Total Variation Regularization Term Research Articles

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.

MULTIPARENT FRACTAL IMAGE CODING-BASED METHODS FOR SALT-AND-PEPPER NOISE REMOVAL

Salt-and-pepper noise consists of outlier pixel values which significantly impair image structure and quality. Multiparent fractal image coding (MFIC) methods substantially exploit image redundancy by utilizing multiple domain blocks to approximate the range block, partially compensating for the information loss caused by noise. Motivated by this, we propose two novel image restoration methods based on MFIC to remove salt-and-pepper noise. The first method integrates Huber M-estimation into MFIC, resulting in an improved anti-salt-and-pepper noise robust fractal coding approach. The second method incorporates MFIC into a total variation (TV) regularization model, including a data fidelity term, an MFIC term and a TV regularization term. An alternative iterative method based on proximity operator is developed to effectively solve the proposed model. Experimental results demonstrate that these two proposed approaches achieve significantly enhanced performance compared to traditional fractal coding methods.

An MTL1TV non-convex regularization model for MR Image reconstruction using the alternating direction method of multipliers

<abstract><p>The acquisition time of magnetic resonance imaging (MRI) is relatively long. To achieve high-quality and fast reconstruction of magnetic resonance (MR) images, we proposed a non-convex regularization model for MR image reconstruction with the modified transformed $ {l_1} $ total variation (MTL1TV) regularization term. We addressed this new model using the alternating direction method of multipliers (ADMM). To evaluate the proposed MTL1TV model, we performed numerical experiments on several MR images. The numerical results showed that the proposed model gives reconstructed images of improved quality compared with those obtained from state of the art models. The results indicated that the proposed model can effectively reconstruct MR images.</p></abstract>

Performance analysis of the convex non-convex total variation denoising model

<p>Total variation (TV) regularization is a powerful tool in image denoising, but it often exhibits limited performance in preserving edges. In contrast, non-convex TV regularization can more effectively preserve edges and contours, albeit posing challenges when solving. Recently, the convex non-convex (CNC) strategy has emerged as a potent approach that allows incorporating non-convex TV regularization terms while maintaining the overall convexity of the objective function. Its superior performance has been validated through various numerical experiments; however, theoretical analysis remains lacking. In this paper, we provided theoretical analysis of the performance of the CNC-TV denoising model. By utilizing the oracle inequality, we derived an improved upper bound on its performance compared to TV regularization. In addition, we devised an alternating direction method of multipliers (ADMM) algorithm to address the proposed model and verified its convergence properties. Our proposed model has been validated through numerical experiments in 1D and 2D denoising, demonstrating its exceptional performance.</p>

Hybrid model of tensor sparse representation and total variation regularization for image denoising

The method based on non-local self-similarity patches has been widely applied in image denoising. For a given image patch, there are similar image patches at neighbor windows in the image. Traditionally, similar patches are vectorized and then rearranged into a matrix, which should have a low rank. Hence the denoising problem can be reformulated into a low rank recovery problem. However, the vectorization process can disrupt the spatial relationships among the patches. To improve this drawback, we rearrange the similar patches into a tensor to keep the spatial relationships. When the higher-order singular value decomposition (HOSVD) is applied to this tensor, the resulting core tensor should be sparse. In this paper, the sparse property of the core tensor is utilized to characterize similar patches, and then an optimization model is derived. Since the commonly used l0 norm regularization is NP-hard, we adopt the MCP function to constrain the sparse property of the core tensor. In addition, a total variation (TV) regularization term for the tensor is added to the model to preserve structural information in the image such as edges. Using the dual form to represent the total variation regularization and transforming the original problem into a saddle point problem, the primal–dual algorithm is applied to overcome the non-differentiability of the TV regularization. Finally, experimental results demonstrate that compared to existing denoising methods, the method proposed can better preserve details and structural information in the image.

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.

A Primal-Dual Finite Element Method for Scalar and Vectorial Total Variation Minimization

Based on the Fenchel duality we build a primal-dual framework for minimizing a general functional consisting of a combined L^1 and L^2 data-fidelity term and a scalar or vectorial total variation regularisation term. The minimization is performed over the space of functions of bounded variations and appropriate discrete subspaces. We analyze the existence and uniqueness of solutions of the respective minimization problems. For computing a numerical solution we derive a semi-smooth Newton method on finite element spaces and highlight applications in denoising, inpainting and optical flow estimation.

A Novel Reconstruction of the Sparse-View CBCT Algorithm for Correcting Artifacts and Reducing Noise

X-ray tomography is often affected by noise and artifacts during the reconstruction process, such as detector offset, calibration errors, metal artifacts, etc. Conventional algorithms, including FDK and SART, are unable to satisfy the sampling theorem requirements for 3D reconstruction under sparse-view constraints, exacerbating the impact of noise and artifacts. This paper proposes a novel 3D reconstruction algorithm tailored to sparse-view cone-beam computed tomography (CBCT). Drawing upon compressed sensing theory, we incorporate the weighted Schatten p-norm minimization (WSNM) algorithm for 2D image denoising and the adaptive steepest descent projection onto convex sets (ASD-POCS) algorithm, which employs a total variation (TV) regularization term. These inclusions serve to reduce noise and ameliorate artifacts. Our proposed algorithm extends the WSNM approach into three-dimensional space and integrates the ASD-POCS algorithm, enabling 3D reconstruction with digital brain phantoms, clinical medical data, and real projections from our portable CBCT system. The performance of our algorithm surpasses traditional methods when evaluated using root mean square error (RMSE), peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM) metrics. Furthermore, our approach demonstrates marked enhancements in artifact reduction and noise suppression.

Sparse deconvolution for background noise suppression with total variation regularization in light field microscopy.

In this Letter, we present a method aiming at background noise removal in the 3D reconstruction of light field microscopy (LFM). Sparsity and Hessian regularization are taken as two prior knowledges to process the original light field image before 3D deconvolution. Due to the noise suppression function of total variation (TV) regularization, we add the TV regularization term to the 3D Richardson-Lucy (RL) deconvolution. By comparing the light field reconstruction results of our method with another state-of-the-art method that is also based on RL deconvolution, the proposed method shows improved performance in terms of removing background noise and detail enhancement. This method will be beneficial to the application of LFM in biological high-quality imaging.

Efficient dual ADMMs for sparse compressive sensing MRI reconstruction

Magnetic Resonance Imaging (MRI) is a kind of medical imaging technology used for diagnostic imaging of diseases, but its image quality may be suffered by the long acquisition time. The compressive sensing (CS) based strategy may decrease the reconstruction time greatly, but it needs efficient reconstruction algorithms to produce high-quality and reliable images. This paper focuses on the algorithmic improvement for the sparse reconstruction of CS-MRI, especially considering a non-smooth convex minimization problem which is composed of the sum of a total variation regularization term and a $$\ell _1$$ -norm term of the wavelet transformation. The partly motivation of targeting the dual problem is that the dual variables are involved in relatively low-dimensional subspace. Instead of solving the primal model as usual, we turn our attention to its associated dual model composed of three variable blocks and two separable non-smooth function blocks. However, the directly extended alternating direction method of multipliers (ADMM) must be avoided because it may be divergent, although it usually performs well numerically. In order to solve the problem, we employ a symmetric Gauss–Seidel (sGS) technique based ADMM. Compared with the directly extended ADMM, this method only needs one additional iteration, but its convergence can be guaranteed theoretically. Besides, we also propose a generalized variant of ADMM because this method has been illustrated to be efficient for solving semidefinite programming in the past few years. Finally, we do extensive experiments on MRI reconstruction using some simulated and real MRI images under different sampling patterns and ratios. The numerical results demonstrate that the proposed algorithms significantly achieve high reconstruction accuracies with fast computational speed.

Two new low rank tensor completion methods based on sum nuclear norm

The low rank tensor completion (LRTC) problem has attracted great attention in computer vision and signal processing. How to acquire high quality image recovery effect is still an urgent task to be solved at present. This paper firstly proposes a new tensor L2,1 norm minimization model (TLNM) that integrates sum nuclear norm (SNN) method, differing from the classical tensor nuclear norm (TNN)-based tensor completion method, with L2,1 norm and QR decomposition for solving the LRTC problem. To improve the utilization rate of the local prior information of the image, a total variation (TV) regularization term is introduced, resulting in another new class of tensor L2,1 norm minimization with total variation model (TLNMTV). Both proposed models are convex and therefore have global optimal solutions. Moreover, we adopt the Alternating Direction Multiplier Method (ADMM) to obtain the closed-form solution of each variable, which makes the algorithm be able to be efficiently implemented. Numerical experiments show that the two proposed algorithms are convergent and outperform compared methods. In particular, our method significantly outperforms the compared methods when the sampling rate is 2.5% for hyperspectral images.

Image-spectral decomposition extended-learning assisted by sparsity for multi-energy computed tomography reconstruction.

Multi-energy computed tomography (CT) provides multiple channel-wise reconstructed images, and they can be used for material identification and k-edge imaging. Nonetheless, the projection datasets are frequently corrupted by various noises (e.g., electronic, Poisson) in the acquisition process, resulting in lower signal-noise-ratio (SNR) measurements. Multi-energy CT images have local sparsity, nonlocal self-similarity in spatial dimension, and correlation in spectral dimension. In this paper, we propose an image-spectral decomposition extended-learning assisted by sparsity (IDEAS) method to fully exploit these intrinsic priors for multi-energy CT image reconstruction. Particularly, a nonlocal low-rank Tucker decomposition (TD) is employed to utilize the correlation and nonlocal self-similarity priors. Moreover, considering the advantages of multi-task tensor dictionary learning (TDL) in sparse representation, an adaptive spatial dictionary and an adaptive spectral dictionary are trained during the iterative reconstruction process. Furthermore, a weighted total variation (TV) regularization term is employed to encourage local sparsity. Numerical simulation, physical phantom, and preclinical mouse experiments are performed to validate the proposed IDEAS algorithm. Specifically, in the simulation experiments, the proposed IDEAS reconstructed high-quality images that are very close to the references. For example, the root mean square error (RMSE) of IDEAS image in energy bin 1 is as low as 0.0672, while the RMSE of other methods are higher than 0.0843. Besides, the structural similarity (SSIM) of IDEAS reconstructed image in energy bin 1 is greater than 0.98. For material decomposition, the RMSE of IDEAS bone component is as low as 0.0152, and other methods are higher than 0.0199. In addition, the computational cost of IDEAS is as low as 98.8 s for one iteration, and the competing tensor decomposition method is higher than 327 s. To further improve the quality of the reconstructed multi-energy CT images, multiple prior regularizations are introduced to the multi-energy CT reconstructed model, leading to an IDEAS method. Both qualitative and quantitative evaluation of our results confirm the outstanding performance of the proposed algorithm compared to the state-of-the-arts.

Arbitrary Order Total Variation for Deformable Image Registration

In this work, we investigate image registration in a variational framework and focus on regularization generality and solver efficiency. We first propose a variational model combining the state-of-the-art sum of absolute differences (SAD) and a new arbitrary order total variation regularization term. The main advantage is that this variational model preserves discontinuities in the resultant deformation while being robust to outlier noise. It is however non-trivial to optimize the model due to its non-convexity, non-differentiabilities, and generality in the derivative order. To tackle these, we propose to first apply linearization to the model to formulate a convex objective function and then break down the resultant convex optimization into several point-wise, closed-form subproblems using a fast, over-relaxed alternating direction method of multipliers (ADMM). With this proposed algorithm, we show that solving higher-order variational formulations is similar to solving their lower-order counterparts. Extensive experiments show that our ADMM is significantly more efficient than both the subgradient and primal-dual algorithms particularly when higher-order derivatives are used, and that our new models outperform state-of-the-art methods based on deep learning and free-form deformation. Our code implemented in both Matlab and Pytorch is publicly available at https://github.com/j-duan/AOTV.

Simulation of GPR B-Scan Data Based on Dense Generative Adversarial Network

Urban subsurface infrastructures, e.g., pipelines and roads, are aging with the expansion of modern cities. Benefiting from the capability of nondestructive detection, ground penetrating radar (GPR) has been widely applied to underground objects or disasters detection, and GPR B-scan images are employed by manual interpretation. While, this way of high subjectivity and uncertainty inevitably results in failure of detection. Meanwhile, the shortage of labelled images greatly impedes the automatization and intelligentization of underground disaster detection based on GPR. Many data simulation techniques, e.g., forward modelling, were used to augment images for training; however, the generated forward images were not similar enough to the real B-scan data, which makes recognition a challenging task. To address this problem, we proposed a novel B-scan image simulation method based on generative adversarial network to generate synthetic images for training detection networks. Our network utilizes DenseNet as the backbone network of generator to extract image features, and a weighted total variation regularization term to regularize the loss function of the network. The comparison and ablation experiments verified that our network could generate simulation images with high similarity to real GPR B-scan images. We believe that this work contributes to the intelligent processing and analysis of GPR data, and improves the efficiency of underground disaster detection.

Correlation-Based Mutual Information Model for Analysis of Lung Cancer CT Image.

Most of the people all over the world pass away from complications related to lung cancer every single day. It is a deadly form of the disease. To improve a person's chances of survival, an early diagnosis is a necessary prerequisite. In this regard, the existing methods of tumour detection, such as CT scans, are most commonly used to recognize infected regions. Despite this, there are certain obstacles presented by CT imaging, so this paper proposes a novel model which is a correlation-based model designed for analysis of lung cancer. When registering pictures of thoracic and abdominal organs with slider motion, the total variation regularization term may correct the border discontinuous displacement field, but it cannot maintain the local characteristics of the image and loses the registration accuracy. The thin-plate spline energy operator and the total variation operator are spatially weighted via the spatial position weight of the pixel points to construct an adaptive thin-plate spline total variation regular term for lung image CT single-mode registration and CT/PET dual-mode registration. The regular term is then combined with the CRMI similarity measure and the L-BFGS optimization approach to create a nonrigid registration procedure. The proposed method assures the smoothness of interior of the picture while ensuring the discontinuous motion of the border and has greater registration accuracy, according to the experimental findings on the DIR-Lab 4D-CT public dataset and the CT/PET clinical dataset.

Fast algorithm for box‐constrained fractional‐order total variation image restoration with impulse noise

In this paper, a novel variational model with box constraints is proposed to restore images corrupted by impulse noise. The proposed model is composed of fractional-order total variation regularization and Lp-fidelity term ( 0 < p < 1 ) $(0<p<1)$ . Moreover, the new model possesses the advantages of preserving sharp edges and removing blocking effect. To solve the proposed model, some auxiliary variables are first introduced to transform it into some easy-to-solve subproblems. Further, the alternating direction method of multipliers, iteratively re-weighted ℓ1 algorithm and fast iteration technique are adopted to solve the related subproblems. Numerical results show that the proposed model performs better in comparison with the several existing methods, in terms of both quantitative evaluation and visual quality.

Suppressing multi-material and streak artifacts with an accelerated 3D iterative image reconstruction algorithm for in-line X-ray phase-contrast computed tomography.

In-line X-ray phase-contrast computed tomography typically contains two independent procedures: phase retrieval and computed tomography reconstruction, in which multi-material and streak artifacts are two important problems. To address these problems simultaneously, an accelerated 3D iterative image reconstruction algorithm is proposed. It merges the above-mentioned two procedures into one step, and establishes the data fidelity term in raw projection domain while introducing 3D total variation regularization term in image domain. Specifically, a transport-of-intensity equation (TIE)-based phase retrieval method is updated alternately for different areas of the multi-material sample. Simulation and experimental results validate the effectiveness and efficiency of the proposed algorithm.

Edge adaptive hybrid regularization model for image deblurring

Parameter selection is crucial to regularization-based image restoration methods. Generally speaking, a spatially fixed parameter for the regularization term does not perform well for both edge and smooth areas. A larger parameter for the regularization term reduces noise better in smooth areas but blurs edge regions, while a small parameter sharpens edge but causes residual noise. In this paper, an automated spatially adaptive regularization model, which combines the harmonic and total variation (TV) terms, is proposed for the image reconstruction from noisy and blurred observation. The proposed model detects the edges and then spatially adjusts the parameters of Tikhonov and TV regularization terms for each pixel according to the edge information. Accordingly, the edge information matrix will also be dynamically updated during the iterations. Computationally, the newly-established model is convex, which can be solved by the semi-proximal alternating direction method of multipliers with a linear convergence rate. Numerical simulation results demonstrate that the proposed model effectively preserves the image edges and eliminates the noise and blur at the same time. In comparison to state-of-the-art algorithms, it outperforms other methods in terms of peak signal to noise ratio, structural similarity index and visual quality.

Efficient Boosted DC Algorithm for Nonconvex Image Restoration with Rician Noise

Image deblurring under Rician noise has attracted considerable attention in imaging science. Frequently appearing in medical imaging, Rician noise leads to an interesting nonconvex optimization problem, termed as the MAP-Rician model, which is based on the Maximum a Posteriori (MAP) estimation approach. As the MAP-Rician model is deeply rooted in Bayesian analysis, we want to understand its mathematical analysis carefully. Moreover, one needs to properly select a suitable algorithm for tackling this nonconvex problem to get the best performance. This paper investigates both issues. Indeed, we first present a theoretical result about the existence of a minimizer for the MAP-Rician model under mild conditions. Next, we aim to adopt an efficient boosted difference of convex functions algorithm (BDCA) to handle this challenging problem. Basically, BDCA combines the classical difference of convex functions algorithm (DCA) with a backtracking line search, which utilizes the point generated by DCA to define a search direction. In particular, we apply a smoothing scheme to handle the nonsmooth total variation (TV) regularization term in the discrete MAP-Rician model. Theoretically, using the Kurdyka--Lojasiewicz (KL) property, the convergence of the numerical algorithm can be guaranteed. We also prove that the sequence generated by the proposed algorithm converges to a stationary point with the objective function values decreasing monotonically. Numerical simulations are then reported to clearly illustrate that our BDCA approach outperforms some state-of-the-art methods for both medical and natural images in terms of image recovery capability and CPU-time cost.