Variation Regularization Method Research Articles

A variational regularization method for solving the non-characteristic Cauchy problem in multiple dimensions

A variational regularization method for solving the non-characteristic Cauchy problem in multiple dimensions

Efficient learning methods for large-scale optimal inversion design

In this work, we investigate various approaches that use learning from training data to solve inverse problems, following a bi-level learning approach. We consider a general framework for optimal inversion design, where training data can be used to learn optimal regularization parameters, data fidelity terms, and regularizers, thereby resulting in superior variational regularization methods. In particular, we describe methods to learn optimal $ p $ and $ q $ norms for $ {\rm L}^p-{\rm L}^q $ regularization and methods to learn optimal parameters for regularization matrices defined by covariance kernels. We exploit efficient algorithms based on Krylov projection methods for solving the regularized problems, both at training and validation stages, making these methods well-suited for large-scale problems. Our experiments show that the learned regularization methods perform well even when there is some inexactness in the forward operator, resulting in a mixture of model and measurement error.

Determine source of Cahn–Hilliard equation with time‐fractional derivative by iterative variational regularization method

In this paper, we consider the inverse problem for the time‐fractional two‐dimensional Cahn–Hilliard equation. The ill‐posedness and a conditional stability of the inverse problem are proved. We present a new iterative variational regularization method to solve it. The convergence rates of the regularized solutions under the a priori regularization parameter choice rule and a posteriori regularization parameter choice rule are obtained. Numerical examples illustrate the effectiveness and stability of our proposed method.

VAEs with structured image covariance applied to compressed sensing MRI

Objective.This paper investigates how generative models, trained on ground-truth images, can be used as priors for inverse problems, penalizing reconstructions far from images the generator can produce. The aim is that learned regularization will provide complex data-driven priors to inverse problems while still retaining the control and insight of a variational regularization method. Moreover, unsupervised learning, without paired training data, allows the learned regularizer to remain flexible to changes in the forward problem such as noise level, sampling pattern or coil sensitivities in MRI. Approach. We utilize variational autoencoders that generate not only an image but also a covariance uncertainty matrix for each image. The covariance can model changing uncertainty dependencies caused by structure in the image, such as edges or objects, and provides a new distance metric from the manifold of learned images. Main results. We evaluate these novel generative regularizers on retrospectively sub-sampled real-valued MRI measurements from the fastMRI dataset. We compare our proposed learned regularization against other unlearned regularization approaches and unsupervised and supervised deep learning methods. Significance. Our results show that the proposed method is competitive with other state-of-the-art methods and behaves consistently with changing sampling patterns and noise levels.

Adaptive Total-Variation and Nonconvex Low-Rank Model for Image Denoising

In recent years, image denoising methods based on total variational regularization have attracted extensive attention. However, the traditional total variational regularization method is an approximate solution based on convex method, and does not consider the particularity of the region with rich details. In this paper, the adaptive total-variation and nonconvex low-rank model for image denoising is proposed, which is a hybrid regularization model. First, the image is decomposed into sparse terms and low rank terms, and then the total variational regularization is used to denoise. At the same time, an adaptive coefficient based on gradient is constructed to adaptively judge the flat area and detail texture area, slow down the denoising intensity of detail area, and then play the role of preserving detail information. Finally, by constructing a nonconvex function, the optimal solution of the function is obtained by using the alternating minimization method. This method not only effectively removes the image noise, but also retains the detailed information of the image. The experimental results show the effectiveness of the proposed model, and SNR and SSIM of the denoised image are improved.

Convergence guarantees for coefficient reconstruction in PDEs from boundary measurements by variational and Newton-type methods via range invariance

Abstract A key observation underlying this paper is the fact that the range invariance condition for convergence of regularization methods for nonlinear ill-posed operator equations—such as coefficient identification in partial differential equations (PDEs) from boundary observations—can often be achieved by extending the searched for parameter in the sense of allowing it to depend on additional variables. This clearly counteracts unique identifiability of the parameter, though. The second key idea of this paper is now to restore the original restricted dependency of the parameter by penalization. This is shown to lead to convergence of variational (Tikhonov type) and iterative (Newton-type) regularization methods. We concretize the abstract convergence analysis in a framework typical of parameter identification in PDEs in a reduced and an all-at-once setting. This is further illustrated by three examples of coefficient identification from boundary observations in elliptic and time-dependent PDEs.

Shape reconstructions by using plasmon resonances with enhanced sensitivity

This paper investigates the shape reconstructions of sub-wavelength objects from near-field measurements in transverse electromagnetic scattering. This geometric inverse problem is notoriously ill-posed and challenging. We develop a novel reconstruction scheme of using plasmon resonances with significantly enhanced sensitivity and resolution. First, by spectral analysis we establish a sharp quantitative relationship between the sensitivity of the reconstruction and the plasmon resonance. It shows that the sensitivity functional blows up when plasmon resonance occurs. Hence, the signal to noise ratio is significantly improved and the robustness and effectiveness of the reconstruction are ensured. Second, a variational regularization method is proposed to overcome the ill-posedness, and an alternating iteration method is introduced to automatically select the regularization parameters. Third, we use Laplace approximation method to capture the statistical information of the target scattering object. Both rigorous theoretical analysis and extensive numerical experiments are conducted to validate the promising features of our method.

A cascading l0 regularization reconstruction method in nonsubsampled contourlet domain for limited-angle CT

In some computed tomography (CT) practical applications, the projection data is incomplete, which makes some limited-angle artifacts presented in the reconstructed image. Total variation (TV) regularization method is used to suppress the artifacts, but the edge of reconstructed image is distorted or some tiny details are missed. To preserve the edge of reconstructed image, a cascading ℓ0 regularization-based reconstruction model in nonsubsampled contourlet transform (NSCT) domain is proposed, which considers both the directional peculiarity of artifacts and the smooth of low frequency of the reconstructed image. In the proposed reconstruction model, a ℓ0 regularization term in the high frequency is used to suppress the artifacts and noise, and a ℓ0 regularization of gradient image in low frequency is used to suppress the limited-angle artifacts, to preserve the edge of object and to smooth the reconstructed image. To solve the proposed model, a proximal alternating linearized Peaceman–Rachford splitting method (PAL-PRSM) is proposed, which uses the Peaceman–Rachford splitting method that updates the Lagrange multiplier twice, one after each minimization of the subproblem, to deal with the optimization problem firstly, and then the proximal linearization is used to avoid computing the inverse of a huge system matrix. In addition, to compensate the proximal linearization, we add an additional term into Lagrange multiplier updating. Real data experiments are investigated to demonstrate the effectiveness of PAL-PRSM method, and the results show that the proposed method outperforms two other CT reconstruction methods on preserving the edge of reconstructed image. On the whole, the proposed method is a compromise between denoising and preserving the edge of reconstructed image.

Reconstruction of conductivity distribution variation with Tikhonov and wavelet frame combined method for electrical impedance tomography

Electrical impedance tomography (EIT) is receiving considerable research interest in a variety of applications. With this technique, it is possible to reconstruct conductivity distribution inside the detected region. Note that reconstruction of conductivity distribution is an ill-posed inverse problem. To deal with this problem, Tikhonov regularization method has been preferred. However, this method suffers from poor reconstruction quality. In this work, a novel approach which combines Tikhonov regularization method with wavelet frame is proposed for image reconstruction of EIT. The proposed method is superior to the popular Tikhonov regularization method in enforcing sparse of the solution and enhancing sharp feature. Simulation work has been conducted to validate the performance of the proposed method. Compared with Landweber method, total variation (TV) regularization method, and Tikhonov regularization method, reconstructions of six different models show that images reconstructed by the proposed method have better quality and are more robust to noise. In addition, image reconstruction is also performed based on phantom experimental data. The results further demonstrate that the proposed method outperforms other three methods since the inclusion has been more accurately reconstructed, and there are almost no artifacts in the reconstructed images.

The 3-D Global Prestack Seismic Inversion in the Time–Frequency Mixed Domain

Traditional seismic inversion has the problems of low lateral resolution and poor anti-noise performance. To address these challenges, the advantages of inversion methods in time and frequency domains are combined; the Lp norm that can preserve more sparsity information is introduced into seismic inversion. Meanwhile, multi-trace simultaneous inversion has become the mainstream of inversion methods, which can effectively improve the quality of pre-stack seismic inversion. However, traditional multi-trace simultaneous inversion methods have high computational costs and are difficult to obtain the 3D result. Ensuring spatial continuity of inversion results is a challenging task. To address this issue, a 3D global pre-stack inversion method in the time-frequency mixed domain is proposed. The proposed method adds a frequency domain fidelity constraint the objective function of traditional time domain inversion to improve the resolution of the inversion results. To improve the computational efficiency of multi-trace simultaneous inversion, the anisotropic total variation regularization method under Lp norm constraint is employed. In addition, the 2D Sylvester equation is extended to 3D space to enable 3D global inversion. The method not only improves computational efficiency, but also ensures spatial continuity of the results. The stability and feasibility of the proposed method have been demonstrated through its application on both the 3D overthrust model and field pre-stack seismic data.

A new difference of anisotropic and isotropic total variation regularization method for image restoration.

Total variation (TV) regularizer has diffusely emerged in image processing. In this paper, we propose a new nonconvex total variation regularization method based on the generalized Fischer-Burmeister function for image restoration. Since our model is nonconvex and nonsmooth, the specific difference of convex algorithms (DCA) are presented, in which the subproblem can be minimized by the alternating direction method of multipliers (ADMM). The algorithms have a low computational complexity in each iteration. Experiment results including image denoising and magnetic resonance imaging demonstrate that the proposed models produce more preferable results compared with state-of-the-art methods.

Space-constrained optimized Tikhonov regularization method for 3D hemorrhage reconstruction by open magnetic induction tomography

Objective. Open magnetic induction tomography (MIT) is a promising technique for detecting the intracranial hemorrhage due to the non-radioactive, non-invasive and portable features. However, severe inhomogeneity of the sensitivity distribution under the open MIT sensor array and the ill-conditioned nature of MIT inverse problem limit the imaging quality in hemorrhage reconstruction. More accurate and robust imaging algorithms are urgently needed in clinical diagnosis. Approach. In this study, the space-constrained optimized Tikhonov regularization (SOTR) method is proposed for 3D hemorrhage reconstruction by open MIT. The sensitivity matrix is optimized according to the characteristics of sensitivity distribution under the open MIT sensor array. To test the performance of the SOTR method, 3D anatomical head models with hemorrhages in different volumes and locations were established. The images of the hemorrhages were reconstructed by the Tikhonov regularization (TR), total variation (TV) regularization, isotropic SOTR, and anisotropic SOTR method. Correlation coefficient localization error and volume error were calculated to evaluate the hemorrhage imaging quality. Main results. Compared with the traditional sensitivity matrix, the optimized sensitivity matrix has smaller column number and better uniformity, which alleviates the under-determined and ill-conditioned problem of MIT. The imaging results indicate that both the isotropic and anisotropic SOTR methods can effectively improve the reconstruction accuracy for the location and volume of the hemorrhages. Moreover, compared with the TR and TV methods, the two SOTR methods are more robust against the measurement noise. Significance. The proposed method improves the imaging quality of the intracranial hemorrhage, which promotes the clinical applications of open MIT.

Hybrid regularization model combining overlapping group sparse second-order total variation and nonconvex total variation

Restoring the original image from an observed noisy or blurred version is a very important issue in image processing. The variational regularization methods obtained by functional minimizing are popular and widely used for image restoration with promising accuracy, but they may suffer from limited performances due to the presence of the conflict between noise-and-blur-removing and detail-and-texture-preserving. To solve this problem, a hybrid variational regularization model is proposed to effectively reconstruct image textures and details while eliminating noise and blur. The proposed model uses an overlapping group of sparse second-order total variation (TV) as the regularizer to restore image textures and details, whereas the nonconvex TV serves as the other regularizer to preserve image edges and contours. An alternating directional multiplier method algorithm is developed to numerically solve the proposed model, where the majorization–minimization and the iterative reweighted l1 (IRL1) algorithm are used to solve the overlapping group and the nonconvex minimization subproblem, respectively. The numerical experiments validate the proposed model and algorithm. In addition, compared with several state-of-the-art image restoration models, the proposed model shows the competitive performance in terms of PSNR, SSIM, and NIQE values.

Feature Reconstruction from Incomplete Tomographic Data without Detour

In this paper, we consider the problem of feature reconstruction from incomplete X-ray CT data. Such incomplete data problems occur when the number of measured X-rays is restricted either due to limit radiation exposure or due to practical constraints, making the detection of certain rays challenging. Since image reconstruction from incomplete data is a severely ill-posed (unstable) problem, the reconstructed images may suffer from characteristic artefacts or missing features, thus significantly complicating subsequent image processing tasks (e.g., edge detection or segmentation). In this paper, we introduce a framework for the robust reconstruction of convolutional image features directly from CT data without the need of computing a reconstructed image first. Within our framework, we use non-linear variational regularization methods that can be adapted to a variety of feature reconstruction tasks and to several limited data situations. The proposed variational regularization method minimizes an energy functional being the sum of a feature dependent data-fitting term and an additional penalty accounting for specific properties of the features. In our numerical experiments, we consider instances of edge reconstructions from angular under-sampled data and show that our approach is able to reliably reconstruct feature maps in this case.

An AtanTV Nonconvex Regularization Model for MRI Reconstruction

Recently, the nonconvex regularization model has attracted much research attention in magnetic resonance imaging (MRI). In comparison with the traditional total variation (TV) regularization method, a nonconvex TV regularizer can effectively improve the fitting performance and prevent bias problems. In this work, we utilize the arctangent function as nonconvex TV regularize term (AtanTV) for MRI reconstruction. With appropriate parameters, the AtanTV model can avoid the systematic underestimation characteristic and maintain the objective function still convex. To address the AtanTV model, we propose an efficient alternating direction method of multipliers (ADMM) method to solve it. Experimental results indicate the superior performance (PSNR, RE, SSIM, etc.) of the proposed MRI reconstruction method.

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.

Diffusion tensor regularization with metric double integrals

AbstractIn this paper, we propose a variational regularization method for denoising and inpainting of diffusion tensor magnetic resonance images. We consider these images as manifold-valued Sobolev functions, i.e. in an infinite dimensional setting, which are defined appropriately. The regularization functionals are defined as double integrals, which are equivalent to Sobolev semi-norms in the Euclidean setting. We extend the analysis of [14] concerning stability and convergence of the variational regularization methods by a uniqueness result, apply them to diffusion tensor processing, and validate our model in numerical examples with synthetic and real data.

Variational Regularization For The Inverse Problem of a Nonlinear Time-Fractional Diffusion Equations

The nonlinear time-fractional diffusion equations were considered in the 2D domain, and the physical information in initial state <inlineformula><texmath id="M2">\begin{document}$ u(0,y,t) $\end{document}</texmath><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="amm-42-0168_M2.jpg"></graphic><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="amm-42-0168_M2.png"></graphic></alternatives></inlineformula> of the material was recovered from the measured data in the final state. This problem is seriously ill-posed, that is, the solution to this problem does not continuously depend on the measured data. Therefore, a variational regularization method was proposed to construct the approximate solution to the problem, and the convergence error estimates of the exact and approximate solutions were obtained under the assumption of the priori bounds on the exact solutions. Finally, a numerical example was given to verify the effectiveness of the proposed method.

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

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

Anisotropic TV Regularization in Electrical Impedance Tomography: An Experimental Study

Total variation (TV) regularization method is a typical method to preserve the discontinuities structure in EIT. Isotropic TV and anisotropic TV are two well-known variants of TV. The main differences between them are that the latter tends to distort the reconstructed internal inhomogeneities along the coordinate axis. In this article, we adopt the alternating direction method of multipliers (ADMM) to overcome the non-differentiability of the anisotropic TV and verify the characteristics of anisotropic TV regularization by the tank experiments.