3D Seismic Edge-preserving Noise Reduction Method Using Cauchy Regularization of Directional Difference

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.

Regularization plays a crucial role in reliably utilizing imaging systems for scientific and medical investigations. It helps to stabilize the process of computationally undoing any degradation caused by physical limitations of the imaging process. In the past decades, total variation regularization played a dominant role in the literature. Two forms of total variation regularizations, namely the first-order and the second-order total variation (TV-1 and TV-2) have been widely used. TV-1 has a disadvantage: it reconstructs images in the form of piece-wise constants when the noise and/or under-sampling is severe, while TV-2 reconstructs natural-looking images under such scenarios. On the other hand, TV-1 can recover sharp jumps better than TV-2. Two forms of generalizations, namely Hessian-Schatten norm (HSN) regularization, and total generalized variation (TGV) regularization, have been proposed and have become significant developments in the area of regularization for imaging inverse problems owing to their performance. While the strength of TGV is that it can combine the advantages of TV-1 and TV-2, HSN has better structure-preserving property. Here, we develop a novel regularization for image recovery that combines the strengths of TGV and HSN. We achieve this by restricting the maximization space in the dual form of HSN in the same way that TGV is obtained from TV-2. We call the new regularization the generalized Hessian-Schatten norm regularization (GHSN). We develop a novel computational method for image reconstruction using the new form of regularization based on the well-known framework called the alternating direction method of multipliers (ADMM). We demonstrate the strength of the GHSN using some reconstruction examples.

The trace-by-trace seismic deconvolution methods for reducing band-limited wavelet interference often cause lateral instability of the estimated reflectivity or impedance, and yield some trails like noodles, corresponding to relatively high wave-number components, when the signal-to-noise ratio of data is low and/or the data include high wave-number noise (or error). In this paper, a modified total variation (TV) regularization by linearly combining the sparseness regularization of model difference along the temporal direction and that along the spatial direction, where spatial regularization can be described as L1 norm of difference model along the interpreted horizontal direction, is introduced to develop a seismic deconvolution method to reduce even overcome these problems. The TV regularization connects the reflectivity at a point with its adjacent points and implicitly with any other points, thus introduces some additional useful information, especially for spatial continuity. In theory, minimizing TV has the ability to preserve edges of model parameters. A synthetic data example and a field data example are adopted to illustrate the performances of the method especially for the inherent role of sparseness regularization of model difference along the temporal and spatial directions.

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

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.

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.

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.

The existence of coherent and incoherent (random) noises in the ground penetration radar (GPR) signals using high-frequency electromagnetic waves is inevitable; therefore, the elimination of noise from GPR data before performing any additional analysis is of great importance to increase the efficiency of the interpretations. We apply the Total Variation De-noising (TVD) and Savitzky-Golay (SG) filter, on the synthetic and real GPR data. For a better perception, we compare the same trace of the data before and after the experiences. The results indicate that the TVD method is more effective than common adaptive filtering in the time domain for reducing noise such as the SG filter which acts as a low-pass filter for smoothing data based on a polynomial least-squares approximation; however, due to the visibility of staircase artifacts using the TVD method, GPR data is first transferred to the empirical mode decomposition (EMD) frame which is useful for non-linear and non-stationary signal processing, then the TVD method is applied on it; ultimately, noise reduction using TVD is compared in the time and EMD domains. The comparison of the outputs shows that the TVD algorithm in the EMD domain, based on the sequential extraction of the energy belonging to the different intrinsic time scales of the signal, provides better noise attenuation than the other algorithms. In addition, TVD-EMD preserves the event forms, signal forms and improves the continuity in sections.

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.

Total variation (TV) denoising is an effective noise suppression method when the derivative of the underlying signal is known to be sparse. TV denoising is defined in terms of a convex optimization problem involving a quadratic data fidelity term and a convex regularization term. A non-convex regularizer can promote sparsity more strongly, but generally leads to a non-convex optimization problem with non-optimal local minima. This letter proposes the use of a non-convex regularizer constrained so that the total objective function to be minimized maintains its convexity. Conditions for a non-convex regularizer are given that ensure the total TV denoising objective function is convex. An efficient algorithm is given for the resulting problem.

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.

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.

In this paper, we propose a variational model for Polarimetric synthetic aperture radar (PolSAR) data speckle reduction. Since the statistical characteristic of covariance matrix can be well described by the complex Wishart distribution, a data fitting term can be derived according to the maximum likelihood rule. Combining it with the total variation (TV) regularization gives a variational model for PolSAR image speckle reduction. An algorithm based on the alternating minimization technique is used to solve the variational problem. Experimental results on real PolSAR images demonstrate the validity of the proposed method.

In the field of magnetic anomaly detection (MAD), the anomaly signal is easily submerged by ambient electromagnetic interference. Though the existing noise suppression methods can effectively improve the signal-to-noise ratio (SNR), there are still some intractable problems, such as signal distortion and boundary blur. To solve these problems, a novel MAD method based on structured low rank (SLR) and total variation (TV) regularization constraints is proposed in this letter. The noise suppression performance is improved by leveraging the structured low rankness of the signal. To preserve clean boundaries of the anomalies, an anisotropic TV regularization constraint is employed in the approach. Comparing the SLR-TV method with four state-of-the-art methods with extensive field tests, the results demonstrate that the proposed SLR-TV method achieves the greatest SNR improvement by about 63.24% and the best structural similarity (SSIM) improvement by about 53.02% over other methods in the range from −40 to 0 dB, showing the utility and high fidelity of the proposed framework in low SNR.

Application of Regularized Richardson-Lucy Algorithm for Deconvolution of Confocal Microscopy Images