Hyperspectral unmixing using weighted sparse regression with total variation regularization

Spectral unmixing is an important technique in hyperspectral image applications. Recently, sparse regression has been widely used in hyperspectral unmixing, but its performance is limited by the high mutual coherence of spectral libraries. To address this issue, a new sparse unmixing algorithm, called double reweighted sparse unmixing and total variation (TV), is proposed in this letter. Specifically, the proposed algorithm enhances the sparsity of fractional abundances in both spectral and spatial domains through the use of double weights, where one is used to enhance the sparsity of endmembers in spectral library, and the other is introduced to improve the sparsity of fractional abundances. Moreover, a TV-based regularization is further adopted to explore the spatial-contextual information. As such, the simultaneous utilization of both double reweighted l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> minimization and TV regularizer can significantly improve the sparse unmixing performance. Experimental results on both synthetic and real hyperspectral data sets demonstrate the effectiveness of the proposed algorithm both visually and quantitatively.

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.

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.

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.

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.

Several bandwise total variation (TV) regularized low-rank (LR)-based models have been proposed to remove mixed noise in hyperspectral images (HSIs). Conventionally, the rank of LR matrix is approximated using nuclear norm (NN). The NN is defined by adding all singular values together, which is essentially a $L_{1}$ -norm of the singular values. It results in non-negligible approximation errors and thus the resulting matrix estimator can be significantly biased. Moreover, these bandwise TV-based methods exploit the spatial information in a separate manner. To cope with these problems, we propose a spatial–spectral TV (SSTV) regularized non-convex local LR matrix approximation (NonLLRTV) method to remove mixed noise in HSIs. From one aspect, local LR of HSIs is formulated using a non-convex $L_{\gamma}$ -norm, which provides a closer approximation to the matrix rank than the traditional NN. From another aspect, HSIs are assumed to be piecewisely smooth in the global spatial domain. The TV regularization is effective in preserving the smoothness and removing Gaussian noise. These facts inspire the integration of the NonLLR with TV regularization. To address the limitations of bandwise TV, we use the SSTV regularization to simultaneously consider global spatial structure and spectral correlation of neighboring bands. Experiment results indicate that the use of local non-convex penalty and global SSTV can boost the preserving of spatial piecewise smoothness and overall structural information.

Sparse unmixing has attracted much attention in recent years. It aims at estimating the fractional abundances of pure spectral signatures in mixed pixels in hyperspectral images. To exploit spatial-contextual information present in the scene, the total variation (TV) regularization is incorporated into the sparse unmixing formulation, promoting adjacent pixels having similar not only endmembers but also fractional abundances, and thus having similar structural sparsity. It is therefore hoped to impose joint sparsity, instead of classic single sparsity, on these adjacent pixels to further improve the unmixing performance. To this end, we include the joint-sparse-blocks regression into the TV spatial regularization framework and present a new unmixing algorithm, termed joint-sparse-blocks unmixing via variable splitting augmented Lagrangian and total variation (JSBUnSAL-TV). In particular, a reweighting strategy is utilized to enhance sparsity along lines within each block. Simulated and real-data experiments show the advantages of the proposed algorithm.

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.

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

The preservation of discontinuous edges in seismic dataset such as faults, fractures, pinch-outs, channels and salt bodies while attenuating random noise is of great significance for the following seismic data interpretation. Noise reduction using inversion with total variation (TV) regularization is a kind of 2D edge-preserving filtering method. However, TV regularization is only limited to 2D data. In this paper, we extend the 2D method to 3D case by firstly taking difference of seismic data along different directions, and then performing Cauchy regularization. A synthetic 3D dataset, a real stacked OBC dataset and a real stacked land dataset are used to illustrate the performance of the method. The results show that the 3D seismic noise-reduction method we proposed can perfectly preserve edges of 3D discontinuous events while reducing random noise. In addition, there is no obvious trail of noise along the spatial directions since 3D noise-reduction scheme is adopted.

The total variation (TV) regularization method has been used to solve the ill-posed inverse problem of electrical resistance tomography (ERT), owing to its good ability to preserve edges. However, the quality of the reconstructed images, especially in the flat region, is often degraded by noise. To optimize the regularization term and the regularization factor according to the spatial feature and to improve the resolution of reconstructed images, a spatially adaptive total variation (SATV) regularization method is proposed. A kind of effective spatial feature indicator named difference curvature is used to identify which region is a flat or edge region. According to different spatial features, the SATV regularization method can automatically adjust both the regularization term and regularization factor. At edge regions, the regularization term is approximate to the TV functional to preserve the edges; in flat regions, it is approximate to the first-order Tikhonov (FOT) functional to make the solution stable. Meanwhile, the adaptive regularization factor determined by the spatial feature is used to constrain the regularization strength of the SATV regularization method for different regions. Besides, a numerical scheme is adopted for the implementation of the second derivatives of difference curvature to improve the numerical stability. Several reconstruction image metrics are used to quantitatively evaluate the performance of the reconstructed results. Both simulation and experimental results indicate that, compared with the TV (mean relative error 0.288, mean correlation coefficient 0.627) and FOT (mean relative error 0.295, mean correlation coefficient 0.638) regularization methods, the proposed SATV (mean relative error 0.259, mean correlation coefficient 0.738) regularization method can endure a relatively high level of noise and improve the resolution of reconstructed images.

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.