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Abstract In this paper, we introduce a new iterative forward–backward splitting algorithm with errors for solving the split monotone variational inclusion problem of the sum of two monotone operators in real Hilbert spaces. We suggest and analyze this method under some mild appropriate conditions imposed on the parameters such that another strong convergence theorem for this problem is obtained. We also apply our main result to image-feature extraction with the multiple-image blends problem, the split minimization problem, and the convex minimization problem, and provide numerical experiments to illustrate the convergence behavior and show the effectiveness of the sequence constructed by the inertial technique.

In this paper, we introduce a new iterative forward–backward splitting algorithm with errors for solving the split monotone variational inclusion problem of the sum of two monotone operators in real Hilbert spaces. We suggest and analyze this method under some mild appropriate conditions imposed on the parameters such that another strong convergence theorem for this problem is obtained. We also apply our main result to image-feature extraction with the multiple-image blends problem, the split minimization problem, and the convex minimization problem, and provide numerical experiments to illustrate the convergence behavior and show the effectiveness of the sequence constructed by the inertial technique. [ABSTRACT FROM AUTHOR]

As an advanced technique, compressed sensing has been used for rapid magnetic resonance imaging in recent years, Two-step Iterative Shrinkage Thresholding Algorithm (TwIST) is a popular algorithm based on Iterative Thresholding Shrinkage Algorithm (ISTA) for fast MR image reconstruction. However TwIST algorithms cannot dynamically adjust shrinkage factor according to the degree of convergence. So it is difficult to balance speed and efficiency. In this paper, we proposed an algorithm which can dynamically adjust the shrinkage factor to rebalance the fidelity item and regular item during TwIST iterative process. The shrinkage factor adjusting is judged by the previous reconstructed results throughout the iteration cycle. It can greatly accelerate the iterative convergence while ensuring convergence accuracy. We used MR images with 2 body parts and different sampling rates to simulate, the results proved that the proposed algorithm have a faster convergence rate and better reconstruction performance. We also used 60 MR images of different body parts for further simulation, and the results proved the universal superiority of the proposed algorithm.

Abstract In this paper, we introduce a new iterative forward-backward splitting method with an error for solving the variational inclusion problem of the sum of two monotone operators in real Hilbert spaces. We suggest and analyze this method under some mild appropriate conditions imposed on the parameters such that another strong convergence theorem for these problem is obtained. We also apply our main result to improve the fast iterative shrinkage thresholding algorithm (IFISTA) with an error for solving the image deblurring problem. Finally, we provide numerical experiments to illustrate the convergence behavior and show the effectiveness of the sequence constructed by the inertial technique to the fast processing with high performance and the fast convergence with good performance of IFISTA.

K-space under-sampling reconstruction technology is an effective means to improve the speed of magnetic resonance imaging. Among its many reconstruction algorithms, split Bregman iteration is an effective method to solve multi-constrained models. This model often contains TV variational regularization terms, the generalized threshold shrinkage operator often used to solve TV constraints subproblem. However, when the generalized threshold shrinkage operator is performing the shrinking operation, it does not consider the inconsistency of the elements in the image matrix, which will cause the loss of image details. In response to this problem, in this paper, a non-uniform threshold shrinkage operator was proposed to solve above TV constraints subproblem, which can dynamically adjust the shrinkage threshold by the residuals of each image element. And introduce this operator when performing Split Bregman iteration to improve the performance of generalized threshold shrinkage. After qualitative and quantitative analysis during the experiments, it can be concluded that compared with the other three methods, the proposed method has better performance in terms of Peak Signal to Noise Ratio (PSNR), Structural Similarity Index Measure (SSIM), Transferred Edge Information(TEI) and Normalized Mutual Information(NMI), and the visual perception is better. Then we also did denoising performance analysis at different noise levels, this method also showed good robustness. The proposed method can improve the reconstruction performance of TV constrained subproblem in split Bregman iteration, and then improve the overall performance of reconstruction algorithm. Moreover, this method also shows good denoising performance at different noise levels. [ABSTRACT FROM AUTHOR]

Various deconvolution algorithms for acoustic source are developed to improve spatial resolution and suppress sidelobe of the conventional beamforming. To improve the computational efficiency and solution convergence of deconvolution, this paper proposes a Fourier-based improved fast iterative shrinkage thresholding algorithm. Simulations and experiments show that Fourier-based improved fast iterative shrinkage thresholding algorithm can achieve excellent acoustic identification performance, with high computational efficiency and good convergence. For Fourier-based improved fast iterative shrinkage thresholding algorithm, the larger the weight coefficient, the narrower the mainlobe width, and the better the convergence, but the spurious source also increases. The recommended weight coefficient for the array described herein is 3. In addition, like other Fourier-based deconvolution algorithms, Fourier-based improved fast iterative shrinkage thresholding algorithm using irregular focus grid can obtain better acoustic source identification performance than using the conventional regular focus grid. [ABSTRACT FROM AUTHOR]

To reduce the magnetic resonance imaging (MRI) data acquisition time and improve the MR image reconstruction performance, reconstruction algorithms based on the iterative shrinkage thresholding algorithm (ISTA) are widely used. However, these traditional algorithms use global threshold shrinkage, which is not efficient. In this paper, a novel algorithm based on local threshold shrinkage, which is called the local shrinkage thresholding algorithm (LSTA), was proposed. The LSTA can shrink differently for different elements from the residual matrix to adjust the shrinkage speed for each element of the image during the iterative process. Then, by taking advantage of the sparser characteristics of the curvelet transform, the LSTA combined with the curvelet transform (CLSTA) can make the construction process more efficient. Finally, compared with ISTA, the generalized thresholding iterative algorithm (GTIA) and the fast iterative shrinkage threshold algorithm (FISTA), when analysing human (brain and cervical) MR images, a conclusion can be drawn that the proposed method has better reconstruction performance in terms of the mean square error (MSE), the peak signal to noise ratio (PNSR), the structural similarity index measure (SSIM), the normalized mutual information (NMI), the transferred edge information (TEI) and the number of iterations. The proposed method can better maintain the detailed information of the reconstructed images and effectively decrease the blurring of the images edges. Graphical abstract. [ABSTRACT FROM AUTHOR]

Abstract In order to accelerate magnetic resonance imaging (MRI) scanning, fast MRI technique based on compressed sensing (CS) was proposed. The shrinkage thresholding algorithm (STA) is an efficient method in related algorithms to decrease the incoherent artifacts produced by the undersampling in k-space directly. The traditional STA uses the fixed iteration step size during the reconstruction progress, and it is not conducive to accelerate the convergence speed. In order to improve global iteration efficiency, in this paper, step adaptive fast iterative shrinkage thresholding algorithm (SAFISTA) was proposed for MRI reconstruction based on STA. It used a feedback to dynamically adjust the iteration step size. The feedback parameter was calculated from the total variations (TV) of two previous iterations. It can effectively improve the efficiency of iteration. Experiments over three kinds of MR images (human head, blood vessels and knee) under different sample ratios indicated that the proposed algorithm SAFISTA showed better reconstruction performance than iterative shrinkage thresholding algorithm (ISTA), fast iterative shrinkage thresholding algorithm (FISTA) and generalized thresholding iterative algorithm (GTIA) in terms of mean square error (MSE), peak signal to noise ratio (PSNR) and structural similarity index measure (SSIM). [ABSTRACT FROM AUTHOR]

As an advanced technique, compressed sensing has been used for rapid magnetic resonance imaging in recent years, Two-step Iterative Shrinkage Thresholding Algorithm (TwIST) is a popular algorithm based on Iterative Thresholding Shrinkage Algorithm (ISTA) for fast MR image reconstruction. However TwIST algorithms cannot dynamically adjust shrinkage factor according to the degree of convergence. So it is difficult to balance speed and efficiency. In this paper, we proposed an algorithm which can dynamically adjust the shrinkage factor to rebalance the fidelity item and regular item during TwIST iterative process. The shrinkage factor adjusting is judged by the previous reconstructed results throughout the iteration cycle. It can greatly accelerate the iterative convergence while ensuring convergence accuracy. We used MR images with 2 body parts and different sampling rates to simulate, the results proved that the proposed algorithm have a faster convergence rate and better reconstruction performance. We also used 60 MR images of different body parts for further simulation, and the results proved the universal superiority of the proposed algorithm.

We present two algorithms to solve the Group Lasso problem (Yuan and Lin in, J R Stat Soc Ser B (Stat Methodol) 68(1):49-67, ). First, we propose a general version of the Block Coordinate Descent (BCD) algorithm for the Group Lasso that employs an efficient approach for optimizing each subproblem exactly. We show that it exhibits excellent performance when the groups are of moderate size. For groups of large size, we propose an extension of ISTA/FISTA SIAM (Beck and Teboulle in, SIAM J Imag Sci 2(1):183-202, ) based on variable step-lengths that can be viewed as a simplified version of BCD. By combining the two approaches we obtain an implementation that is very competitive and often outperforms other state-of-the-art approaches for this problem. We show how these methods fit into the globally convergent general block coordinate gradient descent framework in Tseng and Yun (Math Program 117(1):387-423, ). We also show that the proposed approach is more efficient in practice than the one implemented in Tseng and Yun (Math Program 117(1):387-423, ). In addition, we apply our algorithms to the Multiple Measurement Vector (MMV) recovery problem, which can be viewed as a special case of the Group Lasso problem, and compare their performance to other methods in this particular instance. [ABSTRACT FROM AUTHOR]