Total Variation Regularization Research Articles

Hyperspectral Image Mixed Noise Removal via Double Factor Total Variation Nonlocal Low-Rank Tensor Regularization

A hyperspectral image (HSI) is often corrupted by various types of noise during image acquisition, e.g., Gaussian noise, impulse noise, stripes, deadlines, and more. Thus, as a preprocessing step, HSI denoising plays a vital role in many subsequent tasks. Recently, a variety of mixed noise removal approaches have been developed for HSI, and the methods based on spatial–spectral double factor and total variation (DFTV) regularization have achieved comparable performance. Additionally, the nonlocal low-rank tensor model (NLR) is often employed to characterize spatial nonlocal self-similarity (NSS). Generally, fully exploring prior knowledge can improve the denoising performance, but it significantly increases the computational cost when the NSS prior is employed. To solve this problem, this article proposes a novel DFTV-based NLR regularization (DFTVNLR) model for HSI mixed noise removal. The proposed model employs low-rank tensor factorization (LRTF) to characterize the spectral global low-rankness (LR), introduces 2-D and 1-D TV constraints on double-factor to characterize the spatial and spectral local smoothness (LS), respectively. Meanwhile, the NLR is applied to the spatial factor to characterize the NSS. Then, we developed an algorithm based on proximal alternating minimization (PAM) to solve the proposed model effectively. Particularly, we effectively controlled the computational cost from two aspects, namely taking small-sized double factor as regularization object and putting the time-consuming NLR model before the main loop with fewer iterations to solve it independently. Finally, considerable experiments on simulated and real noisy HSI substantiate that the proposed method is superior to the related state-of-the-art methods in balancing the denoising effect and speed.

Efficient image restoration via non-convex total variation regularization and ADMM optimization

This article presents a novel approach to image restoration utilizing a unique non-convex l1/2-TV regularization model. This model integrates the l1/2-quasi norm as a regularization function, introducing non-convexity to promote sparsity and unevenly penalize elements, thereby enhancing restoration outcomes. To tackle this model, an efficient algorithm named the Alternating Direction Method of Multipliers, based on the Lagrangian multiplier, is introduced. This effectively prevents the penalty parameter from reaching infinity and ensures excellent convergence behavior. The proposed algorithm decomposes the optimization problem into subproblems, for which closed-form solutions are derived, particularly addressing the challenging l1/2 regularization problem. To validate its effectiveness, a comprehensive set of experiments are conducted to compare its performance with existing methods. The experimental results demonstrate that the proposed model performs well in both qualitative and quantitative evaluations. Consequently, the proposed model is not only efficient and stable but also exhibits excellent convergence behavior.

Cascaded combination of total variation regularization and contrast limited adaptive histogram equalization based image dehazing

ABSTRACT Haze and fog can significantly reduce the visibility of distant objects by scattering light and creating blurry, washed-out images lacking in detail. Image dehazing is a technique used in computer vision to enhance the clarity of such obscured images. This study examines the effectiveness of various dehazing methods using real hazy images and synthetic images with artificially created haze. Performance metrics such as PSNR, SSIM, MSE, CII, and computation time are used to evaluate the proposed method. The evaluation is carried out on datasets like RESIDE, GMAN, O-Haze, I-Haze, NH-Haze, and Dense haze to compare the proposed method with existing models. The proposed method attains 27.46%, 20.63% and 21.09% higher PSNR and 12.36%, 23.95% and 36.12% lower Natural Image Quality Evaluator for RESIDE dataset when analyzed to the existing models, such as strategic method towards contrast enhancement by 2D histogram equalization under TV decomposition (CE-TDHE-TVD), multiple level framework basis contrast enhancement for uniform with non-uniform back ground imageries utilizing appropriate histogram equalization (CE-VHE), and Contrast enhancement with brightness preservation of low light pictures with the help of combined CLAHE and BPDHE histogram equalization (CE-CLAHE-BPDHE) respectively.

Parallel ensemble Kalman method with total variation regularization for large-scale field inversion

Field inversion is often encountered in data-driven computational modeling to infer latent spatial–varying parameters from available observations. The ensemble Kalman method is emerging as a useful tool for solving field inversion problems due to its derivative-free merits. However, the method is computationally prohibitive for large-scale field inversion with high-dimensional observation data, which necessitates developing a practical efficient implementation strategy. In this work, we propose a parallel implementation of the ensemble Kalman method with total variation regularization for large-scale field inversion problems. It is achieved by partitioning the computational domain into non-overlapping subdomains and performing local ensemble Kalman updates at each subdomain parallelly. In doing so, the computational complexity of the ensemble-based inversion method is significantly reduced to the level of local subdomains. Further, the total variation regularization is employed to smoothen the physical field over the entire domain, which can reduce the inference discrepancy caused by missing covariances near subdomain interfaces. The capability of the proposed method is demonstrated in three field inversion problems with increasing complexity, i.e., the diffusion problem, the scalar transport problem and the Reynolds averaged Navier-Stokes closure problem. The numerical results show that the proposed method can significantly improve computational efficiency with satisfactory inference accuracy.

Fusion of low-rankness and smoothness under learnable nonlinear transformation for tensor completion

The recently proposed tensor correlated total variation (t-CTV) has achieved success in tensor completion. It utilizes the low-rank structure of the gradient tensor under a unified linear transform to jointly encode low-rankness and smoothness priors. However, fixed linear transforms have inherent limitations in fully characterizing gradient tensors in different directions and adapting them to tensors from diverse categories. In this work, we propose the nonlinear tensor correlated total variation (NTCTV) regularization term that leverages the low-rank correlations of the gradient tensor under the learnable nonlinear transformation, providing a more natural approach to fuse the low-rankness and smoothness priors. Specifically, our approach learns the optimal nonlinear implicit low-rank structure of the gradient tensor along different modes separately, and then achieves the expression of fused prior information in a coupled manner. Furthermore, we propose the NTCTV-based tensor completion model and design the proximal alternating minimization (PAM) algorithm to efficiently solve the optimization model. Moreover, we provide a theoretical proof of the global convergence of the algorithm to a critical point. Comprehensive experimental results for hyperspectral images, medical images, multispectral images, and videos demonstrate that the proposed method achieves substantial quantitative and qualitative improvements over many state-of-the-art tensor completion techniques.

A convex level-set method with multiplicative-additive model for image segmentation

The existing active contour models (ACMs) based on bias field (BF) correction mostly rely on a single BF assumption and lack in-depth discussion on the convexity of the energy functional, often leading to the problem of local minima. To address this issue, this paper introduces a dual BF and proposes a convex level-set (LS) method based on multiplicative-additive (MA) model to achieve global minima. Firstly, a MA model is adopted as the fidelity term, and a kernel function is introduced to adjust the size of the intensity inhomogeneous neighborhood, enhancing the adaptability to intensity inhomogeneity. Then, the convex LS function is embedded in the variational framework to ensure convexity of each variable in the energy functional. This transformation turns the segmentation problem into a convex optimization problem. By introducing the total variation regularization term to smooth the LS function, the model's resistance to noise is effectively enhanced. Finally, by minimizing the proposed energy functional, image segmentation and BF correction are successfully achieved. Experimental results validate the global minima property of our model, while also demonstrating good flexibility in the initial contour. The proposed model achieves superior segmentation results compared to other classical ACMs on various types of images.

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.

Total variation regularization analysis for inverse volatility option pricing problem

We investigate in this paper an inverse problem of recovering the space-dependent volatility in option pricing. To enhance precision across the domain, we transform the original problem into an inverse source problem of a bounded degenerate parabolic equation, utilizing linearization and variable substitution. Unlike classical methods, we apply a total variation regularization combined with a novel generalized finite integration technique. This approach accommodates volatility jumps in an overnight rate scenario. Leveraging an optimal control framework, we demonstrate that the inverse problem can be reformulated as an optimal control problem whose existence, necessary conditions, local uniqueness and stability for the minimizer of control functional are obtained. For numerical verification, we derive the Euler equation and design a discretization algorithm with generalized finite integration technique. Numerical examples showcase the robustness of our approach, highlighting advantages in accuracy and effectiveness over other strategies.

Enhanced Low-Rank Tensor Recovery Fusing Reweighted Tensor Correlated Total Variation Regularization for Image Denoising

Enhanced Low-Rank Tensor Recovery Fusing Reweighted Tensor Correlated Total Variation Regularization for Image Denoising

Multi-scale ground penetrating radar full waveform inversion with hybrid Tikhonov and total-variation regularization for different geometric structure

Abstract Full waveform inversion (FWI) is a high-resolution technique to estimate the parameters of dielectric permittivity (ϵ) and electrical conductivity (σ) and identify the structure of the subsurface for Ground Penetrating Radar (GPR) application. However, permittivity and conductivity parameters can be coupled in bi-parameter GPR inversion. This coupling effect leads to the crosstalk in the FWI result. To solve this problem, we propose a novel approach to use the multi-scale FWI with the hybrid regularization method, which combines Tikhonov and total-variation (TV) regularizers that simultaneously invert the ϵ and the σ parameters, which improve the inversion accuracy and reduce the crosstalk effect. The multi-scale strategy uses the Wiener filtering to process the GPR data in different frequency ranges. Then, the low frequencies signal updates the bottom part and subsequently increases the frequencies to invert for the shallow areas. The Tikhonov regularization stabilizes the reconstruction of the smoothly varying background part. In contrast, Total Variation (TV) regularization can recover the large contrasts associated with the LNAPL model. The new Tikhonov-TV (TT) regularization can mitigate the crosstalk caused by the parameter coupling effect. Numerical tests with typical GPR models demonstrate that the proposed multi-scale TT-FWI strategy can effectively eliminate the crosstalk and improve the reconstruction accuracy when the model parameters have a different structure.

A Sparse SAR Imaging Method for Low-Oversampled Staggered Mode via Compound Regularization

High-resolution wide-swath (HRWS) imaging is the research focus of the modern spaceborne synthetic-aperture radar (SAR) imaging field, with significant relevance and vast application potential. Staggered SAR, as an innovative imaging system, mitigates blind areas across the entire swath by periodically altering the radar pulse repetition interval (PRI), thereby extending the swath width to multiples of that achievable by conventional systems. However, the staggered mode introduces inherent challenges, such as nonuniform azimuth sampling and echo data loss, leading to azimuth ambiguities and substantially impacting image quality. This paper proposes a sparse SAR imaging method for the low-oversampled staggered mode via compound regularization. The proposed method not only effectively suppresses azimuth ambiguities arising from nonuniform sampling without necessitating the restoration of missing echo data, but also incorporates total variation (TV) regularization into the sparse reconstruction model. This enhances the accurate reconstruction of distributed targets within the scene. The efficacy of the proposed method is substantiated through simulations and real data experiments from spaceborne missions.

Impulse Noise Image Restoration Using Nonconvex Variational Model and Difference of Convex Functions Algorithm.

In this article, the problem of impulse noise image restoration is investigated. A typical way to eliminate impulse noise is to use an L1 norm data fitting term and a total variation (TV) regularization. However, a convex optimization method designed in this way always yields staircase artifacts. In addition, the L1 norm fitting term tends to penalize corrupted and noise-free data equally, and is not robust to impulse noise. In order to seek a solution of high recovery quality, we propose a new variational model that integrates the nonconvex data fitting term and the nonconvex TV regularization. The usage of the nonconvex TV regularizer helps to eliminate the staircase artifacts. Moreover, the nonconvex fidelity term can detect impulse noise effectively in the way that it is enforced when the observed data is slightly corrupted, while is less enforced for the severely corrupted pixels. A novel difference of convex functions algorithm is also developed to solve the variational model. Using the variational method, we prove that the sequence generated by the proposed algorithm converges to a stationary point of the nonconvex objective function. Experimental results show that our proposed algorithm is efficient and compares favorably with state-of-the-art methods.

Rapid and Noise‐Resilient Mapping of Photogenerated Carrier Lifetime in Halide Perovskite Thin Films

AbstractHalide perovskite materials offer significant promise for solar energy and optoelectronics yet understanding and enhancing their efficiency and stability require addressing lateral inhomogeneity challenges. While photoluminescence imaging techniques are employed for the measurement of their opto‐electronic and transport properties, going further in terms of precision requires longer acquisition times. Prolonged exposure of perovskites to light, given their high reactivity, can substantially alter these layers, rendering the acquired data less meaningful for analysis. In this paper, a method to extract high‐quality lifetime images from rapidly acquired, noisy time‐resolved photoluminescence images is proposed. This method leverages concepts of the field of constrained reconstruction and includes the Huber loss function and a specific form of total variation regularization. Through both simulations and experiments, it is demonstrated that the approach outperforms conventional pointwise methods. Optimal acceleration and optimization parameters tailored for decay time imaging of perovskite materials, offering new perspectives for accelerated experiments crucial in degradation process characterization are identified. Importantly, this methodology holds the potential for broader applications: it can be extended to explore additional beam‐sensitive materials, and other imaging characterization techniques and employed with more complex physical models to treat time‐resolved decays.

A Fast Total-Variation Driven Network for Sparse Aperture ISAR Imaging

Conventional Fourier transform-based Inverse Synthetic Aperture Radar (ISAR) imaging cannot deal with the cases of incomplete radar echoes, in which the data needs to be specially processed by Sparse Aperture (SA) imaging methods. Most current SA-ISAR imaging employs sparsity-driven optimization methods. However, the assumption that the target reflectivity is sparse in the spatial domain is insufficient to capture the surface and edge features of the target if no sparse representations are found for the target and well incorporated in the imaging process. In view of the edge-preserving capability of Total Variation (TV) and the strong learning ability of the Deep Neural Network (DNN), we propose a TV-driven network to improve the SA ISAR imaging performance. We first develop a fast TV regularization method to perform the imaging where the Gradient Descent (GD) along with the Momentum Acceleration (MA) are incorporated to increase the computational efficiency. Then, we unwrap the iterative fast TV regularization into a cascaded neural network to make the key imaging parameters learnable, leading to improved imaging performance, which we refer to as FGDTV-Net. Experiments under a variety of scenarios show that the proposed FGDTV-Net for SA-ISAR imaging is superior to existing SA imaging algorithms in preserving surface and edge features and is more robust in low signal-to-noise scenarios.

An Iterative Method for the Inverse Eddy Current Problem with Total Variation Regularization

An Iterative Method for the Inverse Eddy Current Problem with Total Variation Regularization

Q-compensated image-domain least-squares reverse time migration through preconditioned point-spread functions

Image-domain least-squares reverse time migration (IDLSRTM) through point-spread functions (PSFs) has been proven to be a feasible approach to improve the spatial resolution and amplitude fidelity of reflection images recovered by reverse time migration (RTM). However, it usually ignores the earth’s Q-effects, which may lead to an unfocused reflection image with an undesired spatial resolution. In this paper, we develop a Q-compensated IDLSRTM approach (denoted as Q-IDLSRTM) through PSFs, in which we use the viscoacoustic wave equation based on the generalized standard linear solid model to simulate inherent subsurface attenuation and the linear inversion to compensate for the amplitude attenuation. The PSFs are estimated by a round of modeling-migration computation and spatial interpolation on the fly. There are two key points in the developed Q-IDLSRTM approach. The first is that we must apply the deblurring filter as a preconditioner to compensate for the attenuation of image amplitude of PSFs and RTM in a viscoacoustic medium, before the iterative solution. The preconditioned PSFs and RTM images can help us to construct a less ill-posed image-domain inverse problem that can produce an improved image quality and a faster convergence rate, compared with the conventional Q-IDLSRTM approach without the deblurring filter. The second key point is that we can impose the L1-norm constraint and total variation regularization on the reflection image to stabilize the solution of the ill-posed inverse problem. Several 2D and 3D experiments verify that the developed approach can achieve better imaging quality in terms of amplitude fidelity and spatial resolution relative to the conventional Q-IDLSRTM and acoustic IDLSRTM approaches.

Boosting MR Image Impulse Noise Removal With Overlapping Group Sparse Fractional-Order Total Variation and Minimax Concave Penalty

The acquisition and transmission of magnetic resonance images are susceptible to noise, particularly impulse noise. Although the method based on the ℓ0-norm and overlapping group sparse total variation (ℓ0-OGSTV) is effective for impulse noise image restoration, it can only mitigate the staircase artifacts to a certain extent. To boost the impulse noise removal performance of ℓ0-OGSTV, we propose a new restoration model that consists of two terms. Specifically, in the first term, we keep using the ℓ0-norm as the data fidelity term to eliminate impulse noise. In the second term, we first introduce an overlapping group sparsity fractional-order total variation regularizer to eliminate staircase artifacts while preserving structural information. Then, we adopt the minimax-concave penalty to further accurately estimate the image edges. Finally, we employ an alternate direction method of multipliers to solve the proposed optimization model. Clinical experiments demonstrate its effectiveness in denoising medical images.

Image restoration based on transformed total variation and deep image prior

Most supervised learning methods require observation data and ground truth pairs as data sets to train the network. However, it is difficult and time-consuming to obtain a large number of high quality data sets, because ground truth is not available in some practical settings, such as medical imaging, dynamic scenes. Deep image prior (DIP) only uses one degraded image for image recovery tasks, which gets rid of the limitation of constructing a large number of training sets but requires an early stop mechanism. In order to further improve the image restoration ability of the DIP model, we combine it with the transformed total variation (TTV) regularization, which is a generalization of the classical total variation (TV) regularization and can achieve better performance for image restoration problems. The proposed method not only uses the deep neural network to capture image prior, but also exploits inherent sparse prior in image gradient domain. In addition, we provide an adaptive weight selection strategy for TTV regularization. The ADMM scheme is employed to solve the proposed models. Numerical results of image restoration illustrate that the proposed methods perform better than the compared methods.

Using a deep learning prior for accelerating hyperpolarized 13 C MRSI on synthetic cancer datasets.

We aimed to incorporate a deep learning prior with k-space data fidelity for accelerating hyperpolarized carbon-13 MRSI, demonstrated on synthetic cancer datasets. A two-site exchange model, derived from the Bloch equation of MR signal evolution, was firstly used in simulating training and testing data, that is, synthetic phantom datasets. Five singular maps generated from each simulated dataset were used to train a deep learning prior, which was then employed with the fidelity term to reconstruct the undersampled MRI k-space data. The proposed method was assessed on synthetic human brain tumor images (N = 33), prostate cancer images (N = 72), and mouse tumor images (N = 58) for three undersampling factors and 2.5% additive Gaussian noise. Furthermore, varied levels of Gaussian noise with SDs of 2.5%, 5%, and 10% were added on synthetic prostate cancer data, and corresponding reconstruction results were evaluated. For quantitative evaluation, peak SNRs were approximately 32 dB, and the accuracy was generally improved for 5 to 8 dB compared with those from compressed sensing with L1-norm regularization or total variation regularization. Reasonable normalized RMS error were obtained. Our method also worked robustly against noise, even on a data with noise SD of 10%. The proposed singular value decomposition + iterative deep learning model could be considered as a general framework that extended the application of deep learning MRI reconstruction to metabolic imaging. The morphology of tumors and metabolic images could be measured robustly in six times acceleration using our method.

Tensor completion via multi-directional partial tensor nuclear norm with total variation regularization

Tensor completion via multi-directional partial tensor nuclear norm with total variation regularization