Truncated iterative thresholding algorithm for compressed sensing via fraction function

Multilevel minimum cross entropy thresholding: A comparative study

The application of deep learning in compressed sensing reconstruction has achieved some excellent results. The deep neural network based on iterative algorithm can not only reflect the excellent performance of deep learning, but also reflect the interpretability of traditional compressed sensing reconstruction algorithm. The existing deep neural networks based on iterative algorithm mainly include learned iterative shrinkage threshold algorithm(LISTA), analytic learned iterative shrinkage threshold algorithm(ALISTA), etc., but each of them has its own shortcomings.We improved the network structure on the basis of predecessors, and proposed a new custom loss function to effectively improve the reconstruction performance of compressed sensing. Experiments show that our proposed neural network reduces the normalized mean square error(NMSE) by 10 dB compared with ALISTA and 15 dB compared with LISTA, and the support set accuracy of the recovered sparse signal can be optimized by our proposed custom loss function by at least 5%, especially LISTA, which has been improved by at least 80%.

Finding the sparset solution of an underdetermined system of linear equations y=Ax has attracted considerable attention in recent years. Among a large number of algorithms, iterative thresholding algorithms are recognized as one of the most efficient and important classes of algorithms. This is mainly due to their low computational complexities, especially for large scale applications. The aim of this paper is to provide guarantees on the global convergence of a wide class of iterative thresholding algorithms. Since the thresholds of the considered algorithms are set adaptively at each iteration, we call them adaptively iterative thresholding (AIT) algorithms. As the main result, we show that as long as A satisfies a certain coherence property, AIT algorithms can find the correct support set within finite iterations, and then converge to the original sparse solution exponentially fast once the correct support set has been identified. Meanwhile, we also demonstrate that AIT algorithms are robust to the algorithmic parameters. In addition, it should be pointed out that most of the existing iterative thresholding algorithms such as hard, soft, half and smoothly clipped absolute deviation (SCAD) algorithms are included in the class of AIT algorithms studied in this paper.

Seismic inversion is a process of imaging or predicting the spatial and physical properties of underground strata. The most commonly used one is sparse-spike seismic inversion with sparse regularization. There are many effective methods to solve sparse regularization, such as L0-norm, L1-norm, weighted L1-norm, etc. This paper studies the sparse-spike inversion with L0-norm. It is usually solved by the iterative hard thresholding algorithm (IHTA) or its faster variants. However, hard thresholding algorithms often lead to a sharp increase or numerical oscillation of the residual, which will affect the inversion results. In order to deal with this issue, this paper attempts the idea of the relaxed optimal thresholding algorithm (ROTA). In the solution process, due to the particularity of the sparse constraints in this convex relaxation model, this model can be considered as a L1-norm problem when dealt with the location of non-zero elements. We use a modified iterative soft thresholding algorithm (MISTA) to solve it. Hence, it forms a new algorithm called the iterative hybrid thresholding algorithm (IHyTA), which combines IHTA and MISTA. The synthetic and real seismic data tests show that, compared with IHTA, the results of IHyTA are more accurate with the same SNR. IHyTA improves the noise resistance.

Regularization algorithms have been investigated extensively to solve the ill-posed inverse problem of electrical tomography. Sparse regularization algorithms with sparsity constrains have become popular in recent years. The iterative shrinkage thresholding algorithms have been applied to deal with the sparse regularization due to their simplicity and low calculation cost. However, the performance of the reconstructed images varies with the thresholding parameter and initial parameters of the iterative thresholding algorithm, which are selected manually. Inspired by the iterative varied thresholding operator, a fast iterative updated thresholding algorithm is proposed for electrical resistance tomography (ERT) and further a new scheme for updating the thresholding parameter adaptively during the iteration process is designed. More penalty is implemented with a larger thresholding parameter when the sparsity is reduced, and less penalty is implemented with a smaller thresholding parameter when the sparsity is increased. In addition, a speedup step is exploited in order to accelerate the progress. This proposed method is verified quantitatively in numerical simulation as well as in experiment test on a practical ERT system. Moreover, the impacts of different initial parameters are discussed in detailed, the simulation results demonstrate that the proposed method is almost unaffected by different initial parameters. The advantage of this method is that a higher spatial resolution image with a faster solving speed can be reconstructed with less iterations. The results indicate that the quality of images reconstructed by this proposed method outperforms that of traditional methods whether in size or location of the inclusion. It also has a stronger ability in preserving edges and noise immunity. Furthermore, the proposed method can be applied to image reconstruction in other kinds of tomography.

The special importance of L1/2 regularization has been recognized in recent studies on sparse modeling (particularly on compressed sensing). The L1/2 regularization, however, leads to a nonconvex, nonsmooth, and non-Lipschitz optimization problem that is difficult to solve fast and efficiently. In this paper, through developing a threshoding representation theory for L1/2 regularization, we propose an iterative half thresholding algorithm for fast solution of L1/2 regularization, corresponding to the well-known iterative soft thresholding algorithm for L1 regularization, and the iterative hard thresholding algorithm for L0 regularization. We prove the existence of the resolvent of gradient of ||x||1/2(1/2), calculate its analytic expression, and establish an alternative feature theorem on solutions of L1/2 regularization, based on which a thresholding representation of solutions of L1/2 regularization is derived and an optimal regularization parameter setting rule is formulated. The developed theory provides a successful practice of extension of the well- known Moreau's proximity forward-backward splitting theory to the L1/2 regularization case. We verify the convergence of the iterative half thresholding algorithm and provide a series of experiments to assess performance of the algorithm. The experiments show that the half algorithm is effective, efficient, and can be accepted as a fast solver for L1/2 regularization. With the new algorithm, we conduct a phase diagram study to further demonstrate the superiority of L1/2 regularization over L1 regularization.

Step adaptive fast iterative shrinkage thresholding algorithm for compressively sampled MR imaging reconstruction

We present a discrepancy-based parameter choice and stopping rule for iterative algorithms performing approximate Tikhonov-functional minimization which adapts the regularization parameter value during the optimization procedure. The suggested parameter choice and stopping rule can be applied to a wide class of penalty terms and iterative algorithms which aim at Tikhonov regularization with a fixed parameter value. It leads, in particular, to computable guaranteed estimates for the regularized exact discrepancy in terms of numerical approximations. Based on these estimates, convergence to a solution is shown. As an example, the developed theory and the algorithm is applied to the case of sparse regularization. We prove order optimal convergence rates in the case of sparse regularization, i.e. weighted ℓp norms, which turn out to be the same as for the a priori parameter choice rule already obtained in the literature as well as for Morozov’s principle applied to exact regularized solutions. Finally, numerical results for two different minimization techniques, iterative soft thresholding algorithm and monotone fast iterative soft thresholding algorithm, are presented, confirming, in particular, the results from the theory.

Terahertz sparse deconvolution based on an iterative shrinkage and thresholding algorithm (ISTA) has been used to characterize multilayered structures with resolution equivalent to or finer than the sampling period of the measurement. However, this method was only studied on thin samples to separate the overlapped echos that can't be distinguished by other deconvolution algorithms. Besides, ISTA heavily depends on the convolution matrix consisting of delayed incident pulse, which is difficult to precisely extricate from the reference signal, and thereby fluctuations caused by noise are occasionally treated as echos. In this work, a terahertz sparse deconvolution based on a learned iterative shrinkage and thresholding algorithm (LISTA) is proposed. The method enclosed the matrix multiplication and soft thresholding in a block and cascaded multiple blocks together to form a deep network. The convolution matrices of the network were updated by stochastic gradient descent to minimize the distance between the output sparse vector and the optimal sparse representation of the signal, and subsequently the trained network made more precise estimation of the echos than ISTA. Additionally, LISTA is notably faster than ISTA, which is important for real-time tomographic-image processing. The algorithm was evaluated on terahertz tomographic imaging of a high-density poly ethylene (HDPE) sample, revealing obvious improvements in detecting defects of different sizes and depths. This technique has potential usage in nondestructive testings of thick samples, where echos reflected by minor defects are not discernible by existed deconvolution algorithms.

Performance analysis for a class of iterative image thresholding algorithms

We propose iterative thresholding algorithms based on theiterated Tikhonov method for image deblurring problems.Our method is similar in idea to the modified linearized Bregman algorithm (MLBA) sois easy to implement. In order to obtain good restorations, MLBA requires an accurate estimate of the regularization parameter $\alpha$ whichis hard to get in real applications.Based on previous results in iterated Tikhonov method, we designtwo nonstationary iterative thresholding algorithms which give nearoptimal results without estimating $\alpha$. One of them is basedon the iterative soft thresholding algorithm and the other is basedon MLBA. We show that the nonstationary methods,if converge, will converge to the same minimizers ofthe stationary variants.Numerical resultsshow that the accuracy and convergence of our nonstationary methods are very robustwith respect to the changes in the parameters and the restorationresults are comparable to those of MLBA with optimal $\alpha$.

Compressively sampled MR image reconstruction using generalized thresholding iterative algorithm

The fast and reliable processing of medical images is of paramount importance to adequately generate data to feed machine learning algorithms that can prevent and diagnose health issues. Here, different compressed sensing techniques applied to magnetic resonance imaging are benchmarked as a means to reduce the acquisition time spent in the collection of data and signals that form the image. It is shown that by using these techniques, it is possible to reduce the number of signals needed and, therefore, substantially decrease the time to acquire the measurements. To this end, different algorithms are considered and compared: the iterative re-weighted least squares, the iterative soft thresholding algorithm, the iterative hard thresholding algorithm, the primal dual algorithm and the log barrier algorithm. Such algorithms have been implemented in different analysis programs that have been used to perform the reconstruction of the images, and it was found that the iterative soft thresholding algorithm gives the optimal results. It is found that the images obtained with this algorithm have lower quality than the original ones, but in any case, the quality should be good enough to distinguish each body structure and detect any health problems under an expert evaluation and/or statistical analysis.

With the continuous expansion of the market of device-free localization in smart cities, the requirements of device-free localization technology are becoming higher and higher. The large amount of high-dimensional data generated by the existing device-free localization technology will improve the positioning accuracy as well as increase the positioning time and complexity. The positions required from single target to multi-targets become a further increasing difficulty for device-free localization. In order to satisfy the practical localizing application in smart city, an efficient multi-target device-free localization method is proposed based on a sparse coding model. To accelerate the positioning as well as improve the localization accuracy, a sparse coding-based iterative shrinkage threshold algorithm (SC-IA) is proposed and a subspace sparse coding-based iterative shrinkage threshold algorithm (SSC-IA) is presented for different practical application requirements. Experiments with practical dataset are performed for single-target and multi-targets localization, respectively. Compared with three typical machine learning algorithms: deep learning based on auto encoder, K-nearest neighbor, and orthogonal matching pursuit, experimental results show that the proposed sparse coding-based iterative shrinkage threshold algorithm and subspace sparse coding-based iterative shrinkage threshold algorithm can achieve high localization accuracy and low time cost simultaneously, so as to be more practical and applicable for the development of smart city.

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.