Generalized Hessian-Schatten Norm Regularization for Image Reconstruction

This paper, we propose a method that exploit total generalized variation (TGV) and Shearlet transform for image restoring. The TGV adaptive regularize different image regions at different levels and the Shearlet transform can efficiently represent image anisotropic features such as edges, curves. A new image restoration model combining TGV and Shearlet transform is proposed for image restoration. The proposed model is solved by splitting variables and applying the alternating direction method of multiplier (ADMM). Experimental results show that the proposed algorithm can effectively restore image and improve the quality.

It is required that input features are represented as vectors or scalars in machine learning for classification, e.g. support vector machine (SVM). However, real world data such as 2D images is naturally represented as matrices or tensors with higher dimensions. Thus, structural information of the data whose dimensions are more than two is not successfully considered. One typical structural information which is useful for the classification task is the spatial relationship of the nearby data points. In this paper, to leverage this kind of structural information, we propose a novel classification method which combines total variational (TV) regularization with SVM, called TV-SVM. Since TV achieves a local smoothing property by penalizing the local discontinuity of data, TV-SVM preserves better local structure than the original SVM due to TV regularization. We solve the objective function of TV-SVM via the alternating direction method of multipliers (ADMM) algorithm. Experimental results on image classification show that TV-SVM is competitive to the state-of-the-art learning method in both classification accuracy and computational complexity.

The use of non-local self-similarity prior between image blocks can improve image reconstruction performance significantly. We propose a compressive sensing image reconstruction algorithm that combines bilateral total variation and nonlocal low-rank regularization to overcome over-smoothing and degradation of edge information which result from the prior reconstructed image. The proposed algorithm makes use of the preservation of image edge information by bilateral total variation operator to enhance the edge details of the reconstructed image. In addition, we use weighted nuclear norm regularization as a low-rank constraint for similar blocks of the image. To solve this convex optimization problem, the Alternating Direction Method of Multipliers (ADMM) is employed to optimize and iterate the algorithm model effectively. Experimental results show that the proposed algorithm can obtain better image reconstruction quality than conventional algorithms with using total variation regularization or considering the nonlocal structure of the image only. At 10% sampling rate, the peak signal-to-noise ratio gain is up to 2.39 dB in noiseless measurements compared with Nonlocal Low-rank Regularization (NLR-CS). Reconstructed image comparison shows that the proposed algorithm retains more high frequency components. In noisy measurements, the proposed algorithm is robust to noise and the reconstructed image retains more detail information.

<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>

We introduce a total variation (TV) regularization model for synthetic aperture radar (SAR) image despeckling. A dual-formulation-based adaptive TV (ATV) regularization method is applied to solve the TV regularization. The parameter adaptation of the TV regularization is performed based on the noise level estimated via wavelets. The TV-regularization-based image restoration model has a good performance in preserving image sharpness and edges while removing noises, and it is therefore effective for edge preserve SAR image despeckling. Experiments have been carried out using optical images contaminated with artificial speckles first and then SAR images. A despeckling evaluation index (DEI) is designed to assess the effectiveness of edge preserve despeckling on SAR images, which is based on the ratio of the standard deviations of two neighborhood areas of different sizes of a pixel. Experimental results show that the proposed ATV method can effectively suppress SAR image speckles without compromising the edge sharpness of image features according to both subjective visual assessment of image quality and objective evaluation using DEI.

Efficient general sparse denoising with non-convex sparse constraint and total variation regularization

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.

To solve the ill-posed problem in electrical impedance tomography (EIT) and overcome the blur effect made by conventional L2 Tikhonov regularization, a new two-step iterative regularization algorithm is applied for the process reconstruction of lung respiration. This algorithm is based on total variation (TV) regularization method, and uses a two-step iterative method with TV denoising operator, so that it can achieve double regularization effect. This algorithm has advantages in both resolution and robustness. Both simulation and experiment results are executed to prove the effectiveness of TV regularization method.

ABSTRACTThis study newly demonstrates a compressed sensing (CS)-based reconstruction scheme for computed tomography (CT) with a translational trajectory. CT with a translational trajectory has many potential applications in the field of non-destructive testing (NDT) and medicine. For example, in the X-ray inspection of a large civil engineering structure, the movements of the source and detector are strictly limited, and a simple straight CT trajectory is preferable. This method could also be applied in X-ray inspection lines in factories and airports. In the medical field, this method would complement the tomosynthesis in mammography. When using a translational trajectory, the dispersion of the fan- or cone-beam creates projection data with the angular information of the sample object. However, the angular information inevitably becomes insufficient when using a translational trajectory, which degrades the reconstruction accuracy. In this case, CS is considered suitable for reconstruction because it has been successfully used for reconstructions from sparse-view and limited-angle data. For reconstruction from the translational trajectory projection, a new concept of directional difference (DD) regularization was proposed. An algorithm was developed based on the alternating direction method of multipliers (ADMM) algorithm to solve the regularization problem. Numerical reconstruction experiments from noisy projection data were conducted and the results were compared with those from other reconstruction methods. The convergence performance of the total variation (TV), DD, and mixed TV-DD regularization methods were examined. The proposed DD and TV-DD methods showed better performance than the TV only regularization. Reconstruction from Monte-Carlo simulated projection data was also demonstrated.

Spectral unmixing aims at identifying the pure spectral signatures in hyperspectral images and simultaneously estimating their proportions in each pixel of the scene. By using an available spectral library as a dictionary, sparse-regression-based approaches aim at finding a subset of the dictionary that can optimally model each pixel in a given hyperspectral image. regularizer has been widely considered as a regularization strategy to exploit the sparsity of the unmixing solution. Further sparsity can be imposed by also using weighting factors. However, most existing strategies focus on the unmixing solution ignoring the gradient information. To account for the gradient information in hyperspectral unmixing, we propose a weighted sparse regression with total variation (WSRTV) unmixing model. The proposed WSRTV model incorporates gradient information in the sparse regression formulation by means of the weighted total variation (WTV) regularizer. The model imposes sparsity on both the solution and the gradient to improve the performance of unmixing. A dual symmetric Gauss-Seidel alternating direction method of multipliers (sGSADMM) is designed to optimize the proposed model. The designed algorithm both handles the anisotropic and isotropic WTV. Simulated and real hyperspectral data demonstrate the effectiveness of the proposed framework.

A common way for solving the denosing problem is to utilize the total variation (TV) regularization. Many efficient numerical algorithms have been developed for solving the TV regularization problem. Condat described a fast direct algorithm to compute the processed 1D signal. Also there exists a direct algorithm with a linear time for 1D TV denoising referred to as the taut string algorithm. The Condat’s algorithm is based on a dual problem to the 1D TV regularization. In this paper, we propose a variant of the Condat’s algorithm based on the direct 1D TV regularization problem. The usage of the Condat’s algorithm with the taut string approach leads to a clear geometric description of the extremal function. Computer simulation results are provided to illustrate the performance of the proposed algorithm for restoration of degraded signals.

This paper proposes a nonlinear weighted anisotropic total variation (NWATV) regularization technique for electrical impedance tomography (EIT). The key idea is to incorporate the internal inhomogeneity information (e.g., edges of the detected objects) into the EIT reconstruction process, aiming to preserve the conductivity profiles (to be detected). We study the NWATV image reconstruction by employing a novel soft thresholding based reformulation included in the alternating direction method of multipliers (ADMM). To evaluate the proposed approach, numerical simulations and human EIT lung imaging are carried out. It is demonstrated that the properties of the internal inhomogeneity are well preserved and improved with the proposed regularization approach, in comparison to traditional total variation (TV) and recently proposed fidelity embedded regularization approaches. Owing to the simplicity of the proposed method, the computational cost is significantly decreased compared with the well established primal-dual algorithm. Precisely, with the proposed algorithm, we are able to alleviate the staircase effect arising in TV regularization and improve the reconstruction accuracy for FER does. To achieve similar accuracy as TV does, the computational times are reduced from 2.311s to 0.629s and 1.733s to 0.428s in 2D and 3D simulations, respectively, and the computational time is reduced from greater than 2s to less than 0.2s in the human experiment. Meanwhile, it was found that the proposed regularization method is quite robust to the measurement noise, which is one of the main uncertainties in EIT.

Total variation regularization is well-known for recovering sharp edges; however, it usually produces staircase artifacts. In this paper, in order to overcome the shortcoming of total variation regularization, we propose a new variational model combining high-order total variation regularization and l1 regularization. The new model has separable structure which enables us to solve the involved subproblems more efficiently. We propose a fast alternating method by employing the fast iterative shrinkage-thresholding algorithm (FISTA) and the alternating direction method of multipliers (ADMM). Compared with some current state-of-the-art methods, numerical experiments show that our proposed model can significantly improve the quality of restored images and obtain higher SNR and SSIM values.

Application of hybrid regularization method for tomographic reconstruction of midlatitude ionospheric electron density

Magnetic resonance imaging (MRI) reconstruction model based on total variation (TV) regularization can deal with problems such as incomplete reconstruction, blurred boundary, and residual noise. In this article, a non‐convex isotropic TV regularization reconstruction model is proposed to overcome the drawback. Moreau envelope and minmax‐concave penalty are firstly used to construct the non‐convex regularization of L2 norm, then it is applied into the TV regularization to construct the sparse reconstruction model. The proposed model can extract the edge contour of the target effectively since it can avoid the underestimation of larger nonzero elements in convex regularization. In addition, the global convexity of the cost function can be guaranteed under certain conditions. Then, an efficient algorithm such as alternating direction method of multipliers is proposed to solve the new cost function. Experimental results show that, compared with several typical image reconstruction methods, the proposed model performs better. Both the relative error and the peak signal‐to‐noise ratio are significantly improved, and the reconstructed images also show better visual effects. The competitive experimental results indicate that the proposed approach is not limited to MRI reconstruction, but it is general enough to be used in other fields with natural images.