High-order Total Variation Research Articles

Image Reconstruction of Conductivity Distribution With Combined L 1-Norm Fidelity and Hybrid Total Variation Penalty

Image reconstruction of conductivity distribution is a highly nonlinear ill-posed inverse problem in electrical impedance tomography (EIT). To solve the problem, Tikhonov regularization and total variation (TV) regularization methods have been generally adopted. However, reconstructed images suffer from excessive smoothness or staircase effect. In this article, a novel approach is proposed for imaging conductivity distribution. Unlike traditional regularization methods, fidelity term based on <inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula>-norm is introduced to stabilize the ill-posed problem and enforce sparsity in the solution. Moreover, a first-order and high-order TV combined hybrid penalty term is introduced to keep the sharp profile and restrain the staircase effect. During image reconstruction, regularization parameter and weighting factor are determined by the adaptive method. In addition, a soft-thresholding operator is introduced to remove artifact and enhance robustness to noise. Numerical simulations have been performed to validate the performance of the proposed approach against Landweber, Tikhonov, TV, high-order TV, and iterated soft-thresholding methods. The impacts of conductivity distribution, conductivity contrast, contact impedance, and boundary shape on reconstruction are studied. Besides, robustness to noise and computation time are evaluated. The performance of the proposed approach is also verified by phantom experiments. Simulation and experimental results demonstrate that the proposed approach performs well in image reconstruction of conductivity distribution. Clearer background is observed and inclusions with sharp or smooth edges can be better reconstructed.

Restoration of Poissonian Images Using Nonconvex Regularizer with Overlapping Group Sparsity

Aimed at achieving the accurate restoration of Poissonian images that exhibit neat edges and no staircase effect, this article develops a novel hybrid nonconvex double regularizer model. The proposed scheme closely takes the advantages of total variation with overlapping group sparsity and nonconvex high-order total variation priors. The overlapping group sparsity is adopted to globally suppress the staircase artifacts, while the nonconvex high-order regularization plays the role of locally preserving the significant image features and edge details. Computationally, a quite efficient alternating direction method of multipliers, associated with the iteratively reweighted ${\ell _{1}}$ algorithm and the majorization-minimization method, is employed to settle the optimization problem iteratively. Finally, exhaustive simulation experiments are executed for recovering Poissonian images, which are made comparisons with several state-of-the-art restoration strategies, indicate the brilliant performance of our model in terms of intuitive effects and accuracy evaluation.

A spatially adaptive hybrid total variation model for image restoration under Gaussian plus impulse noise

In this paper, a spatially adaptive hybrid total variation model is proposed to recover blurred images corrupted by mixed Gaussian-impulse noise. The model consists of a combined L1/L2 data fidelity term and two regularization terms including total variation and high-order total variation. The spatially adaptive parameters with multiple windows are utilized by the model to adequately smooth homogeneous areas while preserving small features. A strategy for adaptively selecting the locally varying parameters together with a solver of the constituted optimisation problem are presented. Experimental results demonstrate the excellent performance of the new approach compared with current state-of-the-art methods with respect to digital indicators and visual quality.

Non-Blind Image Deblurring Method Using Shear High Order Total Variation Norm

In this paper, we propose a shear high-order gradient (SHOG) operator by combining the shear operator and high-order gradient (HOG) operator. Compared with the HOG operator, the proposed SHOG operator can incorporate more directionality and detect more abundant edge information. Based on the SHOG operator, we extend the total variation (TV) norm to shear high-order total variation (SHOTV), and then propose a SHOTV deblurring model. We also study some properties of the SHOG operator, and show that the SHOG matrices are Block Circulant with Circulant Blocks (BCCB) when the shear angle is [see formula in PDF]. The proposed model is solved efficiently by the alternating direction method of multipliers (ADMM). Experimental results demonstrate that the proposed method outperforms some state-of-the-art non-blind deblurring methods in both objective and perceptual quality.

Combined total variation of first and fractional orders for Poisson noise removal in digital images

Introduction: Many methods have been proposed to handle the image restoration problem with Poisson noise. A popular approach to Poissonian image reconstruction is the one based on Total Variation. This method can provide significantly sharp edges and visually fine images, but it results in piecewise-constant regions in the resulting images. Purpose: Developing an adaptive total variation-based model for the reconstruction of images contaminated by Poisson noise, and an algorithm for solving the optimization problem. Results: We proposed an effective way to restore images degraded by Poisson noise. Using the Bayesian framework, we proposed an adaptive model based on a combination of first-order total variation and fractional order total variation. The first-order total variation model is efficient for suppressing the noise and preserving the keen edges simultaneously. However, the first-order total variation method usually causes artifact problems in the obtained results. To avoid this drawback, we can use high-order total variation models, one of which is the fractional-order total variation-based model for image restoration. In the fractional-order total variation model, the derivatives have an order greater than or equal to one. It leads to the convenience of computation with a compact discrete form. However, methods based on the fractional-order total variation may cause image blurring. Thus, the proposed model incorporates the advantages of two total variation regularization models, having a significant effect on the edge-preserving image restoration. In order to solve the considered optimization problem, the Split Bregman method is used. Experimental results are provided, demonstrating the effectiveness of the proposed method. Practical relevance: The proposed method allows you to restore Poissonian images preserving their edges. The presented numerical simulation demonstrates the competitive performance of the model proposed for image reconstruction. Discussion: From the experimental results, we can see that the proposed algorithm is effective in suppressing noise and preserving the image edges. However, the weighted parameters in the proposed model were not automatically selected at each iteration of the proposed algorithm. This requires additional research.

An improved inversion algorithm to reconstruct 2D temperature fields of long sparks with high-speed schlieren technique

The temperature measurement for discharge channels plays an important role in studying the mechanism of long sparks and natural lightning, in which cases the high-speed schlieren technique is usually applied. However, the high-speed schlieren images are noisy, and the discharge channel axis is not discernibly determined from the discrete pixels, which can sometimes cause huge errors using the traditional inversion algorithm. This may be the reason why two-dimensional (2D) temperature fields of long sparks are rarely reported at present. To fill the gap, an improved inversion algorithm is proposed, in which the dichotomy and the high order total variation (TV) regularization methods are introduced. The proposed algorithm is verified to have better performance compared with the traditional ones by numerical tests. At last, 2D temperature fields of discharge channels are reconstructed and discussed.

High-Order Directional Total Variation for Seismic Noise Attenuation

High-amplitude noise could interfere with useful seismic signals, affecting our ability in processing and interpreting seismic data. Thus, attenuating seismic noise is an important task in seismic processing. Total variation (TV) has played an important role in many steps of seismic data processing but always neglected the seismic structural information. Directional TV (DTV), however, considers the structural direction of seismic events but tends to cause the staircasing effect on seismic records based on the first-order formulation. Here, we develop a high-order DTV (HDTV) method for seismic denoising. It considers the local structural direction of the seismic data and calculates the higher order derivatives of seismic images to avoid the staircasing effect. We design several synthetic models that are contaminated by various types of random noise to test the denoising ability. The denoising performance of our new method is compared with the first-order DTV, conventional high-order TV, and TV regularization methods from two aspects, i.e., the signal-to-noise ratio and the effective signal leakage degree. Then, the advantages of the proposed method are further validated via several field seismic data sets.

Gradient-based and wavelet-based compressed sensing approaches for highly undersampled tomographic datasets

Electron tomography is widely employed for the 3D morphological characterization at the nanoscale. In recent years, there has been a growing interest in analytical electron tomography (AET) as it is capable of providing 3D information about the elemental composition, chemical bonding and optical/electronic properties of nanomaterials. AET requires advanced reconstruction algorithms as the datasets often consist of a very limited number of projections. Total variation (TV)-based compressed sensing approaches were shown to provide high-quality reconstructions from undersampled datasets, but staircasing artefacts can appear when the assumption about piecewise constancy does not hold. In this paper, we compare higher-order TV and wavelet-based approaches for AET applications and provide an open-source Python toolbox, Pyetomo, containing 2D and 3D implementations of both methods. A highly sampled STEM-HAADF dataset of an Er-doped porous Si sample and a heavily undersampled STEM-EELS dataset of a Ge-rich GeSbTe (GST) thin film annealed at 450°C are used to evaluate the performance of the different approaches. We show that polynomial annihilation with order 3 (HOTV3) and the Bior4.4 wavelet outperform the classical TV minimization and the related Haar wavelet.

A high-order total-variation regularisation method for full-waveform inversion

Abstract Full-waveform inversion (FWI) is among the most effective methods of velocity modelling in seismic exploration. However, because of the strong nonlinearity of the FWI, if the velocity in the target geobody is not sharply different from that in its surroundings, the total variation (TV) of the model will not be sufficiently sparse. To alleviate this issue, we propose a novel TV-regularised FWI method that can consider the sparsity of the high-order regularisation operator and consequently improve the stability of the inversion process and produce more focused model boundaries. We use a split-Bregman algorithm to solve the inversion optimisation problem while building the TV-regularised objective function. We show that stable model updates can be obtained by this algorithm, which proved to be effective and reliable in the numerical tests. These tests also show that the proposed method converges faster, can model the velocity domain better than conventional methods and can effectively identify layer boundaries with a weak velocity contrast. We conclude that the novel FWI method based on high-order TV regularisation is robust and accurate.

Effect of X-ray $$\mu$$CT Resolution on the Computation of Permeability and Dispersion Coefficient for Granular Soils

X-ray micro-computed tomography ( $$\mu$$ CT) can produce realistic 3D-images of the pore structure of a material. Extracting its geometry enables the computation of effective properties of the material—such as the permeability (k) and the hydrodynamic dispersion coefficient ( $$D_h$$ )—, through the solutions of the Stokes equation (SE) and Advection-Diffusion equation (ADE), respectively. In this study, the effect of the image resolution on these properties is discussed. For such purpose, four different resolutions are evaluated for both a real sample of Fontainebleau sand and a numerically generated sample created by degrading the Fontainebleau image with highest resolution. The SE was computed using the commercial software GeoDict. To solve the ADE, a Finite Volume software was developed which includes a high order total variation diminishing scheme for advection. The analysis of dispersion was based on numerical breakthrough curves. Our model was tested in a large range of Peclet numbers (Pe) and travel distances, accurately describing the transition between diffusion and advection dominated regimes of dispersion. The $$D_h$$ exhibits a linear increase with travel distance for Pe $$> 10$$ . This classical effect increases with increasing Pe. The percentage change on k and $$D_h$$ increases with decreasing resolution in agreement with the corresponding behavior of porosity, specific surface and pore size distributions. The images directly scaled with the $$\mu$$ CT showed more discrepancy than the numerically scaled images. The criteria to estimate the quality of permeability from the pore size distribution proposed on our previous study remains valid. The $$D_h$$ is less sensitive to resolution than k.

Hybrid High-Order and Fractional-Order Total Variation with Nonlocal Regularization for Compressive Sensing Image Reconstruction

Total variation often yields staircase artifacts in the smooth region of the image reconstruction. This paper proposes a hybrid high-order and fractional-order total variation with nonlocal regularization algorithm. The nonlocal means regularization is introduced to describe image structural prior information. By selecting appropriate weights in the fractional-order and high-order total variation coefficients, the proposed algorithm makes the fractional-order and the high-order total variation complement each other on image reconstruction. It can solve the problem of non-smooth in smooth areas when fractional-order total variation can enhance image edges and textures. In addition, it also addresses high-order total variation alleviates the staircase artifact produced by traditional total variation, still smooth the details of the image and the effect is not ideal. Meanwhile, the proposed algorithm suppresses painting-like effects caused by nonlocal means regularization. The Lagrange multiplier method and the alternating direction multipliers method are used to solve the regularization problem. By comparing with several state-of-the-art reconstruction algorithms, the proposed algorithm is more efficient. It does not only yield higher peak-signal-to-noise ratio (PSNR) and structural similarity (SSIM) but also retain abundant details and textures efficiently. When the measurement rate is 0.1, the gains of PSNR and SSIM are up to 1.896 dB and 0.048 dB respectively compared with total variation with nonlocal regularization (TV-NLR).

A simple and fast ASD-POCS algorithm for image reconstruction.

The adaptive steepest descent projection onto convex set (ASD-POCS) algorithm is a promising algorithm for constrained total variation (TV) type norm minimization models in computed tomography (CT) image reconstruction using sparse and/or noisy data. However, in ASD-POCS algorithm, the existing gradient expression of the TV-type norm appears too complicated in the implementation code and reduces image reconstruction speed. To address this issue, this work aims to develop and test a simple and fast ASD-POCS algorithm. Since the original algorithm is not derived thoroughly, we first obtain a simple matrix-form expression by thorough derivation via matrix representations. Next, we derive the simple matrix expressions of the gradients of TV, adaptive weighted TV (awTV), total p-variation (TpV), high order TV (HOTV) norms by term combinations and matrix representations. The deep analysis is then performed to identify the hidden relations of these terms. The TV reconstruction experiments by use of sparse-view projections via the Shepp-Logan, FORBILD and a real CT image phantoms show that the simplified ASD-POCS (S-ASD-POCS) using the simple matrix-form expression of TV gradient achieve the same reconstruction accuracy relative to ASD-POCS, whereas it enables to speed up the whole ASD process 1.8-2.7 time fast. The derived simple matrix expressions of the gradients of these TV-type norms may simplify the implementation of the ASD-POCS algorithm and speed up the ASD process. Additionally, a general gradient expression suitable to all the sparse transform-based optimization models is demonstrated so that the ASD-POCS algorithm may be tailored to extended image reconstruction fields with accelerated computational speed.

Non-Convex High Order Total Variation With Overlapping Group Sparsity Denoising Model Under Cauchy Noise

It is widely known that the total variation regularization model preserves the edges well in the restored images but has some staircase effects. We consider using non-convex high-order total variation and overlapping group sparsity as a hybrid regularization to present a new denoising model. The proposed model can well preserve edges and reduce the staircase effect in the smooth region of the restored images. In order to solve the proposed hybrid model, we develop an efficient alternating minimization method. Compared with other models for removing Cauchy noise, numerical experimental results demonstrate that the superiority of the proposed model and algorithm, both in terms of visual and quantitative measures.

Image retinex based on the nonconvex TV-type regularization

&lt;p style='text-indent:20px;'&gt;Retinex theory is introduced to show how the human visual system perceives the color and the illumination effect such as Retinex illusions, medical image intensity inhomogeneity and color shadow effect etc.. Many researchers have studied this ill-posed problem based on the framework of the variation energy functional for decades. However, to the best of our knowledge, the existing models via the sparsity of the image based on the nonconvex &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \ell^p $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-quasinorm were limited. To deal with this problem, this paper considers a TV&lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ _p $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-HOTV&lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ _q $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-based retinex model with &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ p, q\in(0, 1) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Specially, the TV&lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ _p $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; term based on the total variation(TV) regularization can describe the reflectance efficiently, which has the piecewise constant structure. The HOTV&lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ _q $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; term based on the high order total variation(HOTV) regularization can penalize the smooth structure called the illumination. Since the proposed model is non-convex, non-smooth and non-Lipschitz, we employ the iteratively reweighed &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ \ell_1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; (IRL1) algorithm to solve it. We also discuss some properties of our proposed model and algorithm. Experimental experiments on the simulated and real images illustrate the effectiveness and the robustness of our proposed model both visually and quantitatively by compared with some related state-of-the-art variational models.&lt;/p&gt;

Infrared Image Deblurring via High-Order Total Variation and Lp-Pseudonorm Shrinkage

The quality of infrared images is affected by various degradation factors, such as image blurring and noise pollution. Anisotropic total variation (ATV) has been shown to be a good regularization approach for image deblurring. However, there are two main drawbacks in ATV. First, the conventional ATV regularization just considers the sparsity of the first-order image gradients, thus leading to staircase artifacts. Second, it employs the L1-norm to describe the sparsity of image gradients, while the L1-norm has a limited capacity of depicting the sparsity of sparse variables. To address these limitations of ATV, a high-order total variation is introduced in the ATV deblurring model and the Lp-pseudonorm is adopted to depict the sparsity of low- and high-order total variation. In this way, the recovered image can fit the image priors with clear edges and eliminate the staircase artifacts of the ATV model. The alternating direction method of multipliers is used to solve the proposed model. The experimental results demonstrate that the proposed method does not only remove blurs effectively but is also highly competitive against the state-of-the-art methods, both qualitatively and quantitatively.

Hybrid higher‐order total variation model for multiplicative noise removal

As an important, challenging, and difficult problem in image processing, multiplicative noise removal (MNR) has attracted great attention. To this end, many variational methods have been effectively proposed in the past few decades. Among these variational methods, total variation (TV) and its higher-order extensions are very effective, where the former can preserve sharp edges but cause some undesirable staircase effects and the latter can better reduce the staircase effects but sometimes smooth the image details. To overcome the drawbacks while taking full use of their merits, the authors propose a novel hybrid higher-order TV regularisation model for MNR, in which the novelty of the proposed model consists of combining the image prior information of first-order and second-order derivatives to propose a novel higher-order regulariser, named as hybrid higher-order TV (HHTV). More specifically, a more preferable equivalent formulation of HHTV is derived. Then, they use the derived equivalent formulation to design an efficient alternating iterative algorithm to solve the proposed model. Finally, the experimental results demonstrate that the proposed HHTV method outperforms several state-of-the-art methods in terms of image quality and convergence speed.

An adaptive method for image restoration based on high-order total variation and inverse gradient

The total variation (TV) regularization model for image restoration is widely utilized due to its edge preservation properties. Despite its advantages, the TV regularization can obtain spurious oscillations in flat regions of digital images and thus recent works advocate high-order TV regularization models. In this work, we propose an adaptive image restoration method based on a combination of first-order and second-order total variations regularization with an inverse-gradient-based adaptive parameter. The proposed model removes noise effectively and preserves image structures. Due to the adaptive parameter estimation based on the inverse gradient, it avoids the staircasing artifacts associated with TV regularization and its variant models. Experimental results indicate that the proposed method obtains better restorations in terms of visual quality as well as quantitatively. In particular, our proposed adaptive higher-order TV method obtained (19.3159, 0.7172, 0.90985, 0.79934, 0.99838) PSNR, SSIM, MS-SSIM, F-SIM, and P-SIM values compared to related models such as the TV-Bounded Hessian (18.9735, 0.6599, 0.8718, 0.73833, 0.99767), and TV-Laplacian (19.0345, 0.6719, 0.88198, 0.75405, 0.99789).

An Improved Mapping with Super-Resolved Multispectral Images for Geostationary Satellites

Super-resolution (SR) technology has shown great potential for improving the performance of the mapping and classification of multispectral satellite images. However, it is very challenging to solve ill-conditioned problems such as mapping for remote sensing images due to the presence of complicated ground features. In this paper, we address this problem by proposing a super-resolution reconstruction (SRR) mapping method called the mixed sparse representation non-convex high-order total variation (MSR-NCHOTV) method in order to accurately classify multispectral images and refine object classes. Firstly, MSR-NCHOTV is employed to reconstruct high-resolution images from low-resolution time-series images obtained from the Gaofen-4 (GF-4) geostationary orbit satellite. Secondly, a support vector machine (SVM) method was used to classify the results of SRR using the GF-4 geostationary orbit satellite images. Two sets of GF-4 satellite image data were used for experiments, and the MSR-NCHOTV SRR result obtained using these data was compared with the SRR results obtained using the bilinear interpolation (BI), projection onto convex sets (POCS), and iterative back projection (IBP) methods. The sharpness of the SRR results was evaluated using the gray-level variation between adjacent pixels, and the signal-to-noise ratio (SNR) of the SRR results was evaluated by using the measurement of high spatial resolution remote sensing images. For example, compared with the values obtained using the BI method, the average sharpness and SNR of the five bands obtained using the MSR-NCHOTV method were higher by 39.54% and 51.52%, respectively, and the overall accuracy (OA) and Kappa coefficient of the classification results obtained using the MSR-NCHOTV method were higher by 32.20% and 46.14%, respectively. These results showed that the MSR-NCHOTV method can effectively improve image clarity, enrich image texture details, enhance image quality, and improve image classification accuracy. Thus, the effectiveness and feasibility of using the proposed SRR method to improve the classification accuracy of remote sensing images was verified.

Study of CT image reconstruction algorithm based on high order total variation

The traditional total variation (TV) minimization algorithm is an image reconstruction algorithm based on compressed sensing, which can accurately reconstruct images from sparse data or highly noisy data and has been widely used in low-dose computed tomography (CT). Sometimes it may lead to staircase effect if the reconstructed image has not obvious piecewise constant feature. Recently, researches in the field of image processing suggested that the high order total variation (HOTV) can effectively suppress staircase effect. Whereas the HOTV reconstruction algorithm has not been carried out deeply and extensively in image reconstruction. Herein, we propose a HOTV reconstruction model and design its adaptive steepest descent-projection onto convex sets (ASD-POCS) solving algorithm, and then characterize its reconstruction performance. We construct the second order TV norm using the second order gradient, and design the data fidelity constrained, second order TV minimization model, and derive the its ASD-POCS algorithm. In order to characterize its reconstruction performance, we use the Shepp-Logan simulation phantom of wavy background, the gray-changing simulation phantom and the chest X-ray image simulation phantom to perform reconstructions. Sparse reconstruction results show that compared to the traditional TV algorithm, the HOTV algorithm can effectively suppress the staircase effect and improve the reconstruction accuracy. Noisy-projections reconstruction results show that the traditional TV algorithm and the HOTV algorithm both have good denoising effect but the HOTV algorithm is slightly better for it can better protect the image edge information. The HOTV algorithm is a better reconstruction algorithm than the TV algorithm in image reconstruction if the piecewise constant feature of the reconstructed image is not so obvious but the grayscale fluctuation feature is prominent. The proposed HOTV algorithm can be extended to various CT configurations and other imaging modalities.

Higher-Order Total Directional Variation: Analysis

We analyze a new notion of total anisotropic higher-order variation which, differently from total generalized variation in [K. Bredies, K. Kunisch, and T. Pock, SIAM J. Imaging Sci., 3 (2010), pp. 492--526], quantifies for possibly nonsymmetric tensor fields their variations at arbitrary order weighted by possibly inhomogeneous, smooth elliptic anisotropies. We prove some properties of this total variation and of the associated spaces of tensors with finite variations. We show the existence of solutions to a related regularity-fidelity optimization problem. We also prove a decomposition formula which appears to be helpful for the design of numerical schemes, as shown in a companion paper, where several applications to image processing are studied.