Total Variation Method Research Articles

Discrete Total Variation-Based Non-Local Means Filter for Denoising Magnetic Resonance Images

Magnetic resonance (MR) images suffer from noise introduced by various sources. Due to this noise, diagnosis remains inaccurate. Thus, removal of noise becomes a very important task when dealing with MR images. In this paper, a denoising method has been discussed that makes use of non-local means filter and discrete total variation method. The proposed approach has been compared with other noise removal techniques like non-local means filter, anisotropic diffusion, total variation, and discrete total variation method, and it proves to be effective in reducing noise. The performance of various denoising methods is compared on basis of metrics such as peak signal-to-noise ratio (PSNR), mean square error (MSE), universal image quality index (UQI), and structure similarity index (SSIM) values. This method has been tested for various noise levels, and it outperformed other existing noise removal techniques, without blurring the image.

Fast Total Variation Method Based on Iterative Reweighted Norm for Airborne Scanning Radar Super-Resolution Imaging

The total variation (TV) method has been applied to realizing airborne scanning radar super-resolution imaging while maintaining the outline of the target. The iterative reweighted norm (IRN) approach is an algorithm for addressing the minimum Lp norm problem by solving a sequence of minimum weighted L2 norm problems, and has been applied to solving the TV norm. However, during the solving process, the IRN method is required to update the weight term and result term in each iteration, involving multiplications and the inversion of large matrices. Consequently, it suffers from a huge calculation load, which seriously restricts the application of the TV imaging method. In this work, by analyzing the structural characteristics of the matrix involved in iteration, an efficient method based on suitable matrix blocking is proposed. It transforms multiplications and the inversion of large matrices into the computation of multiple small matrices, thereby accelerating the algorithm. The proposed method, called IRN-FTV method, is more time economical than the IRN-TV method, especially for high dimensional observation scenarios. Numerical results illustrate that the proposed IRN-FTV method enjoys preferable computational efficiency without performance degradation.

4D-AirNet: a temporally-resolved CBCT slice reconstruction method synergizing analytical and iterative method with deep learning

Four-dimensional (4D) cone-beam CT (CBCT) reconstructs temporally-resolved phases of 3D volumes often with the same amount of projection data that are meant for reconstructing a single 3D volume. 4D CBCT is a sparse-data problem that is very challenging for high-quality 4D CBCT image reconstruction. Here we develop a new method, namely 4D-AirNet, that synergizes analytical and iterative method with deep learning for high-quality temporally-resolved CBCT slice reconstruction. 4D-AirNet is an unrolling method using the optimization framework of fused analytical and iterative reconstruction (AIR), which is based on proximal forward-backward splitting (PFBS). Three different strategies are developed for 4D-AirNet: random-phase (RP), prior-guided (PG), and all-phase (AP). RP-AirNet and PG-AirNet utilize phase-by-phase training and reconstruction, while PG-AirNet also uses a prior image reconstructed with all-phase projection data. Dense connectivity is built into 4D-AirNet networks for improved reconstruction quality. In contrast, AP-AirNet trains and reconstructs all phases simultaneously. In addition, the joint regularization method of DL and conventional spatiotemporal total variation (TV) is investigated. 4D-AirNet methods were evaluated in comparison with conventional iterative (TV) and deep learning (LEARN) methods, using simulated 2D-t CBCT scans from a lung dataset with various sparse-data levels. The reconstruction results suggest 4D-AirNet methods outperform TV and LEARN, and AP-AirNet provides the best reconstruction quality overall.

An optimized SVM based possibilistic fuzzy c-means clustering algorithm for tumor segmentation

To design an efficient partial differential equation-based total variation method for denoising and possibilistic fuzzy c-means clustering algorithm for segmentation and these methods presented the more detailed information of the MRI medical images compared to traditional methods. In this article, the pipeline of the proposed method described by two modules like pre-processing and segmentation. In pre-processing, noisy image is decomposed using nonsubsampled contourlet transform and it contains highpass contourlet coefficient (i.e., noisy coefficient) is removed by the threshold method as well. After reconstruction, the primary denoised image is enhanced by an improved partial differential equation-based total variation method in terms of image details like edges, boundaries, etc. In segmentation, the enhanced primary denoised image is segmented by an improved possibilistic fuzzy c-means clustering algorithm that avoids limitations in possibilistic c-means, fuzzy c-means, and K-means clustering. Next, a support vector machine classifier is utilized to identify brain tissues into gray matter, white matter, cerebrospinal fluid, and tumor part. The parameters were optimally selected by a grey wolf optimization algorithm for the classification of brain tissues. The performance of the proposed method is computed with reference to peak signal-to-noise ratio, mean square error, structural similarity index, sensitivity, specificity, and accuracy. The experimental results claimed that the proposed method is better than the traditional methods.

Analytical and Simple Form of Shrinkage Functions for Non-Convex Penalty Functions in Fused Lasso Algorithm

In some circumstances, the performance of machine learning (ML) tasks are based on the quality of signal (data) that is processed in these tasks. Therefore, the pre-processing techniques, such as reconstruction and denoising methods, are important techniques in ML tasks. In reconstructed (estimated) method, the fused lasso algorithm with non-convex penalty function is an efficient method when the signal corrupted by additive white Gaussian noise (AWGN) is considered. Therefore, this paper proposes new shrinkage functions for non-convex penalty functions, modified arctangent and exponential models, in fused lasso formulation. A lot of works present the shrinkage function for arctangent penalty function. Unfortunately, there is no closed-form solution. The numerical solution is required for shrinkage function of this penalty function. However, the analytical solution is derived in this paper. Moreover, the shrinkage function of modified exponential penalty function is proposed. This shrinkage function obtains from simple iterative method, fixed-point algorithm. We demonstrate the proposed methods through simulations with standard one-dimensional signals contaminated by AWGN. The proposed techniques are compared with traditional estimation methods, such as total variation (TV) and wavelet denoising methods. In experimental results, our proposed methods outperform several exiting methods both visual quality and in terms of root mean square error (RMSE). In fact, the proposed methods can better preserve the feature of noise-free signal than the compared methods. The denoised signals produced by the proposed methods are less smooth than the denoised signals produced by the compared methods.

Electrical Resistance Tomography Image Reconstruction With Densely Connected Convolutional Neural Network

Electrical resistance tomography (ERT) is an effective visualization and analysis tool for multiphase flow process through an array of boundary electrodes. However, the existing deep learning network for the image reconstruction of ERT with the sparse information flow and gradient flow is not well trained, and the accuracy of reconstructed images cannot meet the increasing demands. In order to solve this problem and improve the accuracy of the reconstructed image, a new imaging algorithm that consists of initial imaging module, feature extraction module, image reconstruction module, and residual module is proposed based on dense connections. Four dense blocks with dense connections are adopted to fully fuse the features between feature extraction module and image reconstruction module, which mitigate the problems of information vanishing of the higher layer and gradient vanishing of the lower layer. From the perspective of information flow and gradient flow, why dense connections have a good performance of promoting a deep network training is explained, which provides a theoretical support for the application and development of dense connections. Compared with the Tikhonov regularization method, total variation method, and the networks without dense connections using extensive numerical simulation and experiment results, the imaging accuracy of the new proposed algorithm is improved. Besides, for the ERT cases considered here, the spatial resolution of V Dense Net (VD-Net) (3.12%–4%) is better than that of the traditional image reconstruction (10%–15%).

Noninvasive electrocardiographic imaging with low-rank and non-local total variation regularization

Abstract The reconstruction of epicardial and endocardial extracellular potentials (EEP) by noninvasive methods has become a significant topic in cardiac electrophysiology over recent years. It is of great importance for the diagnosis of arrhythmia and for guidance of radiofrequency ablation, based on the difference in potentials between different locations on the heart's surface. In this study, we propose a non-local regularization of total variation (TV) in a low-rank (LR) and sparse decomposition framework, suitable for the rank-deficient problem of EEP reconstruction. LR and sparse decomposition can be utilized to extract the spatial-temporal information resulting from the sparse properties of EEP data, and the non-local similarities in the LR part can be a constraint for a non-local total variation regularization. The proposed method is implemented in simulated myocardial infarction (MI), interventional, and clinical premature ventricular contraction (PVC) experiments to verify its feasibility and reliability. Compared with the existing LR and TV methods, the proposed method performs better at potential reconstruction as well as PVC localization, particularly in the boundary of the MI region, while the results of this method are also consistent with those of invasive measurements using an EnSite 3000 system in the clinical experiment.

Application of dictionary learning to denoise LIGO’s blip noise transients

Data streams of gravitational-wave detectors are polluted by transient noise features, or "glitches", of instrumental and environmental origin. In this work, we investigate the use of total-variation methods and learned dictionaries to mitigate the effect of those transients in the data. We focus on a specific type of transient, "blip" glitches, as this is the most common type of glitch present in the LIGO detectors and their waveforms are easy to identify. We randomly select 80 blip glitches scattered in the data from advanced LIGO's O1 run, as provided by the citizen-science project Gravity Spy. Our results show that dictionary-learning methods are a valid approach to model and subtract most of the glitch contribution in all cases analyzed, particularly at frequencies below $\sim 1$ kHz. The high-frequency component of the glitch is best removed when a combination of dictionaries with different atom length is employed. As a further example, we apply our approach to the glitch visible in the LIGO-Livingston data around the time of merger of binary neutron star signal GW170817, finding satisfactory results. This paper is the first step in our ongoing program to automatically classify and subtract all families of gravitational-wave glitches employing variational methods.

Adaptive Weighted High Frequency Iterative Algorithm for Fractional-Order Total Variation with Nonlocal Regularization for Image Reconstruction

We propose an adaptive weighted high frequency iterative algorithm for a fractional-order total variation (FrTV) approach with nonlocal regularization to alleviate image deterioration and to eliminate staircase artifacts, which result from the total variation (TV) method. The high frequency gradients are reweighted in iterations adaptively when we decompose the image into high and low frequency components using the pre-processing technique. The nonlocal regularization is introduced into our method based on nonlocal means (NLM) filtering, which contains prior image structural information to suppress staircase artifacts. An alternating direction multiplier method (ADMM) is used to solve the problem combining reweighted FrTV and nonlocal regularization. Experimental results show that both the peak signal-to-noise ratios (PSNR) and structural similarity index (SSIM) of reconstructed images are higher than those achieved by the other four methods at various sampling ratios less than 25%. At 5% sampling ratios, the gains of PSNR and SSIM are up to 1.63 dB and 0.0114 from ten images compared with reweighted total variation with nuclear norm regularization (RTV-NNR). The improved approach preserves more texture details and has better visual effects, especially at low sampling ratios, at the cost of taking more time.

Nonlocal Total Variation Using the First and Second Order Derivatives and Its Application to CT Image Reconstruction.

We propose a new class of nonlocal Total Variation (TV), in which the first derivative and the second derivative are mixed. Since most existing TV considers only the first-order derivative, it suffers from problems such as staircase artifacts and loss in smooth intensity changes for textures and low-contrast objects, which is a major limitation in improving image quality. The proposed nonlocal TV combines the first and second order derivatives to preserve smooth intensity changes well. Furthermore, to accelerate the iterative algorithm to minimize the cost function using the proposed nonlocal TV, we propose a proximal splitting based on Passty’s framework. We demonstrate that the proposed nonlocal TV method achieves adequate image quality both in sparse-view CT and low-dose CT, through simulation studies using a brain CT image with a very narrow contrast range for which it is rather difficult to preserve smooth intensity changes.

Hyperspectral Image Restoration Using Adaptive Anisotropy Total Variation and Nuclear Norms

Random Gaussian noise and striping artifacts are common phenomena in hyperspectral images (HSI). In this article, an effective restoration method is proposed to simultaneously remove Gaussian noise and stripes by merging a denoising and a destriping submodel. A denoising submodel performs a multiband denoising, i.e., Gaussian noise removal, considering Gaussian noise variations between different bands, to restore the striped HSI from the corrupted image, in which the striped HSI is constrained by a weighted nuclear norm. For the destriping submodel, we propose an adaptive anisotropy total variation method to adaptively smoothen the striped HSI, and we apply, for the first time, the truncated nuclear norm to constrain the rank of the stripes to 1. After merging the above two submodels, an ultimate image restoration model is obtained for both denoising and destriping. To solve the obtained optimization problem, the alternating direction method of multipliers (ADMM) is carefully schemed to perform an alternative and mutually constrained execution of denoising and destriping. Experiments on both synthetic and real data demonstrate the effectiveness and superiority of the proposed approach.

Magnetic Induction Tomography Spectroscopy for Structural and Functional Characterization in Metallic Materials.

Magnetic induction tomography (MIT) is a powerful imaging system for monitoring the state of metallic materials. Tomographic methods enable automatic inspection of metallic samples making use of multi-sensor measurements and data processing of eddy current-based sensing from mutual inductances. This paper investigates a multi-frequency MIT using both amplitude and phase data. The image reconstruction algorithm is based on a novel spectrally-correlative total variation method allowing an efficient and all-in-one spectral reconstruction. Additionally, the paper shows the rate of change in spectral images with respect to the excitation frequencies. Using both spectral maps and their spectral derivative maps, one can derive key structural and functional information regarding the material under test. This includes their type, size, number, existence of voids and cracks. Spectral maps can also give functional information, such as mechanical strains and their thermal conditions and composition.

Texture preservation and speckle reduction in poor optical coherence tomography using the convolutional neural network.

For a poor quality optical coherence tomography (OCT) image, quality enhancement is limited to speckle residue and edge blur as well as texture loss, especially at the background region near edges. To solve this problem, in this paper we propose a de-speckling method based on the convolutional neural network (CNN). In the proposed method, we use a deep nonlinear CNN mapping model in the serial architecture, here named as OCTNet. Our OCTNet in the proposed method can fully utilize the deep information on speckles and edges as well as fine textures of an original OCT image. And also we construct an available pertinent dataset by combining three existing methods to train the model. With the proposed method, we can accurately get the speckle noise from an original OCT image. We test our method on four experimental human retinal OCT images and also compare it with three state-of-the-art methods, including the adaptive complex diffusion (ACD) method and the curvelet shrinkage (Curvelet) method as well as the shearlet-based total variation (STV) method. The performance of these methods is quantitatively evaluated in terms of image distinguishability, contrast, smoothness and edge sharpness, and also qualitatively analyzed at aspects of speckle reduction, texture protection and edge preservation. The experimental results show that our OCTNet can reduce the speckle noise and protect the structural information as well as preserve the edge features effectively and simultaneously, even where the background region near edges. And also our OCTNet has full advantages on excellent generalization, adaptiveness, robust and batch performance. These advantages make our method be suitable to process a great mass of different images rapidly without any parameter fine-turning under a time-constrained real-time situation.

Consensus optimization of total variation–based reverse time migration

A common method of imaging with seismic data is reverse time migration (RTM) in which recorded data are back-propagated to form an image of the subsurface. Least-squares RTM (LSRTM) extends this method to an iterative method that minimizes a data misfit term. An appropriate regularization term such as a sparsity-promoting functional (e.g., total variation (TV)) is required to stabilize the LSRTM solution. In this paper, in order to efficiently solve such regularized LSRTM via distributed optimization algorithms, we first reformulate the problem into a consensus form. The alternating direction method of multipliers (ADMM) allows us to split the problem into separate subproblems, resulting to a two-step iteration which efficiently solves the original problem. The most time-consuming step is due to LSRTM of the data associated with each source, separately, which is carried out in parallel via a set of workers. The resulting local images are sent to a master that is responsible for generating (via a proximal mapping) a unique global image that is close to the mean of local images while minimizing the regularizing function. This loop gives a general framework for the parallel computation of regularized LSRTM and allows different regularizing functions to be employed by using appropriate proximal operators. We demonstrate the performance of the proposed algorithm with a set of numerical examples. The results show that TV methods can generate more accurate images compared with the l1-norm sparse LSRTM, l2-norm LSRTM, and traditional RTM. Furthermore, the new algorithm shows improvements over the traditional formulations due to the consensus parallelization.

A Total Variation Denoising Method Based on Median Filter and Phase Consistency

The total variation method is widely used in image noise suppression. However, this method is easy to cause the loss of image details, and it is also sensitive to parameters such as iteration time. In this work, the total variation method has been modified using a diffusion rate adjuster based on the phase congruency and a fusion filter of median filter and phase consistency boundary, which is called the MPC-TV method. Experimental results indicate that MPC-TV method is effective in noise suppression, especially for the removing of speckle noise, and it can also improve the robustness of iteration time of TV method on noise with different variance.

Simultaneous denoising and enhancement for X-ray angiograms by employing spatial-frequency filter

X-ray angiograms, which suffer from low-contrast and noise, need to be improved by both the image enhancement and denoising techniques. However, the goals of these two tasks usually conflict, which makes it difficult to efficiently combine the enhancement and denoising in one scheme. To deal with this problem, we propose a novel spatial-frequency filtering (SFF) scheme for high-contrast X-ray cardiovascular angiogram image restoration. The proposed scheme includes the following four parts: (1) low-quality X-ray angiogram images are decomposed into the base layer with strong structures and redundant parts by employing a relative total variation (RTV) method. (2) the base layer is enhanced by a proposed improved histogram equalization (IHE) method. (3) The detail layer with weak structures is extracted by using a short-time Fourier transform (STFT) and is further enhanced by a proposed adaptive correction parameter. (4) The improved image is the combination of results obtained by the base layer and the detail layer. Experimental results in both qualification and quantitation validate that the proposed method performs better than the state-of-the-arts on contrast enhancement, structure preservation, and noise reduction.

Low-Dose X-ray CT Reconstruction Algorithm Using Shearlet Sparse Regularization

Computed tomography (CT) scans produce ionizing radiation in the body, and high-dose CT scans may increase the risk of cancer. Therefore, reducing the CT radiation dose is particularly important in clinical diagnosis, which is achieved mainly by reducing projection views and tube current. However, the projection data are incomplete in the case of sparse views, which may cause stripe artifacts in the image reconstructed by the filtered back projection (FBP) algorithm, thereby losing the details of the image. Low current intensity also increases the noise of the projection data, degrading the quality of the reconstructed image. This study aimed to use the alternating direction method of multipliers (ADMM) to address the shearlet-based sparse regularization problem, which is subsequently referred to as ADMM-shearlet method. The low-dose projection data were simulated by adding Gaussian noise with zero mean to high-dose projection data. Then FBP, simultaneous algebraic reconstruction technique, total variation, and ADMM-shearlet methods were used to reconstruct images. Normalized mean square error, peak signal-to-noise ratio, and universal quality index were used to evaluate the performance of different reconstruction algorithms. Compared with the traditional reconstruction algorithms, the ADMM-shearlet algorithm performed well in suppressing the noise due to the low dose while maintaining the image details.

Full-Spectrum-Knowledge-Aware Tensor Model for Energy-Resolved CT Iterative Reconstruction.

Energy-resolved computed tomography (ErCT) with a photon counting detector concurrently produces multiple CT images corresponding to different photon energy ranges. It has the potential to generate energy-dependent images with improved contrast-to-noise ratio and sufficient material-specific information. Since the number of detected photons in one energy bin in ErCT is smaller than that in conventional energy-integrating CT (EiCT), ErCT images are inherently more noisy than EiCT images, which leads to increased noise and bias in the subsequent material estimation. In this work, we first deeply analyze the intrinsic tensor properties of two-dimensional (2D) ErCT images acquired in different energy bins and then present a F ull- S pectrum-knowledge-aware Tensor analysis and processing (FSTensor) method for ErCT reconstruction to suppress noise-induced artifacts to obtain high-quality ErCT images and high-accuracy material images. The presented method is based on three considerations: (1) 2D ErCT images obtained in different energy bins can be treated as a 3-order tensor with three modes, i.e., width, height and energy bin, and a rich global correlation exists among the three modes, which can be characterized by tensor decomposition. (2) There is a locally piecewise smooth property in the 3-order ErCT images, and it can be captured by a tensor total variation regularization. (3) The images from the full spectrum are much better than the ErCT images with respect to noise variance and structural details and serve as external information to improve the reconstruction performance. We then develop an alternating direction method of multipliers algorithm to numerically solve the presented FSTensor method. We further utilize a genetic algorithm to tackle the parameter selection in ErCT reconstruction, instead of manually determining parameters. Simulation, preclinical and synthesized clinical ErCT results demonstrate that the presented FSTensor method leads to significant improvements over the filtered back-projection, robust principal component analysis, tensor-based dictionary learning and low-rank tensor decomposition with spatial-temporal total variation methods.

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).

Seafloor elastic-parameter estimation based on a regularized AVO inversion method

Seafloor properties play important roles in both the civilian and the military domains. In this paper, we propose to use a regularization-based AVO (amplitude variation with offset) inversion method to estimate the elastic parameters of seafloor sediment along a seismic survey line. Conventional AVO research focuses mainly on the strata beneath seafloor sediment; accordingly, the AVO response of the interface between the seawater and seafloor sediment is typically ignored. Several studies on the estimation of seafloor elastic parameters have been published based on AVO theory. However, these earlier studies only consider one CRP (common reflection point) gather in the inversion process. When dealing with a long seismic survey with many CRPs, earlier methods involve clear random inversion errors along the survey line. To suppress the inversion errors, we try to regularize each of the elastic parameters along the seismic survey by a total-variation method. Synthetic tests demonstrate that the regularization-based method can effectively suppress the inversion errors along the seismic survey and yield much cleaner inversion results. Finally, we apply the regularization-based method to a real two-dimensional marine seismic survey and obtain satisfactory results, which further proves the efficacy of the method.