Iterative Thresholding Algorithm Research Articles

Group-Based Sparse Representation for Compressed Sensing Image Reconstruction with Joint Regularization

Achieving high-quality reconstructions of images is the focus of research in image compressed sensing. Group sparse representation improves the quality of reconstructed images by exploiting the non-local similarity of images; however, block-matching and dictionary learning in the image group construction process leads to a long reconstruction time and artifacts in the reconstructed images. To solve the above problems, a joint regularized image reconstruction model based on group sparse representation (GSR-JR) is proposed. A group sparse coefficients regularization term ensures the sparsity of the group coefficients and reduces the complexity of the model. The group sparse residual regularization term introduces the prior information of the image to improve the quality of the reconstructed image. The alternating direction multiplier method and iterative thresholding algorithm are applied to solve the optimization problem. Simulation experiments confirm that the optimized GSR-JR model is superior to other advanced image reconstruction models in reconstructed image quality and visual effects. When the sensing rate is 0.1, compared to the group sparse residual constraint with a nonlocal prior (GSRC-NLR) model, the gain of the peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) is up to 4.86 dB and 0.1189, respectively.

A Nonconvex Method to Low-Rank Matrix Completion

In recent years, the problem of recovering a low-rank matrix from partial entries, known as low-rank matrix completion problem, has attracted much attention in many applications. However, it is a NP-hard problem due to the nonconvexity nature of the matrix rank function. In this paper, a rank Laplace function is studied to recover the low-rank matrices. Firstly, we propose an iterative Laplace thresholding algorithm to solve the regularized Laplace low-rank matrix completion problem. Secondly, some other iterative thresholding algorithms are designed to recover the low-rank matrices. Finally, we provide a series of numerical simulations to test the proposed algorithms on some low-rank matrix completion and image inpainting problems, and the results show that our algorithms perform better than some state-of-art methods in recovering the low-rank matrices.

Fast Independent Component Analysis Denoising for Magnetotelluric Data Based on a Correlation Coefficient and Fast Iterative Shrinkage Threshold Algorithm

Magnetotelluric (MT) sounding data are easily contaminated by various noise sources, especially noise with a long duration (even full-time noise), which makes it difficult to obtain accurate values when calculating the weight factors of the response function, resulting in distortion of the response results. Based on blind source separation theory, fast independent component analysis (FastICA) can separate this kind of noise. However, this method is challenged by the number of separated field sources and the unequal signal amplitude before and after processing. We have developed a novel signal noise separation method, improving the traditional FastICA, where a correlation coefficient is used to compute the number of field sources, and the fast iterative shrinkage threshold algorithm (FISTA) is used to adjust the signal amplitude problem before and after FastICA decomposition. Compared with common field source division methods and the traditional FastICA, the experimental results indicate that our method can separate and remove noise with a long duration, increase the signal-to-noise ratio of the data, and improve the MT response curves. Meanwhile, case studies of measured data illustrate that our method obtains a more robust magnetotelluric response than the conventional robust method and FastICA.

Model-Driven Enhanced Analytic Learned Iterative Shrinkage Threshold Algorithm

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%.

Hybrid ISTA: Unfolding ISTA With Convergence Guarantees Using Free-Form Deep Neural Networks.

It is promising to solve linear inverse problems by unfolding iterative algorithms (e.g., iterative shrinkage thresholding algorithm (ISTA)) as deep neural networks (DNNs) with learnable parameters. However, existing ISTA-based unfolded algorithms restrict the network architectures for iterative updates with the partial weight coupling structure to guarantee convergence. In this paper, we propose hybrid ISTA to unfold ISTA with both pre-computed and learned parameters by incorporating free-form DNNs (i.e., DNNs with arbitrary feasible and reasonable network architectures), while ensuring theoretical convergence. We first develop HCISTA to improve the efficiency and flexibility of classical ISTA (with pre-computed parameters) without compromising the convergence rate in theory. Furthermore, the DNN-based hybrid algorithm is generalized to popular variants of learned ISTA, dubbed HLISTA, to enable a free architecture of learned parameters with a guarantee of linear convergence. To our best knowledge, this paper is the first to provide a convergence-provable framework that enables free-form DNNs in ISTA-based unfolded algorithms. This framework is general to endow arbitrary DNNs for solving linear inverse problems with convergence guarantees. Extensive experiments demonstrate that hybrid ISTA can reduce the reconstruction error with an improved convergence rate in the tasks of sparse recovery and compressive sensing.

Feature Enhancement Based on Regular Sparse Model for Planetary Gearbox Fault Diagnosis

The planetary gearbox, widely used in many machinery fields, suffers from harmful vibration excited by bearings fault, which always causes machine breakdowns. Thus, fault diagnosis is the necessary approach to keeping machines safe, which often takes fault features, which are extracted from signals, as critical information. However, limited to various interferences caused by gear meshing and background noise, fault characteristic information is always weak and difficult to identify, bringing an increasing emphasis on feature enhancement. In this paper, based on the feature enhancement problem under strong interferences, a sparse regular diagnosis algorithm is studied. Firstly, a new fault sensitivity indicator is constructed to distinguish different periodic impulse signals and enhance the energy of the fault impulse. Combined with tunable Q-factor wavelet transform multi-scale representation, the guided enhanced vector is introduced to guide the directional gathering of the fault frequency band. Then a weighted regular term is constructed to establish the Guided Enhanced Regular Sparse (GERS) model. Iterative soft threshold algorithm is employed to solve the model. Compared with state-of-the-art methods, through numerical simulation and fault experiment, the effectiveness and superiority of the proposed algorithm is verified successfully.

Experimental Validation of the DBIM-TwIST Algorithm for Brain Stroke Detection and Differentiation Using a Multi-Layered Anatomically Complex Head Phantom

We present an experimental validation of the distorted Born iterative method with the two-step iterative shrinkage thresholding (DBIM-TwIST) algorithm for the problem of brain stroke detection and differentiation, using an anatomically accurate, multi-layer head phantom. To this end, we have developed a gelatine-based, anatomically complex head phantom which mimics various brain tissues and also includes a target mimicking hemorrhagic or ischemic stroke. We simulated the model and setup using CST Microwave Studio and then used our experimental imaging setup to collect numerical and measured data, respectively. We then used our DBIM-TwIST algorithm to reconstruct the dielectric properties of the imaging domain for both simulated and measured data. Results from our CST simulations showed that we are able to locate and reconstruct the permittivity of different stroke targets using an approximate initial guess. Our experimental results demonstrated the potential and challenges for successful detection and differentiation of the stroke targets.

Interpretable Neural Network via Algorithm Unrolling for Mechanical Fault Diagnosis

Artificial neural network (ANN) has achieved great success in mechanical fault diagnosis and has been widely used. However, traditional ANN is still opaque in terms of interpretability, making it difficult for users to understand and trust the diagnosis results. This paper proposes an interpretable neural network to provide high-performance and credible mechanical fault diagnosis results. The proposed network is mainly generated by unrolling the nested iterative soft thresholding algorithm (NISTA) for a sparse coding model and it is named NISTA-Net. Therefore, the network architecture of NISTA-Net has a clear theoretical basis and users know how it is designed. Additionally, we propose a visualization method for NISTA-Net to examine whether the network has learned meaningful features. This method helps users better understand how NISTA-Net performs classifications. These two aspects of transparency/interpretability allow NISTA-Net to be more credible when applied for mechanical fault diagnosis. We carried out simulations and two experiments of fault diagnosis to verify the performance of NISTA-Net. The results reveal that NISTA-Net can well extract the fault features of the concerned bearings and gears. As a consequence, it achieves the best performance compared with other advanced networks. Given the success of NISTA-Net, a systematic approach is finally summarized to help design interpretable fault diagnosis networks, aiming to inspire more related research.

γ-Net: Superresolving SAR Tomographic Inversion via Deep Learning

Synthetic aperture radar tomography (TomoSAR) has been extensively employed in 3-D reconstruction in dense urban areas using high-resolution SAR acquisitions. Compressive sensing (CS)-based algorithms are generally considered as the state-of-the art in super-resolving TomoSAR, in particular in the single look case. This superior performance comes at the cost of extra computational burdens, because of the sparse reconstruction, which cannot be solved analytically, and we need to employ computationally expensive iterative solvers. In this article, we propose a novel deep learning-based super-resolving TomoSAR inversion approach, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {\gamma }$ </tex-math></inline-formula> -Net, to tackle this challenge. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {\gamma }$ </tex-math></inline-formula> -Net adopts advanced complex-valued learned iterative shrinkage thresholding algorithm (CV-LISTA) to mimic the iterative optimization step in sparse reconstruction. Simulations show the height estimate from a well-trained <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {\gamma }$ </tex-math></inline-formula> -Net approaches the Cramér-Rao lower bound (CRLB) while improving the computational efficiency by one to two orders of magnitude comparing to the first-order CS-based methods. It also shows no degradation in the super-resolution power comparing to the state-of-the-art second-order TomoSAR solvers, which are much more computationally expensive than the first-order methods. Specifically, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {\gamma }$ </tex-math></inline-formula> -Net reaches more than 90% detection rate in moderate super-resolving cases at 25 measurements at 6 dB SNR. Moreover, simulation at limited baselines demonstrates that the proposed algorithm outperforms the second-order CS-based method by a fair margin. Test on real TanDEM-X data with just six interferograms also shows high-quality 3-D reconstruction with high-density detected double scatterers.

Simultaneous Seismic Data Interpolation and Denoising Based on Nonsubsampled Contourlet Transform Integrating With Two-Step Iterative Log Thresholding Algorithm

Seismic data interpolation and denoising play vital roles in obtaining complete and clean data in seismic data processing. Seismic data usually misses along various spatial axes and always mix with random noise. In order to obtain complete and clean seismic data, reconstruction technology can interpolate missing data and attenuate random noise. Nonsubsampled contourlet transform is an effective transform to obtain multi-scale and multi-direction sparse domain data for compression sensing interpolation and denoising. However, conventional iterative shrinkage/thresholding cannot handle ill-posed and ill-conditioned equations for solving linear inverse problem. We present a two-step iterative log thresholding method to overcome ill-posed and ill-conditioned problems and improve the convergence rate and solution accuracy, which can interpolate and denoise seismic data simultaneously in the nonsubsampled contourlet transform framework. First, we use nonsubsampled contourlet transform to convert the seismic missing data with random noise to sparse domain. Then, we apply two-step iterative log thresholding algorithm to interpolate and denoise data in sparse domain. The result of each iteration is based on the results of the previous two iterations, which can accelerate convergence rate. In addition, log thresholding can further improve convergence rate and solution accuracy. Finally, we use inverse nonsubsampled contourlet transform to obtain the interpolated and denoised seismic data. The new method can reconstruct the irregularly missing data and attenuate random noise to obtain complete and clean seismic data with high accuracy, which is crucial for seismic imaging and inversion. We demonstrate the applicability and effectiveness of this simultaneous interpolation and denoising technique with successful applications to both synthetic and field data examples.

Parametric Iterative Soft Thresholding Algorithm for Refocusing of Moving Targets in SAR Images

As a classical sparse reconstruction algorithm, iterative soft thresholding (IST) is often used in SAR sparse imaging applications. However, for moving targets, the radar echo contains unknown phase error, so it is difficult to directly apply the IST algorithm to reconstruct the SAR moving target image. In this paper, a parametric iterative soft thresholding (P-IST) algorithm is proposed for refocusing of moving targets in synthetic aperture radar (SAR) images. In this algorithm, the echo phase error is modeled as a function of the phase compensation factor, and in the iteration process, a phase compensation factor optimization selection step based on residual energy minimization is added. Compared with the existing algorithms, the proposed P-IST algorithm can obtain the global optimal solution, and has better efficiency and robustness. Both simulated data and measured data are used to validate the effectiveness of the proposed algorithm.

A New L₁ Multi-Kernel Learning Support Vector Regression Ensemble Algorithm With AdaBoost

This paper proposes a new multi-kernel learning ensemble algorithm, called Ada-<inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula>MKL-WSVR, which can be regarded as an extension of multi-kernel learning (MKL) and weighted support vector regression (WSVR). The first novelty is to add the <inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula> norm of the weights of the combined kernel function to the objective function of WSVR, which is used to adaptively select the optimal base models and their parameters. In addition, an accelerated method based on fast iterative shrinkage thresholding algorithm (FISTA) is developed to solve the weights of the combined kernel function. The second novelty is to propose an integrated learning framework based on AdaBoost, named Ada-<inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula>MKL-WSVR. In this framework, we integrate FISTA into AdaBoost. At each iteration, we optimize the weights of the combined kernel function and update the weights of the training samples at the same time. Then an ensemble regression function of a set of regression functions is output. Finally, two groups of the experiments are designed to verify the performance of our algorithm. On the first group of the experiments including eight datasets from UCI machine learning repository, the MAEs and RMSEs of Ada-<inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula>MKL-WSVR are reduced by 11.14&#x0025; and 9.08&#x0025; on average, respectively. Furthermore, on the second group of the experiments including the COVID-19 epidemic datasets from eight countries, the MAEs and RMSEs of Ada-<inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula>MKL-WSVR are reduced by 31.19&#x0025; and 29.98&#x0025; on average, respectively.

Evaluation of Optimization Algorithms and Noise Robustness of Sparsity-Promoting Dynamic Mode Decomposition

In the present study, we organize the existing sparsity-promoting dynamic mode decomposition (DMDsp) in terms of noise robustness, propose faster optimization algorithm for DMDsp, and evaluate its characteristics. Two kinds of DMDsp, namely system-based DMDsp (sDMDsp) and observation-based DMDsp (oDMDsp), combined with three kinds of optimization algorithm, namely the fast iterative shrinkage thresholding algorithm (FISTA), the alternating direction method of multipliers (ADMM), and a greedy algorithm, are investigated. For both sDMDsp and oDMDsp, FISTA yields the shortest processing time. The processing time for sDMDsp with FISTA is shorter than that for oDMDsp with FISTA. The original data reconstruction errors for sDMDsp and oDMDsp are similar among the three optimization algorithms. The noise robustness for sDMDsp and oDMDsp is evaluated. sDMDsp and oDMDsp have similar robustness to observation noise, except for a system with large system and observation noise, for which oDMDsp outperforms sDMDsp.

1-Bit Radar Imaging Based on Adversarial Samples

Radar imaging with 1-bit data is attractive thanks to its low storage and transmission burden. Existing 1-bit radar imaging methods cannot satisfactorily suppress the artifacts in the imaging result induced by 1-bit quantization error and noise. In this article, we propose a new 1-bit compressive sensing (CS) based algorithm, i.e., the adversarial-sample-based binary iterative hard thresholding (AS-BIHT) algorithm, to improve the 1-bit radar imaging performance. First, we formulate a parametric model for 1-bit radar imaging with a new adjustable quantization level parameter. The parametric 1-bit radar imaging model updates the imaging scene and the quantization level parameter in an iterative fashion based on adversarial samples. Then, we design a mechanism to generate adversarial samples by attacking the 1-bit radar imaging model to resist the quantization consistency condition, such that forcing quantization consistent reconstruction on adversarial samples mitigates the quantization error and noise. The quantization level parameter is then tuned based on the adversarial samples. In this way, the ability of the model to adapt to echo data contaminated by noise and quantization error is enhanced, and the artifacts are well suppressed. Simulation and experimental results on real radar data demonstrate the effectiveness of the proposed AS-BIHT algorithm in 1-bit radar imaging.

Online Tensor Robust Principal Component Analysis

Online robust principal component analysis (RPCA) algorithms recursively decompose incoming data into low-rank and sparse components. However, they operate on data vectors and cannot directly be applied to higher-order data arrays (e.g. video frames). In this paper, we propose a new online robust PCA algorithm that preserves the multi-dimensional structure of data. Our algorithm is based on the recently proposed tensor singular value decomposition (T-SVD). We develop a convex optimization-based approach to recover the sparse component; and subsequently, update the low-rank component using incremental T-SVD. We propose an efficient tensor convolutional extension to the fast iterative shrinkage thresholding algorithm (FISTA) to produce a fast algorithm to solve this optimization problem. We demonstrate tensor-RPCA with the application of background foreground separation in a video stream. The foreground component is modeled as a sparse signal. The background component is modeled as a gradually changing low-rank subspace. Extensive experiments on real-world videos are presented and results demonstrate the effectiveness of our online tensor robust PCA.

Basis Pursuit Denoising via Recurrent Neural Network Applied to Super-Resolving SAR Tomography

Finding sparse solutions of underdetermined linear systems commonly requires the solving of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> regularized least squares minimization problem, which is also known as the basis pursuit denoising (BPDN). They are computationally expensive since they cannot be solved analytically. An emerging technique known as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">deep unrolling</i> provided a good combination of the descriptive ability of neural networks, explainable, and computational efficiency for BPDN. Many unrolled neural networks for BPDN, e.g. learned iterative shrinkage thresholding algorithm and its variants, employ shrinkage functions to prune elements with small magnitude. Through experiments on synthetic aperture radar tomography (TomoSAR), we discover the shrinkage step leads to unavoidable information loss in the dynamics of networks and degrades the performance of the model. We propose a recurrent neural network (RNN) with novel sparse minimal gated units (SMGUs) to solve the information loss issue. The proposed RNN architecture with SMGUs benefits from incorporating historical information into optimization, and thus effectively preserves full information to the final output. Taking TomoSAR inversion as an example, extensive simulations demonstrated that the proposed RNN outperforms the state-of-the-art deep learning-based algorithm in terms of super-resolution power as well as generalization ability. It achieved 10% to 20% higher double scatterers detection rate and is less sensitive to phase and amplitude ratio difference between scatterers. Test on real TerraSAR-X spotlight images also shows high-quality 3-D reconstruction of test site.

SAR Imaging Based on Deep Unfolded Network With Approximated Observation

Compressed sensing (CS) based synthetic aperture radar (SAR) imaging methods are showing superior potential in imaging performance over classical matched filtering based methods. However, the CS-based methods require much more computational cost to solve the iterative optimization composed of large-scale matrix operators. To hold the improvement of imaging performance and reduce the computational cost, in this paper, we propose a novel SAR imaging method by Deep Unfolded Network (DUN) of Iterative Shrinkage Threshold Algorithm (ISTA) with the approximated observation of Range-Doppler Algorithm (RDA) operator. The proposed method takes the radar echoes as the input to learn the imaging procedure. Firstly, the approximated observation is utilized in SAR imaging model to reduce the size of the DUN. Moreover, we use ISTA as an example to introduce how to establish DUN with approximated observation, in which the detailed structure to handle the complex-valued radar echoes is also designed. Finally, the auto-encoder is utilized to calculate the difference of the echoes rather than the imaging results so that we can train the proposed network by unsupervised learning. The experiments of both point targets, surface targets, and real scenes show that the proposed imaging method is superior in terms of imaging performance and computing efficiency.

PSCSC-Net: A Deep Coupled Convolutional Sparse Coding Network for Pansharpening

Given a low-resolution multispectral (MS) image and a high-resolution panchromatic image, the task of pansharpening is to generate a high-resolution MS image. Deep learning (DL)-based methods receive extensive attention recently. Different from the existing DL-based methods, this article proposes a novel deep neural network for pansharpening inspired by the learned iterative soft thresholding algorithm. First, a coupled convolutional sparse coding-based pansharpening (PSCSC) model and related traditional optimization algorithm are proposed. Then, following the procedures of traditional algorithm for solving PSCSC, an interpretable end-to-end deep pansharpening network is developed using a deep unfolding strategy. The designed deep architecture can also be understood in the view of details injection (DI)-based scheme. This work offers a solution that integrates the DL-, DI-, and variational optimization-based schemes into a framework. The experimental results on the reduced- and full-scale datasets demonstrate that the proposed deep pansharpening network outperforms popular traditional methods and some current DL-based methods.

Amplitude-Phase CNN-Based SAR Target Classification via Complex-Valued Sparse Image

It is known that a synthetic aperture radar (SAR) image obtained by matched filtering (MF)-based algorithms always suffers from serious noise, sidelobes, and clutters. However, the improvement of the image quality means the complexity of the SAR system will increase, which affects the application of the SAR image. The introduction of the sparse signal processing technique into SAR imaging proposes a new way to solve this problem. Sparse SAR image obtained by sparse recovery algorithms shows a better image performance than the typical complex SAR image with lower sidelobes and higher signal-to-noise ratio. As the most widely applied field of the SAR image, target classification relies on the SAR image with high quality. Therefore, a novel target classification model based on the amplitude and phase information of the sparse SAR image is introduced in this article. First, complex sparse image dataset is constructed by a novel iterative soft thresholding (BiIST) algorithm. Compared with typical regularization-based sparse recovery algorithms, BiIST not only can improve the quality of recovered image, but also can obtain a nonsparse solution with retaining phase information and background statistical distribution of the SAR image. Then, targets are classified by the proposed amplitude-phase convolutional neural network (AP-CNN). Typical SAR target classification networks imitate those on optical image, just using amplitude data. However, considering the particularity of the SAR image, the AP-CNN uses both amplitude and phase for training, which theoretically improves the classification accuracy. Experimental results show that the AP-CNN outperforms the typical amplitude-based CNN in target classification, both under standard operating conditions (SOCs) and extended operating conditions (EOCs). Results under SOC demonstrate that the AP-CNN improves the classification accuracy by 11.46&#x0025; with only 1000 training samples. Even under EOC, the accuracy gap between the AP-CNN and CNN can reach 6.6&#x0025; in the case of 800 training samples. This means that even if the number of samples is limited, the AP-CNN can obtain optimal classification result.

Learning From Noisy Data: An Unsupervised Random Denoising Method for Seismic Data Using Model-Based Deep Learning

For seismic random noise attenuation, deep learning has attracted much attention and achieved promising performance. However, compared with conventional methods, the denoising performance of supervised learning-based methods heavily depends on massive training samples with high-quality labeled data, which makes their generalization capabilities limited. Even though deep neural networks (DNNs) usually outperform the conventional denoising methods, their performance is not guaranteed since neural networks still lack good mathematical interpretability at present. To alleviate the dependency on labeled data and explore insights into the denoising system, we proposed an unsupervised denoising method based on model-based deep learning, which combined domain knowledge and a data-driven method. We designed a network based on the modified iterative soft threshold algorithm (ISTA), which omitted the soft threshold to alleviate uncertainties introduced by empirically selected thresholds. In this network, we set the dictionary and code as trainable parameters. A loss function with a smooth penalty was designed to ensure that the network training can be implemented in an unsupervised manner. In the proposed method, we set the denoised result by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f-x$ </tex-math></inline-formula> deconvolution as the input for our network, and the further denoised data can be obtained after each epoch of the training, which means that our method does not need the testing procedure. Experiments on synthetic and field seismic data demonstrate that our method exhibits competitive performance compared to the conventional, supervised, and unsupervised methods, including <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f-x$ </tex-math></inline-formula> deconvolution, curvelet, the Denoising Convolutional Neural Network (DnCNN), and the integration of neural network and Block-matching and 3-D filtering method (NN + BM3D).