreover, the new model possesses the advantages of preserving sharp edges and removing blocking effect. To solve the proposed model, some auxiliary variables are first introduced to transform it into some easy-to-solve subproblems. Further, the alternating direction method of multipliers, iteratively re-weighted ℓ1 algorithm and fast iteration technique are adopted to solve the related subproblems. Numerical results show that the proposed model performs better in comparison with the several existing methods, in terms of both quantitative evaluation and visual quality.

Speckle Reduction in Ultrasound Images of the Common Carotid Artery Based on Integer and Fractional-Order Total Variation.

Designing a technique with higher speckle noise suppressing capability, better edge preserving performance, and lower time complexity is a research objective for the common carotid artery (CCA) ultrasound despeckling. Total variation based techniques have been widely used in the image denoising and have good performance in preserving the edges in the images. However, the total variation based filters can produce the staircase artifacts. To address this issue, second-order total variation based techniques have been proposed for the image denoising. However, the previous study has been proved that the fractional differential model has better performance in reducing the speckles in ultrasound despeckling compared with the second-order model. Thus, to improve the performance of ultrasound despeckling and edge preserving, a novel despeckling model based on integer and fractional-order total variation (IFOTV) is proposed for CCA ultrasound images. Moreover, the minimization problems in our despeckling model are solved by the alternating direction method of multiplier (ADMM). In results with synthetic images, the edge preservation index (EPI) values of proposed method are 0.9524, 0.8797, and 0.7351 as well as 0.9137, 0.8253, and 0.6847 under three different levels of noise, which are the highest among four advanced methods. In results with simulated CCA ultrasound images, the speckle suppression and mean preservation indices of proposed method are 0.5596, 0.6571, and 0.8106 under three different levels of noise, which are the best among four advanced methods. In results with clinical images, the average absolute error of intima-media thickness measurements of proposed method is 0.0660 ± 0.0679 (mean ± std in mm), which is the lowest among four advanced methods. In conclusion, the IFOTV method has improved performance in suppressing the speckle noise and preserving the edge, and is thus a potential alternative to the current filters for the CCA ultrasound despeckling.