High-order Total Variation Regularization Research Articles

Remove the salt and pepper noise based on the high order total variation and the nuclear norm regularization

This paper proposes a new model to remove the salt and pepper (SAP) noise problem. In the proposed method, we combine the high order total variation regularization with the nuclear norm regularization in order to keep details and structures of the restored images. Since the proposed model is convex and separable, the classic alternating direction method of multipliers can be employed to solve it by introducing some auxiliary variables to transform the original problem into the saddle point problem. Theoretically, we establish the convergence analysis of the proposed numerical algorithm. Final experimental comparisons are provided to show the satisfactory performance of the proposed model, which outperforms other related competitive methods in both the SNR and the SSIM.

SAR Image Speckle Reduction Based on Nonconvex Hybrid Total Variation Model

Speckle noise inherent in synthetic aperture radar (SAR) images seriously affects the visual effect and brings great difficulties to the postprocessing of the SAR image. Due to the edge-preserving feature, total variation (TV) regularization-based techniques have been extensively utilized to reduce the speckle. However, the strong scatters in SAR image with radiometry several orders of magnitude larger than their surrounding regions limit the effectiveness of TV regularization. Meanwhile, the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm first-order TV regularization sometimes causes staircase artifacts as it favors solutions that are piecewise constant, and it usually underestimates high-amplitude components of image gradient as the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm uniformly penalizes the amplitude. To overcome these shortcomings, a new hybrid variation model, called Fisher-Tippett (FT) distribution-ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> -norm first-and second-order hybrid TVs (HTpVs), is proposed to reduce the speckle after removing the strong scatters. Especially, the FT-HTpV inherits the advantages of the distribution based data fidelity term, the nonconvex regularization, and the higher order TV regularization. Therefore, it can effectively remove the speckle while preserving point scatters and edges and reducing staircase artifacts well. To efficiently solve the nonconvex minimization problem, an iterative framework with a nonmonotone-accelerated proximal gradient (nmAPG) method and a matrix-vector acceleration strategy are used. Extensive experiments on both the simulated and real SAR images demonstrate the effectiveness of the proposed method.

On the Taut String Interpretation and Other Properties of the Rudin\u2013Osher\u2013Fatemi Model in One Dimension

We study the one-dimensional version of the Rudin–Osher–Fatemi (ROF) denoising model and some related TV-minimization problems. A new proof of the equivalence between the ROF model and the so-called taut string algorithm is presented, and a fundamental estimate on the denoised signal in terms of the corrupted signal is derived. Based on duality and the projection theorem in Hilbert space, the proof of the taut string interpretation is strictly elementary with the existence and uniqueness of solutions (in the continuous setting) to both models following as by-products. The standard convergence properties of the denoised signal, as the regularizing parameter tends to zero, are recalled and efficient proofs provided. The taut string interpretation plays an essential role in the proof of the fundamental estimate. This estimate implies, among other things, the strong convergence (in the space of functions of bounded variation) of the denoised signal to the corrupted signal as the regularization parameter vanishes. It can also be used to prove semi-group properties of the denoising model. Finally, it is indicated how the methods developed can be applied to related problems such as the fused lasso model, isotonic regression and signal restoration with higher-order total variation regularization.

Remote sensing images destriping using unidirectional hybrid total variation and nonconvex low-rank regularization

In this paper, we propose a novel model for remote sensing images destriping, which includes the Schatten 1 ∕ 2 -norm and the unidirectional first-order and high-order total variation regularization. The main idea is that the stripe layer is low-rank, and the desired image possesses smoothness across stripes. Therefore, we use the Schatten 1 ∕ 2 -norm regularization to depict the low-rankness of stripes, and use the unidirectional total variation and the unidirectional high-order total variation to guarantee the smoothness of the underlying image. We develop the alternating direction method of multipliers algorithm to solve the proposed model. Extensive experiments on synthetic and real data are reported to show the superiority of the proposed method over state-of-the-art methods in terms of both quantitative and qualitative assessments.

Restoration of remote sensing images based on nonconvex constrained high-order total variation regularization

Convex total variation (TV) regularization models have been widely used in remote sensing image restoration problems; however, these models tend to produce staircase effects. We consider a nonconvex second-order TV regularization model with linear constraints for remote sensing image restoration. To solve the nonconvex second-order TV regularization model, we propose an efficient alternating minimization algorithm based on generalized iterated shrinkage algorithm and alternating direction method of multipliers. Experimental results demonstrate the effectiveness of the proposed model, which can reduce staircase effects while preserving edges. In terms of signal-to-noise ratio and structural similarity index measure, the experimental results show that our proposed model and algorithm can give better performance compared with some state-of-the-art methods.

AVO inversion with high-order total variation regularization

With ill-posed inverse problems such as AVO inversion, regularization schemes are required to obtain stable and unique solutions. The total variation (TV) regularization method is commonly used to resolve sharp interfaces and obtain blocky solutions, where edges and discontinuities are preserved. TV regularization accomplishes these goals by imposing sparsity on the first-order difference (gradient) of the model parameters. In this paper, we first explore the use of TV regularization in the three-term AVO inversion problem. Then, we illustrate that the discretization and sampling in data processing may yield complex shapes of models, whereas conventional TV regularization is not applicable to such situation. Therefore, we adaptively refine the original TV regularization by updating the order of difference operator and propose a high-order TV regularization scheme for non-block-structured models. The method can effectively use the structural information of the models, which ultimately results in better-quality solutions. The rationality of the approach is also confirmed by using sparse theory and a reconstruction algorithm. Moreover, to solve the inversion problem, we improve an existing optimization algorithm via a step search process to obtain higher efficiency and lower residual error. Tests on both synthetic and real data show that our method can provide a credible and high-resolution estimation of the subsurface models.

Image Restoration by a Mixed High-Order Total Variation and l1 Regularization Model

Total variation regularization is well-known for recovering sharp edges; however, it usually produces staircase artifacts. In this paper, in order to overcome the shortcoming of total variation regularization, we propose a new variational model combining high-order total variation regularization and l1 regularization. The new model has separable structure which enables us to solve the involved subproblems more efficiently. We propose a fast alternating method by employing the fast iterative shrinkage-thresholding algorithm (FISTA) and the alternating direction method of multipliers (ADMM). Compared with some current state-of-the-art methods, numerical experiments show that our proposed model can significantly improve the quality of restored images and obtain higher SNR and SSIM values.

Higher order total variation regularization for EIT reconstruction.

Electrical impedance tomography (EIT) attempts to reveal the conductivity distribution of a domain based on the electrical boundary condition. This is an ill-posed inverse problem; its solution is very unstable. Total variation (TV) regularization is one of the techniques commonly employed to stabilize reconstructions. However, it is well known that TV regularization induces staircase effects, which are not realistic in clinical applications. To reduce such artifacts, modified TV regularization terms considering a higher order differential operator were developed in several previous studies. One of them is called total generalized variation (TGV) regularization. TGV regularization has been successively applied in image processing in a regular grid context. In this study, we adapted TGV regularization to the finite element model (FEM) framework for EIT reconstruction. Reconstructions using simulation and clinical data were performed. First results indicate that, in comparison to TV regularization, TGV regularization promotes more realistic images. Graphical abstract Reconstructed conductivity changes located on selected vertical lines. For each of the reconstructed images as well as the ground truth image, conductivity changes located along the selected left and right vertical lines are plotted. In these plots, the notation GT in the legend stands for ground truth, TV stands for total variation method, and TGV stands for total generalized variation method. Reconstructed conductivity distributions from the GREIT algorithm are also demonstrated.

EIT reconstruction using higher order TV regularization

Electrical Impedance Tomography (EIT) reconstructs the conductivity distribution of a domain from the electrical boundary conditions. This is an ill-posed inverse problem; its solution is very unstable. Total Variation (TV) regularization is one of the common techniques employed to stabilize the reconstructions. However, TV regularization is known to induce staircase effects. These effects are not realistic in clinical applications. To reduce such artifacts, we introduced higher order generalized TV, Total Generalized Variation (TGV) regularization. Simulations showed that reconstruction with TGV regularization produced less distorted images when the conductivity distribution in the domain is not blockwise constant.

Composite SAR imaging using sequential joint sparsity

This paper investigates accurate and efficient ℓ1 regularization methods for generating synthetic aperture radar (SAR) images. Although ℓ1 regularization algorithms are already employed in SAR imaging, practical and efficient implementation in terms of real time imaging remain a challenge. Here we demonstrate that fast numerical operators can be used to robustly implement ℓ1 regularization methods that are as or more efficient than traditional approaches such as back projection, while providing superior image quality. In particular, we develop a sequential joint sparsity model for composite SAR imaging which naturally combines the joint sparsity methodology with composite SAR. Our technique, which can be implemented using standard, fractional, or higher order total variation regularization, is able to reduce the effects of speckle and other noisy artifacts with little additional computational cost. Finally we show that generalizing total variation regularization to non-integer and higher orders provides improved flexibility and robustness for SAR imaging.

Cartoon-texture composite regularization based non-blind deblurring method for partly-textured blurred images with Poisson noise

In recent years, deblurring image with Poisson noise has attracted more and more attention in many areas such as astronomy and biological imaging. This is an ill-posed problem and can be regularized to improve the quality of the solution. Fractional-order total variation regularization can eliminate the staircase effect caused by the total variation regularization and avoid over-smoothing at the edges caused by the high-order total variation regularization, but it cannot preserve textures well. Non-local regularization can preserve textures well but often causes extra artifacts especially in the non-texture regions. In this paper, we deal with the cartoon component and the texture component differently and propose a cartoon-texture composite regularization based non-blind deblurring method. We utilize fractional-order total variation regularization for the cartoon component to eliminate the staircase effect and avoid over-smoothing at the edges, and use the non-local total variation regularization for the texture component to preserve the details and eliminate the artifacts in the non-textured regions. We develop an alternating direction method of multipliers based algorithm to solve the proposed model. Experiments results show that our method can improve the result both visually and in terms of the peak signal to noise ratio and structural similarity index efficiently.

非凸高阶全变差正则化自然光学图像盲复原

非凸高阶全变差正则化自然光学图像盲复原

High-order total variation regularization approach for axially symmetric object tomography from a single radiograph

In this paper, we consider tomographic reconstruction for axially symmetric objects from a single radiograph formed by fan-beam X-rays. All contemporary methods are based on the assumption that the density is piecewise constant or linear. From a practical viewpoint, this is quite a restrictive approximation. The method we propose is based on high-order total variation regularization. Its main advantage is to reduce the staircase effect while keeping sharp edges and enable the recovery of smoothly varying regions. The optimization problem is solved using the augmented Lagrangian method which has been recently applied in image processing. Furthermore, we use a one-dimensional (1D) technique for fan-beam X-rays to approximate 2D tomographic reconstruction for cone-beam X-rays. For the 2D problem, we treat the cone beam as fan beam located at parallel planes perpendicular to the symmetric axis. Then the density of the whole object is recovered layer by layer. Numerical results in 1D show that the proposed method has improved the preservation of edge location and the accuracy of the density level when compared with several other contemporary methods. The 2D numerical tests show that cylindrical symmetric objects can be recovered rather accurately by our high-order regularization model.

Splines in Higher Order TV Regularization

Splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces. In this paper, we show in a strictly discrete setting that splines of degree m?1 solve also a minimization problem with quadratic data term and m-th order total variation (TV) regularization term. In contrast to problems with quadratic regularization terms involving m-th order derivatives, the spline knots are not known in advance but depend on the input data and the regularization parameter ?. More precisely, the spline knots are determined by the contact points of the m---th discrete antiderivative of the solution with the tube of width 2? around the m-th discrete antiderivative of the input data. We point out that the dual formulation of our minimization problem can be considered as support vector regression problem in the discrete counterpart of the Sobolev space W 2,0 m . From this point of view, the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines.