Total Generalized Variation Research Articles

Dyadic Partition-Based Training Schemes for TV/TGV Denoising.

Due to their ability to handle discontinuous images while having a well-understood behavior, regularizations with total variation (TV) and total generalized variation (TGV) are some of the best-known methods in image denoising. However, like other variational models including a fidelity term, they crucially depend on the choice of their tuning parameters. A remedy is to choose these automatically through multilevel approaches, for example by optimizing performance on noisy/clean image pairs. In this work, we consider such methods with space-dependent parameters which are piecewise constant on dyadic grids, with the grid itself being part of the minimization. We prove existence of minimizers for fixed discontinuous parameters under mild assumptions on the data, which lead to existence of finite optimal partitions. We further establish that these assumptions are equivalent to the commonly used box constraints on the parameters. On the numerical side, we consider a simple subdivision scheme for optimal partitions built on top of any other bilevel optimization method for scalar parameters, and demonstrate its improved performance on some representative test images when compared with constant optimized parameters.

Spatially adaptive oscillation total generalized variation for image restoration with structured textures

Cartoon and texture are the main components of an image, and decomposing them has gained much attention in various image restoration tasks. In this paper, we propose a novel infimal convolution type model based on total generalized variation (TGV) and spatially adaptive oscillation TGV to address cartoon-texture restoration problems. The proposed spatially adaptive oscillation TGV regularizer is capable of capturing structured textures with different orientations and frequencies in localized regions. Additionally, we incorporate the second-order tensor TGV to regularize the orientation and frequency parameter. The lower semicontinuity of the new functional is established and the existence of the solutions to the proposed model is analyzed. Furthermore, we discuss suitable discretizations of the proposed model, and introduce the alternating minimization algorithm where each subproblem can be implemented by the primal-dual method. Numerical experiments on image decomposition, denoising and inpainting demonstrate that the proposed model excels in preserving textures and is competitive against existing variational and learning-based models.

Adaptive prior image constrained total generalized variation for low-dose dynamic cerebral perfusion CT reconstruction

BACKGROUND: Dynamic cerebral perfusion CT (DCPCT) can provide valuable insight into cerebral hemodynamics by visualizing changes in blood within the brain. However, the associated high radiation dose of the standard DCPCT scanning protocol has been a great concern for the patient and radiation physics. Minimizing the x-ray exposure to patients has been a major effort in the DCPCT examination. A simple and cost-effective approach to achieve low-dose DCPCT imaging is to lower the x-ray tube current in data acquisition. However, the image quality of low-dose DCPCT will be degraded because of the excessive quantum noise. OBJECTIVE: To obtain high-quality DCPCT images, we present a statistical iterative reconstruction (SIR) algorithm based on penalized weighted least squares (PWLS) using adaptive prior image constrained total generalized variation (APICTGV) regularization (PWLS-APICTGV). METHODS: APICTGV regularization uses the precontrast scanned high-quality CT image as an adaptive structural prior for low-dose PWLS reconstruction. Thus, the image quality of low-dose DCPCT is improved while essential features of targe image are well preserved. An alternating optimization algorithm is developed to solve the cost function of the PWLS-APICTGV reconstruction. RESULTS: PWLS-APICTGV algorithm was evaluated using a digital brain perfusion phantom and patient data. Compared to other competing algorithms, the PWLS-APICTGV algorithm shows better noise reduction and structural details preservation. Furthermore, the PWLS-APICTGV algorithm can generate more accurate cerebral blood flow (CBF) map than that of other reconstruction methods. CONCLUSIONS: PWLS-APICTGV algorithm can significantly suppress noise while preserving the important features of the reconstructed DCPCT image, thus achieving a great improvement in low-dose DCPCT imaging.

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 Mathematical Algorithm for Improving the Medical Image

In this paper, we present a hybrid model based on total generalized variation (TGV) and shearlet with non-quadratic fidelity data terms for blurred images corrupted by impulsive and Poisson noises. Numerical experiments demonstrate that the proposed can reduce the staircase effect while preserving edges and outperform classical TV-based models in the peak signal-to-noise ratio (PSNR).

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.

Photoacoustic image reconstruction with anobjective function using TGV and ESTGV asaregularization functional.

Photoacoustic tomographic imaging is a non-invasive medical diagnostic technology for visualizing biological tissue. However, the inverse problem and noise in photoacoustic signals often cause blurred images. Existing regularization methods struggle with staircasing artifacts and edge preservation. To overcome this, an objective function incorporating total generalized variation (TGV) is proposed. However, it failed with high-density Gaussian noise. To address this, an extended version called edge-guided second-order TGV (ESTGV) is introduced. For sparsification, wavelet transform and discrete cosine transform are introduced, while the fast-composite-splitting algorithm is employed for the inverse problem solution. Experimental validation demonstrates the potential of these approaches.

Combined non-convex second-order total variation with overlapping group sparsity for full waveform inversion

Full waveform inversion (FWI) can provide an accurate velocity model by matching observed and simulated seismograms. Mathematically, FWI is a highly ill-posed inverse problem that the observed data are independent of the model and cannot be inverted accurately. For a stable and reasonable inversion, proper regularization methods have to be taken into account. We propose a novel composite regularization for frequency-domain FWI problem, which uses a non-convex second-order total variation (TV) term and an overlapping group sparse TV (OGS-TV) regularization term. Compared with the conventional TV regularization, our method has better accuracy and robustness, and could effectively make use of the sparsity in the velocity model. Furthermore, the alternating direction multiplier algorithm with the adaptive selection of the penalty parameters is developed to solve this composite constraint problem, which can improve the stability of the FWI process. To illustrate the superior method both visually and quantitatively, we experimentally compare the proposed method with the conventional TV regularized FWI and the second-order total generalized variation regularized FWI and overlapping group sparsity regularized FWI on the well-known geological models.

Sinogram Upsampling via Sub-Riemannian Diffusion with Adaptive Weighting

Computed tomography is a versatile imaging technique used to enable seeing internal structures of objects without opening or destroying them. This is possible through a process called tomographic reconstruction, which reconstructs images from projections of the object that are obtained by penetrating the object with beams of radiation, such as X-rays, from different angles. These projection data are often limited to low-resolution data in terms of projection angles. These limited or subsampled data make it difficult to obtain high-quality reconstruction results. Hence, upsampling projection data is necessary. In this paper, we propose a sinogram upsampling method via the sub-Riemannian diffusion process. We first lift the data into a feature space, and we fill in the missing angle parts by propagating information from the observed data to the missing parts. We observe that the sinogram with limited angle data has high directional dependency, and based on this observation, we suggest an adaptive weighting scheme to keep information propagating toward the missing regions. This adaptive weighting allows for diffusing toward the desired directions. The experimental results show the effectiveness of the proposed method in some scenarios regarding inpainting fine details, when compared to the existing model-based methods, such as Plug-and-Play and total generalized variation.

A Fractional-Order Fidelity-Based Total Generalized Variation Model for Image Deblurring

Image deblurring is a fundamental image processing task, and research for efficient image deblurring methods is still a great challenge. Most of the currently existing methods are focused on TV-based models and regularization term construction; little efforts are paid to model proposal and correlated algorithms for the fidelity term in fractional-order derivative space. In this paper, we propose a novel fractional-order variational model for image deblurring, which can efficiently address three different blur kernels. The objective functional contains a fractional-order gradient fidelity term and a total generalized variation (TGV) regularization term, and it highlights the ability to preserve details and eliminate the staircase effect. To solve the problem efficiently, we provide two numerical algorithms based on the Chambolle-Pock primal-dual method (PD) and the alternating direction method of multipliers (ADMM). A series of experiments show that the proposed method achieves a good balance between detail preservation and deblurring compared with several existing advanced models.

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.

Template-based CT reconstruction with optimal transport and total generalized variation

x-ray computed tomography (CT) has been widely used in clinical diagnosis as a modality of medical imaging. To decrease the radiation dose patients suffering from, sparse-view CT has gained much attention in medical imaging field. In this paper, we propose to design a variational model based on dynamic optimal transportation and total generalized variation for CT reconstruction. This is a joint task involving inverse problem and template registration. The final state image of the optimal transport problem is unknown and needs to be reconstructed through CT inversion, while the given initial state can be regarded as a template to provide some structural information for the final one. Moreover, the existence and stability of minimizers to our proposed model are shown in continuous space. In discretization with the continuity equation, we utilize the well-known staggered grid in fluid mechanics and develop a first-order algorithm based on primal-dual method for numerically solving the proposed model. Finally, numerical experiments for sparse-view CT reconstruction are exhibited to show the performance of our proposed model in recovering images with high quality and structure preservation.

A non–convex total generalized variation model for image denoising

We propose an effective regularization model based on second–order total generalized variation for image restoration with mixed Poisson–Gaussian noise. An efficient alternating minimization algorithm is employed to solve the considered model. Finally, provided numerical results show that our proposed model can preserve more details and get higher image visual quality than recent state-of-the-art 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.

Simultaneous spatial and temporal regularization in low-dose dynamic contrast-enhanced CT cerebral perfusion studies.

To apply total generalized variation (TGV) and its combination with low-rank and sparse decomposition (LRSD) (LTGV) to cerebral perfusion studies using low-dose dynamic contrast-enhanced (DCE) CT and to quantitatively evaluate their performances through comparisons with those without any regularizers and those of total variation (TV) and its combination with LRSD (LTV) using simulation and clinical data. The simulation study used a realistic digital brain phantom. Low-dose DCE-CT images were reconstructed using the regularizers and primal-dual algorithm. Subsequently, cerebral perfusion parameter (CPP) images were generated from them. Thereafter, their quality was evaluated based on the peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM). Further, the accuracy of CPP estimation was evaluated through a linear regression analysis between the CPP values obtained by the above regularizers and those obtained from the noise-free DCE-CT images. In addition, the mean and standard deviation of the CPP were calculated (region analysis). In the clinical study, low-dose DCE-CT images were generated using normal-dose images acquired from a patient, and CPP images were generated from them similar to that in the simulation study. When using LTV and LTGV, both PSNR and SSIM were higher than those of the other methods with increasing regularization parameter values. The results of the linear regression and region analyses demonstrated that TGV generally exhibited the best performance, followed by LTGV, and finally that of TV was significantly different from those of the other regularizers. Despite an overall consistency between the simulation and clinical results, certain inconsistencies appeared owing to the difference in generating low-dose DCE-CT images. The results implied that TGV and LTGV were useful in improving the accuracy of CPP estimation using low-dose DCE-CT. This study provides an improved understanding of the performance of regularizers and is expected to aid in the selection of a suitable regularizer for low-dose DCE-CT perfusion studies.

Total Generalized Variation for Piecewise Constant Functions on Triangular Meshes with Applications in Imaging

We propose a novel discrete concept for the total generalized variation (TGV), which was originally derived to reduce the staircasing effect in classical total variation regularization, in image denoising problems. We describe discrete, second-order TGV for piecewise constant functions on triangular meshes, thus allowing the TGV functional to be applied to more general data structures than pixel images, and in particular in the context of finite element discretizations. Particular attention is given to the description of the kernel of the TGV functional, which, in the continuous setting, consists of linear polynomials. We discuss how to take advantage of this kernel structure using piecewise constant functions on triangular meshes. Numerical experiments include denoising and inpainting problems for images defined on nonstandard grids, including data from a three-dimensional scanner.

Comparison of quantitative susceptibility mapping methods for iron-sensitive susceptibility imaging at 7T: An evaluation in healthy subjects and patients with Huntington's disease

Quantitative susceptibility mapping (QSM) is a promising tool for investigating iron dysregulation in neurodegenerative diseases, including Huntington's disease (HD). Many diverse methods have been proposed to generate accurate and robust QSM images. In this study, we evaluated the performance of different dipole inversion algorithms for iron-sensitive susceptibility imaging at 7T on healthy subjects of a large age range and patients with HD. We compared an iterative least-squares-based method (iLSQR), iterative methods that use regularization, single-step approaches, and deep learning-based techniques. Their performance was evaluated by comparing: (1) deviations from a multiple-orientation QSM reference; (2) visual appearance of QSM maps and the presence of artifacts; (3) susceptibility in subcortical brain regions with age; (4) regional brain susceptibility with published postmortem brain iron quantification; and (5) susceptibility in HD-affected basal ganglia regions between HD subjects and healthy controls. We found that single-step QSM methods with either total variation or total generalized variation constraints (SSTV/SSTGV) and the single-step deep learning method iQSM generally provided the best performance in terms of correlation with iron deposition and were better at differentiating between healthy controls and premanifest HD individuals, while deep learning QSM methods trained with multiple-orientation susceptibility data created QSM maps that were most similar to the multiple orientation reference and with the best visual scores.

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.