Total Variation Model Research Articles

Constrained Total Variation Based Three-Dimension Single Particle Reconstruction in Cryogenic Electron Microscopy

The single particle reconstruction (SPR) in cryogenic electron microscopy is considered in this paper. This is an emerging technique for determining the three-dimensional (3D) structure of biological specimens from a limited number of the micrographs. Because the micrographs are modulated by contrast transfer functions and corrupted by heavy noise, the number of micrographs might be limited, in general it is a serious ill-posed problem to reconstruct the original particle. In this paper, we propose a constrained total variation (TV) model for single particle reconstruction. The TV norm is represented by the dual formulation that changes the SPR problem into a minimax one. The primal-dual method is applied to find the saddle point of the minimax problem, and the convergence condition is given. Numerical results show that the proposed model is very effective in reconstructing the particle.

Block-Based Refitting in $$\ell _{12}$$ Sparse Regularization

In many linear regression problems, including ill-posed inverse problems in image restoration, the data exhibit some sparse structures that can be used to regularize the inversion. To this end, a classical path is to use $$\ell _{12}$$ block-based regularization. While efficient at retrieving the inherent sparsity patterns of the data—the support—the estimated solutions are known to suffer from a systematical bias. We propose a general framework for removing this artifact by refitting the solution toward the data while preserving key features of its structure such as the support. This is done through the use of refitting block penalties that only act on the support of the estimated solution. Based on an analysis of related works in the literature, we introduce a new penalty that is well suited for refitting purposes. We also present a new algorithm to obtain the refitted solution along with the original (biased) solution for any convex refitting block penalty. Experiments illustrate the good behavior of the proposed block penalty for refitting solutions of total variation and total generalized variation models.

Edge-guided second-order total generalized variation for Gaussian noise removal from depth map

Total generalized variation models have recently demonstrated high-quality denoising capacity for single image. In this paper, we present an accurate denoising method for depth map. Our method uses a weighted second-order total generalized variational model for Gaussian noise removal. By fusing an edge indicator function into the regularization term of the second-order total generalized variational model to guide the diffusion of gradients, our method aims to use the first or second derivative to enhance the intensity of the diffusion tensor. We use the first-order primal–dual algorithm to minimize the proposed energy function and achieve high-quality denoising and edge preserving result for depth maps with high -intensity noise. Extensive quantitative and qualitative evaluations in comparison to bench-mark datasets show that the proposed method provides significant higher accuracy and visual improvements than many state-of-the-art denoising algorithms.

Color image multiplicative noise and blur removal by saturation-value total variation

In this paper, we propose and develop a novel Saturation-Value Total Variation (SVTV) model for multiplicative noise and blur removal of color images. In the proposed model, SVTV regularization term is applied to model the target color image in HSV color space instead of RGB color space, and the fidelity term is well-adapted to multiplicative noise. We investigate into the existence and uniqueness of the minimizer of the proposed minimization problem. We study and show the convergence of an implicit scheme of the associated evolution problem for the numerical solution of the proposed SVTV model. Numerical examples are presented to demonstrate the performance of the proposed SVTV model is significantly better than that of other testing methods in terms of some criteria such as PSNR, SSIM and S-CIELAB color error.

Improved partial differential equation-based total variation approach to non-subsampled contourlet transform for medical image denoising

This article proposes an improved partial differential equation (PDE)-based total variation (TV) model that enhances grey and coloured brain tumour images obtained by magnetic resonance imaging. A nonsubsampled contourlet transform was applied to images from standard databases that converted into lowpass and highpass (or bandpass) contourlet coefficients. An improved version of the power-law transform method was used on the lowpass contourlet coefficients, and an adaptive threshold method was applied to the highpass (or bandpass) contourlet coefficients. The inverse contourlet transform was performed on all the enhanced contourlet coefficients to generate a complete brain tumour image. Finally, the PDE-based TV model was applied to this image to produce the denoised image. The performance of the suggested method was calculated in terms of the peak signal-to-noise ratio, mean square error, and structural similarity index. This method achieved the best peak signal-to-noise ratio, mean square error, and structural similarity index of 77.9846 dB, 0.00012612, and 97.895%, respectively, compared to the conventional PDE+modified transform-based gamma correction, adaptive PDE+generalized cross-validation, parallel magnetic resonance imaging, and Berkeley wavelet transform+support vector machine methods.

A deep unrolling network inspired by total variation for compressed sensing MRI

Compressed Sensing theory breaks through the limitation of the Nyquist sampling law and provides theoretical support for accelerating the imaging process of MRI while reconstructing high-quality images. It can use less sampling data through the reconstructed algorithm to restore the original signal. Reconstruction time and reconstruction algorithms play important roles in compressed sensing. However, it is difficult to apply the iterative reconstruction method in clinical applications because of the long reconstruction time. In addition, hand-crafted image prior which is widely used in traditional iterative method lacks of adaptivity for MRI reconstruction. In this paper, inspired by the total variation model, we propose a novel deep network for CSMRI, dubbed as TV-Inspired Network (TVINet), which incorporates the deep priors into the traditional iterative algorithm. We apply a general compressed sensing reconstruction framework inspired by TV regularization and solve it by combines the iterative method with deep learning. The design of TVINet comes from the inference process on the basis of Primal Dual Hybrid Gradient algorithm, which makes the network has preferable interpretability. The linear operator and regularization term are learned from the training dataset by using a convolution network. The proposed approach trains the network end-to-end and reconstructs the desired image directly from the under-sampling data. Experimental results show that the proposed method has better PSNR or SSIM than state-of-the-art methods, and maintains more complex and fine details in the reconstructed image.

Speckle Noise Removal by Energy Models with New Regularization Setting

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.

A Non-Convex L₁-Norm Penalty-Based Total Generalized Variation Model for Reconstruction of Conductivity Distribution

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.

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.

Algorithmic versatility of SPF-regularization methods

Sparsity promoting functions (SPFs) are commonly used in optimization problems to find solutions which are sparse in some basis. For example, the [Formula: see text]-regularized wavelet model and the Rudin–Osher–Fatemi total variation (ROF-TV) model are some of the most well-known models for signal and image denoising, respectively. However, recent work demonstrates that convexity is not always desirable in SPFs. In this paper, we replace convex SPFs with their induced nonconvex SPFs and develop algorithms for the resulting model by exploring the intrinsic structures of the nonconvex SPFs. These functions are defined as the difference of the convex SPF and its Moreau envelope. We also present simulations illustrating the performance of a special SPF and the developed algorithms in image denoising.

Fast stripe noise removal from hyperspectral image via multi-scale dilated unidirectional convolution

Hyperspectral images (HSIs) are often contaminated by noises due to the multi-detector imaging systems, which greatly affects the subsequent HSIs interpretation and application. The 3D HSIs deliver extra spectral information, which makes the most existing destriping algorithms hardly satisfied, and the complete stripes removal and less test time consuming remain to be overcome. To meet these challenges, we present a multi-scale dilated unidirectional convolution network (MsDUC) with the following contributions. First, the deep learning-based method can fully exploit and preserve spatial-spectral correlations in 3D HSIs while the conventional methods failed to realize it. Second, different dilated convolution learns different scale features, so the introduced multi-scale dilated convolution could get more contextual information for the final restoration. Third, the clear directional signature of stripe noise and the unidirectional total variation (UTV) model inspired us to put forward the unidirectional convolution to capture the directional signature of stripe, meanwhile, the less trainable parameters and the utilized residual strategy speed up the learning process. Experimental results have shown that our method outperforms many of the state-of-the-art methods in both image restoration performance and test running time. Our code can be download from .

RETRACTED: A compensating total variation image denoising model combining L1 and L2 norm

To overcome the “staircase effect” while preserving the structural information such as image edges and textures quickly and effectively, we propose a compensating total variation image denoising model combining L1 and L2 norm. A new compensating regular term is designed, which can perform anisotropic and isotropic diffusion in image denoising, thus making up for insufficient diffusion in the total variation model. The algorithm first uses local standard deviation to distinguish neighborhood types. Then, the anisotropic diffusion based on L1 norm plays the role of edge protection in the strong edge region. The anisotropic and the isotropic diffusion simultaneously exist in the smooth region, so that the weak textures can be protected while overcoming the “staircase effect” effectively. The simulation experiments show that this method can effectively improve the peak signal-to-noise ratio and obtain the higher structural similarity index and the shorter running time.

Hyperspectral Image Mixed Noise Removal Based on Multidirectional Low-Rank Modeling and Spatial–Spectral Total Variation

Conventional low-rank (LR)-based hyperspectral image (HSI) denoising models generally convert high-dimensional data into 2-D matrices or just treat this type of data as 3-D tensors. However, these pure LR or tensor low-rank (TLR)-based methods lack flexibility for considering different correlation information from different HSI directions, which leads to the loss of comprehensive structure information and inherent spatial-spectral relationship. To overcome these shortcomings, we propose a novel multidirectional LR modeling and spatial-spectral total variation (MLR-SSTV) model for removing HSI mixed noise. By incorporating the weighted nuclear norm, we obtain the weighted sum of weighted nuclear norm minimization (WSWNNM) and the weighted sum of weighted tensor nuclear norm minimization (WSWTNNM) to estimate the more accurate LR tensor, especially, to remove the dead-line noise better. Gaussian noise is further denoised and the local spatial-spectral smoothness is preserved effectively by SSTV regularization. We develop an efficient algorithm for solving the derived optimization based on the alternating direction method of multipliers (ADMM). Extensive experiments on both synthetic data and real data demonstrate the superior performance of the proposed MLR-SSTV model for HSI mixed noise removal.

A modified total variation regularization approach based on the Gauss-Newton algorithm and split Bregman iteration for magnetotelluric inversion

The conductivity distribution of subsurface often contains segmented structure with sharp interfaces, although the magnetotelluric (MT) inversions based on Tikhonov regularization are usually stable, the smoothing interfaces tomography result from high-resistance artifacts and boundary blurring still post a great challenge for MT inversion. On the contrary, Total variation (TV) regularization can deal with the problem of high-impedance artifact and boundary blurring, but its stability is difficult to be guaranteed. In order to invert the sharp interface of conductivity distribution and improve the accuracy of MT inversion, this paper proposes an MT inversion method based on a new modified total variation (MTV) regularization. The improved method transforms the TV model constraint into two different sub-constraints: one is the standard MT inversion constraint with L2 norm, and the other is the standard L2-TV model constraint, then the Gauss-Newton algorithm and split Bregman iteration are employed to separately solve these two different sub-constraint problems. In the inversion experiments, we design a wedge-shaped model and an undulating surface model, the comparison results of accuracy and efficiency show that, among Tikhonov, TV and MTV regularization methods at the same iterations, the proposed regularization method can achieve the lowest reconstruction error and data misfit, thereby eliminating high-resistance artifacts and, in particular, can solve the problem of smoothing interfaces tomography. To this end, a field data testing is also presented to verify the effectiveness and the practicability of the new modified method.

Directional fractional-order total variation hybrid regularization for image deblurring

To overcome the staircase effect and texture loss of total variation (TV), a nonblind image deblurring method that combines the multidirectional fractional-order TV and the traditional TV is proposed. First, a four-directional fractional-order TV (FOTV4) model is put forward to explore more directional information of an image. Then the FOTV4 model and the traditional TV model are related in a unified regularization framework. Finally, the split Bregman algorithm and fast Fourier transform theory are utilized to optimize the proposed hybrid regularization model. Numerical experiments demonstrate that the proposed method can protect the texture and edge while eliminating the staircase effect effectively.

Adaptive total variation and second-order total variation-based model for low-rank tensor completion

Recently, low-rank regularization has achieved great success in tensor completion. However, only considering the global low-rankness is not sufficient, especially for a low sampling rate (SR). Total variation (TV) is introduced into low-rank tensor completion (LRTC) problem to promote the local smoothness by incorporating the first-order derivatives information. However, TV usually leads to undesirable staircase effects. To alleviate these staircase effects, we suggest a first- and second-order TV-based parallel matrix factorization model for LRTC problem, which integrates the local smoothness and global low-rankness by simultaneously exploiting the first- and second-order derivatives information. To solve the proposed model, an efficient proximal alternating optimization (PAO)-based algorithm is developed with theoretical guarantee. Moreover, we suggest a regularization parameter selection strategy to automatically update two regularization parameters, which is able to take advantage of the best properties of each of the two regularization terms. Extensive experiments on different tensor data show the superiority of the proposed method over other methods, particularly for extremely low SRs.

An image denoising iterative approach based on total variation and weighting function

Image denoising is an important technology for image preprocessing. In recent years, the image denoising technology based on total variation (TV) has been rapidly developed. However, However, although it can preserve image details well, which generates obvious staircase effects. This is due to the traditional TV-based image denoising technology only applies the gradient information and ignored the local variance of the image. In order to suppress staircase effect, in this paper, a novel image denoising approach based on TV model and weighting function is proposed. First, the theory mechanism of staircase effect brought by the traditional TV model is analyzed. Second, the effects of weighting function on edge regions, flat regions, and gradation and detail regions are also analyzed. Third, based on the above analysis, an improved TV model is proposed. Finally, the image denoising approach is implemented by an iterative algorithm. The experimental results show that, compared with various state-of-the-art models denoising models, the proposed image denoising approach can effectively suppress the staircase effect of the traditional TV model in most cases, preserve the image details, and improve the image denoising performance.

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.

A Total Fractional-Order Variation Model for Image Super-Resolution and Its SAV Algorithm

Single-image super-resolution reconstruction aims to obtain a high-resolution image from a low-resolution image. Since the super-resolution problem is ill-posed, it is common to use a regularization technique. However, the choice of the fidelity and regularization terms is not obvious, and it plays a major role in the quality of the desired high resolution image. In this paper, a hybrid single-image super-resolution model integrated with total variation (TV) and fractional-order TV is proposed to provide an effective reconstruction of the HR image. We develop an efficient numerical scheme for this model using the scalar auxiliary variable approach with an adaptive time stepping strategy. Thorough experimental results suggest that the proposed model and numerical scheme can reconstruct high quality results both quantitatively and perceptually.

Image denoising algorithm based on boosting high order non-convex total variation model

Image denoising algorithm based on boosting high order non-convex total variation model