Total Generalized Variation Regularization Research Articles

Sparse-View Spectral CT Reconstruction Based on Tensor Decomposition and Total Generalized Variation

Spectral computed tomography (CT)-reconstructed images often exhibit severe noise and artifacts, which compromise the practical application of spectral CT imaging technology. Methods that use tensor dictionary learning (TDL) have shown superior performance, but it is difficult to obtain a high-quality pre-trained global tensor dictionary in practice. In order to resolve this problem, this paper develops an algorithm called tensor decomposition with total generalized variation (TGV) for sparse-view spectral CT reconstruction. In the process of constructing tensor volumes, the proposed algorithm utilizes the non-local similarity feature of images to construct fourth-order tensor volumes and uses Canonical Polyadic (CP) tensor decomposition instead of pre-trained tensor dictionaries to further explore the inter-channel correlation of images. Simultaneously, introducing the TGV regularization term to characterize spatial sparsity features, the use of higher-order derivatives can better adapt to different image structures and noise levels. The proposed objective minimization model has been addressed using the split-Bregman algorithm. To assess the performance of the proposed algorithm, several numerical simulations and actual preclinical mice are studied. The final results demonstrate that the proposed algorithm has an enormous improvement in the quality of spectral CT images when compared to several existing competing algorithms.

Improved quantification in CEST-MRI by joint spatial total generalized variation.

In this work, the use of joint Total Generalized Variation (TGV) regularization to improve Multipool-Lorentzian fitting of chemical exchange saturation transfer (CEST) Spectra in terms of stability and parameter signal-to-noise ratio (SNR) was investigated. The joint TGV term was integrated into the nonlinear parameter fitting problem. To increase convergence and weight the gradients, preconditioning using a voxel-wise singular value decomposition was applied to the problem, which was then solved using the iteratively regularized Gauss-Newton method combined with a Primal-Dual splitting algorithm. The TGV method was evaluated on simulated numerical phantoms, 3T phantom data and 7T in vivo data with respect to systematic errors and robustness. Three reference methods were also implemented: The standard nonlinear fitting, a method using a nonlocal-means filter for denoising and the pyramid scheme, which uses downsampled images to acquire accurate start values. The proposed regularized fitting method showed significantly improved robustness (compared to the reference methods). In testing, over a range of SNR values the TGV fit outperformed the other methods and showed accurate results even for large amounts of added noise. Parameter values found were closer or comparable to the ground truth. For in vivo datasets, the added regularization increased the parameter map SNR and prevented instabilities. The proposed fitting method using TGV regularization leads to improved results over a range of different data-sets and noise levels. Furthermore, it can be applied to all Z-spectrum data, with different amounts of pools, where the improved SNR and stability can increase diagnostic confidence.

Evaluation of low-dose computed tomography reconstruction using spatial-radon domain total generalized variation regularization

The x-ray radiation dose in computed tomography (CT) examination has been a major concern for patients. Lowing the tube current and exposure time in data acquisition is a straightforward and cost-effective strategy to reduce the x-ray radiation dose. However, this will inevitably increase the noise fluctuations in measured projection data, and the corresponding CT image quality will be severely degraded if noise suppression is not performed during image reconstruction. To reconstruct high-quality low-dose CT image, we present a spatial-radon domain total generalized variation (SRDTGV) regularization for statistical iterative reconstruction based on penalized weighted least-squares (PWLS) principle, which is called PWLS-SRDTGV for simplicity. The presented PWLS-SRDTGV model can simultaneously reconstruct high-quality CT image in space domain and its corresponding projection in radon domain. An efficient split Bregman algorithm was applied to minimize the cost function of the proposed reconstruction model. Qualitative and quantitative studies were performed to evaluate the effectiveness of the PWLS-SRDTGV image reconstruction algorithm using a digital 3D XCAT phantom and an anthropomorphic torso phantom. The experimental results demonstrate that PWLS-SRDTGV algorithm achieves notable gains in noise reduction, streak artifact suppression, and edge preservation compared with competing reconstruction approaches.

A non-local total generalized variation regularization reconstruction method for sparse-view x-ray CT

Sparse-view x-ray computed tomography (CT) reconstruction, employing total generalized variation (TGV), effectively mitigates the stepwise artifacts associated with total variation (TV) regularization while preserving structural features within transitional regions of the reconstructed image. Despite TGV surpassing TV in reconstruction quality, it neglects the non-local self-similarity prior, recognized for its efficacy in restoring details during CT reconstruction. This study introduces a non-local TGV (NLTGV) to address the limitation of TGV regularization method. Specifically, we propose an NLTGV-regularized method for sparse-view CT reconstruction, utilizing non-local high-order derivative information to maintain image features and non-local self-similarity for detail recovery. Owing to the non-differentiability of the NLTGV regularized, we employ an alternating direction method of multipliers optimization method, facilitating an efficient solution by decomposing the reconstruction model into sub-problems. The proposed method undergoes evaluation using both simulated and real-world projection data. Simulation and experimental results demonstrate the efficacy of the proposed approach in enhancing the quality of reconstructed images compared to other competitive variational reconstruction methods. In conclusion, the simultaneous incorporation of sparsity priors of high-order TV and non-local similarity proves advantageous for structural detail recovery in sparse-view CT reconstruction.

Multiplicative noise removal using complementary total generalized variation and nonlocal low rank regularization

The problem of multiplicative image denoising is a typical ill-posed inverse problem which has been extensively studied in the past few decades and is still an open problem. Exploiting appropriate image priors is critical to solve ill-posed inverse image restoration problem. The existing total generalized variation (TGV) based approaches have achieved significant performance in removing the multiplicative noise. However, these TGV regularized approaches only consider the local smoothness prior of the potential clean image, which limits their ability for image restoration. Other than the local smoothness property, the nonlocal self-similarity prior of images can also be exploited to offer helpful remedy for better image reconstruction. In this paper, to simultaneously make use of both the local smoothness prior and nonlocal self-similarity prior, we integrate the TGV regularizer with the nonlocal low rank regularizer to propose a novel variational model for multiplicative image denoising. Specifically, in the objective function, the nonlocal low rank regularization is expressed by the celebrated weighted nuclear norm minimization (WNNM) which has recently been applied to image denoising and presents state-of-art result. Then, we apply the split Bregman algorithm to efficiently solve the resulting model. Numerical experiments demonstrate that the proposed method achieve superior performance over the competing methods, including the state-of-the-art SAR-BM3D method.

Non-convex TGV regularized [formula omitted]-norm fidelity model for impulse noise removal

Aiming at preserving the structural features of images from degradations contaminated by impulse noise, this article proposes a novel strategy that inherits the characteristics of ℓ0-norm data fidelity and non-convex total generalized variation regularizer. The ℓ0-norm fidelity is more conducive to removing impulse noise, while the non-convex total generalized variation penalty term is both good at eliminating the staircase artifacts and maintaining neat contours. In addition, a modified alternating minimization algorithm is constructed to optimize the solution of the developed model, which merges the popular iteratively reweighted ℓ1 algorithm and primal-dual approach. Meanwhile, the convergence property of the designed algorithm is briefly presented. Finally, in the experimental part, exhaustive simulation experiments are carried out to demonstrate the effectiveness of the introduced scheme for removing impulse noise. As observed in the numerical results, our scheme possesses obvious superiorities in both visual effects and quantitative comparisons, in contrast to some existing popular image restoration methods.

Impulse noise removal by using a nonconvex TGV regularizer and nonconvex fidelity

With the aim of acquiring high-quality restored images, this paper derives a novel nonconvex variational model for the removal of impulse noise. The investigated solver integrates the superiorities of nonconvex second-order total generalized variation (TGV) regularization and nonconvex data fidelity. More precisely, the usage of a nonconvex TGV regularizer helps to eliminate the staircase artifacts and simultaneously preserve edge details. Nonconvex fidelity, which enhances sparsity, is adopted to effectively detect impulse noise. Computationally, incorporating the popular iteratively reweighted ℓ1 algorithm and variable splitting method, we propose to adopt an efficient alternating direction method of multipliers for the purpose of rapidly resolving the designed optimization problem. Additionally, simulation examples have been executed to compare our method with several state-of-the-art techniques. Experimental results and quantitative comparisons confirm that the developed strategy outperforms other competitors for suppressing impulse noise (even high density) in both visual outcomes and recovery accuracy.

Mesh Total Generalized Variation for Denoising.

Recent studies have shown that the Total Generalized Variation (TGV) is highly effective in preserving sharp features as well as smooth transition variations for image processing tasks. However, currently there is no existing work that is suitable for applying TGV to 3D data, in particular, triangular meshes. In this article, we develop a novel framework for discretizing second-order TGV on triangular meshes. Further, we propose a TGV-based variational method for the denoising of face normal fields on triangular meshes. The TGV regularizer in our method is composed of a first-order term and a second-order term, which are automatically balanced. The first-order term allows our TGV regularizer to locate and preserve sharp features, while the second-order term allows our regularizer to recognize and recover smoothly curved regions. To solve the optimization problem, we introduce an efficient iterative algorithm based on variable-splitting and augmented Lagrangian method. Extensive results and comparisons on synthetic and real scanning data validate that the proposed method outperforms the state-of-the-art visually and numerically.

A Derivative Fidelity-Based Total Generalized Variation Method for Image Restoration

Image edge is the most indicative feature that forms a significant role in image analysis and image understanding, but edge-detail preservation is a difficult task in image restoration due to noise and blur during imaging. The balance between edge preservation and noise removal has always been a difficult problem in image restoration. This paper proposes a derivative fidelity-based total generalized variation method (D-TGV) to improve this balance. First, an objective function model that highlights the ability to maintain details is proposed for the image restoration problem, which is combined with a fidelity term in derivative space and a total generalized variation regularization term. This is designed to achieve the advantage of preserving details in derivative space and eliminate the staircase effect caused by traditional total variation. Second, the alternating direction method of the multipliers (ADMM) is used to solve the model equations by decomposing the original, highly complex model into several simple sub-problems to attain rapid convergence. Finally, a series of experiments conducted on standard grayscale images showed that the proposed method exhibited a good balance between detail preservation and denoising but also reached completion with the fewest iterations compared with the currently established methods.

A Mismatch Correction Method for Electrode Offset in Electrical Impedance Tomography

Electrical impedance tomography (EIT) has attracted research interests in imaging of brain function, detection of breast cancer, monitoring of lung function and other medical fields. For EIT, its image reconstruction is largely affected by variation of electrode placement arranged around the detected object. Note that electrode offset is unavoidable in some medical applications, leading to low-quality reconstructed images or even failure of reconstruction. In this paper, a novel method is proposed to improve reconstruction quality in case of electrode offset. By identifying which electrode moves, mismatched boundary voltage caused by movement of electrode is corrected. After mismatch correction, image reconstruction is conducted with total generalized variation regularization method. To demonstrate performance of the proposed method in suppressing impact of electrode offset on reconstruction, extensive work is carried out. Mismatch corrections for four models under a series of counterclockwise or clockwise movements of a certain electrode are studied. The reconstruction results show that the proposed method can effectively improve the image quality under electrode offset. Even in presence of larger electrode offset and noise, reconstructed images have been greatly improved. Quantitative evaluation is also conducted by calculating relative blur radius (RBR). It is shown that RBR values are much closer to 1 compared with results under mismatch conditions. Image reconstruction when two electrodes move is also investigated which further validate the performance of the proposed method. This method is potential for mismatch correction in some medical applications.

3D nanoscale elemental mapping of precipitates in steel: Evaluation of analytical electron tomography and comparison to atom probe tomography

To identify different types of precipitates and their composition, analytical electron tomography analysis on a needle-shaped sample was performed. Three-dimensional chemical maps from each element in a high alloyed steel are acquired and the resulting elemental maps are jointly reconstructed. Since analytical electron tomography data suffers from noise, total generalized variation regularization is used to improve the reconstruction quality compared to conventional reconstruction techniques. We analyse the influence of regularization parameters on these reconstructions in terms of elemental quantification, and compare these results to atom probe tomography measurements done on another sample of the same material.

Generalized Hessian-Schatten Norm Regularization for Image Reconstruction

Regularization plays a crucial role in reliably utilizing imaging systems for scientific and medical investigations. It helps to stabilize the process of computationally undoing any degradation caused by physical limitations of the imaging process. In the past decades, total variation regularization played a dominant role in the literature. Two forms of total variation regularizations, namely the first-order and the second-order total variation (TV-1 and TV-2) have been widely used. TV-1 has a disadvantage: it reconstructs images in the form of piece-wise constants when the noise and/or under-sampling is severe, while TV-2 reconstructs natural-looking images under such scenarios. On the other hand, TV-1 can recover sharp jumps better than TV-2. Two forms of generalizations, namely Hessian-Schatten norm (HSN) regularization, and total generalized variation (TGV) regularization, have been proposed and have become significant developments in the area of regularization for imaging inverse problems owing to their performance. While the strength of TGV is that it can combine the advantages of TV-1 and TV-2, HSN has better structure-preserving property. Here, we develop a novel regularization for image recovery that combines the strengths of TGV and HSN. We achieve this by restricting the maximization space in the dual form of HSN in the same way that TGV is obtained from TV-2. We call the new regularization the generalized Hessian-Schatten norm regularization (GHSN). We develop a novel computational method for image reconstruction using the new form of regularization based on the well-known framework called the alternating direction method of multipliers (ADMM). We demonstrate the strength of the GHSN using some reconstruction examples.

A two-stage image reconstruction strategy for electrical impedance tomography

Electrical impedance tomography (EIT) is a potential and promising tomographic technique. Based on a reconstruction strategy, conductivity distribution can be imaged by processing boundary measurements. It should be noticed that the process of image reconstruction involves the solution of a nonlinear ill-posed inverse problem. To tackle this problem, a novel two-stage image reconstruction strategy is proposed in this work. It combines the advantages of total generalized variation regularization method and tight wavelet approach. The solution of the proposed method is acquired by employing alternating minimization algorithm and spilt Bregman algorithm. In the numerical simulation, reconstruction of five models is studied. Aside from the visual observation, we have also validated the proposed method with quantitative comparison. Meanwhile, the impact of noise on the reconstruction is considered. Furthermore, the proposed method is evaluated by phantom experimental data. The simulation and experimental results have demonstrated the superior performance of the proposed method in visualizing conductivity distribution.

Adaptive regularization parameter for nonconvex TGV based image restoration

Overcoming the staircase effect and simultaneously preserving edge details is an important and challenging issue in image processing. To this aim, this paper investigates the nonconvex total generalized variation regularization model for image restoration. Numerically, a highly efficient alternating direction method of multipliers is constructed to deal with the optimization problem in detail, which closely incorporates the superiorities of iteratively reweighted ℓ1 algorithm and variable splitting technique. The proposed strategy can efficiently achieve image restoration and adaptive parameter estimation by applying Morozov’s discrepancy principle. Furthermore, the convergence property of our novel algorithm with the variable regularization parameter is presented. Finally, numerical simulations concertedly illustrate that our approach outperforms several state-of-the-art regularization models, in terms of reconstruction accuracy and edge-preserving ability.

Joint multi-field T1 quantification for fast field-cycling MRI.

PurposeRecent developments in hardware design enable the use of fast field‐cycling (FFC) techniques in MRI to exploit the different relaxation rates at very low field strength, achieving novel contrast. The method opens new avenues for in vivo characterizations of pathologies but at the expense of longer acquisition times. To mitigate this, we propose a model‐based reconstruction method that fully exploits the high information redundancy offered by FFC methods.MethodsThe proposed model‐based approach uses joint spatial information from all fields by means of a Frobenius ‐ total generalized variation regularization. The algorithm was tested on brain stroke images, both simulated and acquired from FFC patients scans using an FFC spin echo sequences. The results are compared to three non‐linear least squares fits with progressively increasing complexity.ResultsThe proposed method shows excellent abilities to remove noise while maintaining sharp image features with large signal‐to‐noise ratio gains at low‐field images, clearly outperforming the reference approach. Especially patient data show huge improvements in visual appearance over all fields.ConclusionThe proposed reconstruction technique largely improves FFC image quality, further pushing this new technology toward clinical standards.

Non-linear fitting with joint spatial regularization in arterial spin labeling.

Multi-Delay single-shot arterial spin labeling (ASL) imaging provides accurate cerebral blood flow (CBF) and, in addition, arterial transit time (ATT) maps but the inherent low SNR can be challenging. Especially standard fitting using non-linear least squares often fails in regions with poor SNR, resulting in noisy estimates of the quantitative maps. State-of-the-art fitting techniques improve the SNR by incorporating prior knowledge in the estimation process which typically leads to spatial blurring. To this end, we propose a new estimation method with a joint spatial total generalized variation regularization on CBF and ATT. This joint regularization approach utilizes shared spatial features across maps to enhance sharpness and simultaneously improves noise suppression in the final estimates. The proposed method is evaluated at three levels, first on synthetic phantom data including pathologies, followed by in vivo acquisitions of healthy volunteers, and finally on patient data following an ischemic stroke. The quantitative estimates are compared to two reference methods, non-linear least squares fitting and a state-of-the-art ASL quantification algorithm based on Bayesian inference. The proposed joint regularization approach outperforms the reference implementations, substantially increasing the SNR in CBF and ATT while maintaining sharpness and quantitative accuracy in the estimates.

Second-Order Total Generalized Variation Regularization for Pansharpening

Pansharpening can be regarded as an inverse problem, where a high-resolution multispectral (MS) image is estimated given a low-resolution MS image and a panchromatic (Pan) image. Considering the effectiveness of total generalized variation (TGV) to regularize ill-posed inverse problems, this letter proposes a variational model in accordance with the observation model based on the satellite imaging system and the second-order TGV. Furthermore, a primal–dual algorithm is adopted to resolve the variational model by splitting it into a sequence of simpler sub-problems, which leads to a more efficient algorithm compared to the other variational methods. In addition, the exploitation of TGV in the variational model allows the presence of the very fine details of the Pan image in the sharpened MS image whereas it is beyond the capabilities of the total variation (TV)-based models. The proposed algorithm is compared with some recent classical and variational pansharpening methods by experiments on two data sets at full and reduced resolution.

MR-Based Electrical Conductivity Imaging Using Second-Order Total Generalized Variation Regularization

Electrical properties provide essential information for cancer detection and specific absorption rate (SAR) estimation. Magnetic resonance electrical properties tomography (MREPT) is an approach to retrieve the distribution of electrical properties. The conventional method suffers from the locally homogeneous assumption and amplification of noise. In this study, a novel approach was introduced to improve the accuracy and the noise robustness of conductivity imaging. The proposed approach reformulated the central equation of the gradient-based method to avoid the calculation of the Laplacian operator. The equation was regularized using the second-order total generalized variation, which formulates an objective function. The optimization problem was solved by the alternating direction method of multipliers (ADMM) method. The proposed method was validated by the simulation data of the cylindrical phantom and Ella head model, and the performance was compared with existing methods. The results demonstrated that the proposed method reconstructed an accurate conductivity image and alleviated the noise effects. Furthermore, phantom and healthy volunteer experiments were implemented at a 3 Tesla (T) magnetic resonance imaging (MRI) scanner. The results suggested that the developed method can provide solutions for improved conductivity reconstruction and show potential for clinical application.

Total generalized variation-based Retinex image decomposition

Human visual system (HVS) can perceive color under varying illumination conditions, and Retinex theory is precisely aimed to simulate and explain how the HVS perceives reflectance regardless of different illumination conditions. In this paper, we introduce a reflectance and illumination decomposition model for the Retinex problem via total generalized variation regularization and $$H^{1}$$ decomposition. The total generalized variation regularization ameliorates the staircasing artifacts that appear in the reflectance component of existing total variation-based models and $$H^{1}$$ norm guarantees smoother illumination. We analyze the existence and uniqueness of the proposed model and employ an alternating minimization scheme based on split Bregman iteration. We present numerous numerical experiments on both grayscale and color images to make comparisons with several state-of-the-art methods and demonstrate that our method is comparable both quantitatively and qualitatively.

Variational regularization of inverse problems for manifold-valued data

Abstract In this paper, we consider the variational regularization of manifold-valued data in the inverse problems setting. In particular, we consider total variation and total generalized variation regularization for manifold-valued data with indirect measurement operators. We provide results on the well-posedness and present algorithms for a numerical realization of these models in the manifold setup. Further, we provide experimental results for synthetic and real data to show the potential of the proposed schemes for applications.