MsDC‐DEQ‐Net: Deep Equilibrium Model (DEQ) with Multiscale Dilated Convolution for Image Compressive Sensing (CS)

Compressed sensing (CS) reconstruction of nonsparse scenes is one of the difficulties in synthetic aperture radar (SAR) imaging technology. Although the conventional CS method with sparse representation has proven applicable for nonsparse SAR reconstruction, its disadvantages are unsatisfactory imaging quality and high computational complexity under downsampling. In this paper, a novel deep learning approach for nonsparse SAR scene reconstruction is proposed based on sparse representation and the iterative shrinkage threshold algorithm (ISTA). Specifically, we first develop a sparse representation-based imaging model associated with the ℓ₁ sparse regularizer in nonlinear transform domains. Then, the advantages of the recurrent neural network (RNN) and convolutional neural network (CNN) are incorporated into an ISTA-inspired deep unfolded network (DUN) called SR-ISTA-Net, in which all the parameters are layer-varied rather than handcrafted. The experiments verify that the proposed SR-ISTA-Net can provide high-quality reconstruction results under nonsparse scenes while substantially reducing imaging time.

The fast compressed sensing (CS) reconstruction for natural image processing aims to infer the original pixel vector from randomized measurements as correctly as possible. Current first-order proximal mainstream schemes, including alternating direction methods of multipliers (ADMM) and the iterative shrinkage thresholding algorithm (ISTA), have advantages in convergence and stability at the expense of computing speed and complex tuning strategy. This paper proposes the deep iterative shrinkage network for real-time applications, where network layers replace the iterative optimization steps. The hyperparameters associated with shrinkage thresholds and step sizes are adjusted by standard deep learning updates. Numerical results suggest that the proposed method has flexibility and effectiveness in image reconstruction with different compression ratios while preserving the computational simplicity of end-to-end methods.

This paper focuses on processing low oversampling echo data of staggered synthetic aperture radar (SAR). In staggered mode, the non-uniformly sampling and echo data loss cause severe azimuth ambiguity. To solve this problem, we propose a method combining the compressed sensing (CS) tool and the conformal Fourier transform (CFT) algorithm. First, the 2-D-fast iterative shrinkage thresholding algorithm (FISTA) is used to efficiently solve the optimization problem based on CS theory to recovery the missing data. Second, the CFT algorithm is used to accurately compute the spectrum of the non-uniformly sampled echo data. Unlike other methods suiting fast pulse repetition interval (PRI) change only, this method performs well for both fast and slow PRI change. Plus, the proposed method can be well applied to both point and distributed targets. Simulation results demonstrate that the proposed method can effectively and robustly suppress the azimuth ambiguity for staggered SAR data.

Sparse-representation-based synthetic aperture radar (SAR) imaging technology has shown superior potential in the reconstruction of nonsparse scenes. However, many existing compressed sensing (CS) methods with sparse representation cannot obtain an optimal sparse basis and only apply to the sensing matrix obtained by exact observation, resulting in a low image quality occupying more storage space. To reduce the computational cost and improve the imaging performance of nonsparse scenes, we formulate a deep learning SAR imaging method based on sparse representation and approximated observation deduced from the chirp-scaling algorithm (CSA). First, we incorporate the CSA-derived approximated observation model and a nonlinear transform function within a sparse reconstruction framework. Second, an iterative shrinkage threshold algorithm is adopted to solve this framework, and the solving process is unfolded as a deep SAR imaging network. Third, a dual-path convolutional neural network (CNN) block is designed in the network to achieve the nonlinear transform, dramatically improving the sparse representation capability over conventional transform-domain-based CS methods. Last, we improve the CNN block to develop an enhanced version of the deep SAR imaging network, in which all the parameters are layer-varied and trained by supervised learning. The experiments demonstrate that our proposed two imaging networks outperform conventional CS-driven and deep-learning-based methods in terms of computing efficiency and reconstruction performance of nonsparse scenes.

The most common technique for generating B-mode ultrasound (US) images is delay-and-sum (DAS) beamforming, where the signals received at the transducer array are sampled before an appropriate delay is applied. This necessitates sampling rates exceeding the Nyquist rate and the use of a large number of antenna elements to ensure sufficient image quality. Recently, we proposed methods to reduce the sampling rate and the array size relying on image recovery using iterative algorithms based on compressed sensing (CS) and the finite rate of innovation (FRI) frameworks. Iterative algorithms typically require a large number of iterations, making them difficult to use in real time. In this article, we propose a reconstruction method from sub-Nyquist samples in the time and spatial domain, which is based on unfolding the iterative shrinkage thresholding algorithm (ISTA), resulting in an efficient and interpretable deep network. The inputs to our network are the subsampled beamformed signals after summation and delay in the frequency domain, requiring only a subset of the US signal to be stored for recovery. Our method allows reducing the number of array elements, sampling rate, and computational time while ensuring high-quality imaging performance. Using in vivo data, we demonstrate that the proposed method yields high-quality images while reducing the data volume traditionally used up to 36 times. In terms of image resolution and contrast, our technique outperforms previously suggested methods as well as DAS and minimum-variance (MV) beamforming, paving the way to real-time applicable recovery methods.

With the aim of developing a fast yet accurate algorithm for compressive sensing (CS) reconstruction of natural images, we combine in this paper the merits of two existing categories of CS methods: the structure insights of traditional optimization-based methods and the speed of recent network-based ones. Specifically, we propose a novel structured deep network, dubbed ISTA-Net, which is inspired by the Iterative Shrinkage-Thresholding Algorithm (ISTA) for optimizing a general $$ norm CS reconstruction model. To cast ISTA into deep network form, we develop an effective strategy to solve the proximal mapping associated with the sparsity-inducing regularizer using nonlinear transforms. All the parameters in ISTA-Net (e.g. nonlinear transforms, shrinkage thresholds, step sizes, etc.) are learned end-to-end, rather than being hand-crafted. Moreover, considering that the residuals of natural images are more compressible, an enhanced version of ISTA-Net in the residual domain, dubbed ISTA-Net+, is derived to further improve CS reconstruction. Extensive CS experiments demonstrate that the proposed ISTA-Nets outperform existing state-of-the-art optimization-based and network-based CS methods by large margins, while maintaining fast computational speed. Our source codes are available: http://jianzhang.tech/projects/ISTA-Net.

We present two model-based neural network architectures purposed for sporadic user detection and channel estimation in massive machine-type communications. In the scenario under consideration, a base station assigns the users a set of pilot sequences that is linearly dependent, but because user activity is sporadic the detection/estimation problem is amenable to sparse recovery algorithms. Further, we consider a millimeter-wave wireless channel, so that the channel vectors are sparse in a known dictionary. We apply the deep unfolding framework to design custom neural network layers by unrolling two iterative optimization algorithms: (1) linearized alternating direction method of multipliers, which we apply to a constrained convex problem, and (2) vector approximate message passing featuring a novel denoiser based on the iterative shrinkage thresholding algorithm. The networks thus inherit domain knowledge as encapsulated by the signal model, and suitable operations as informed by the algorithms—in the same spirit as convolutional networks that exploit structure inherent in images and audio, except grounded in optimization and statistics. The networks, trained on synthetic data generated from the block-fading millimeter-wave multiple access channel model, offer improved complexity and accuracy relative to their iterative counterparts, and are potentially a boon to cell-free MIMO systems.

Compressed sensing (CS) is a technique to sample compressible signals below the Nyquist rate, whilst still allowing near optimal reconstruction of the signal. In this paper, we apply the iterative hard thresholding (IHT) algorithm for compressed sensing on the speech signal. The interested speech signal is transformed to the frequency domain using Discrete Fourier Transform (DCT) and then compressed sensing is applied to that signal. The compressed signal can be reconstructed using the recently introduced Iterative Hard Thresholding (IHT) algorithm and also by the tradditional \( \ell_{1} \) minimization (basic pursuit) for comparison. It is shown that the compressed sensing can provide better root mean square error (RMSE) than the tradition DCT compression method, given the same compression ratio.

Synthetic aperture radar tomography (TomoSAR) has been widely employed in 3-D urban mapping. However, state-of-the-art super-resolving TomoSAR algorithms are computationally expensive, because conventional numerical solvers need to solve the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$l_{2^{-}}l_{1}$</tex> mix norm minimization. This paper proposes a computationally efficient super-resolving To-moSAR inversion algorithm based on deep learning. We studied the potential of deep learning to mimic a conventional <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$l_{2}-l_{1}$</tex> mix norm solver, i.e. iterative shrinkage thresholding algorithm (ISTA), and proposed several improvements of the complex-valued learned ISTA for TomoSAR inversion. Investigation on the super-resolution ability and estimator efficiency of the proposed algorithm shows that the proposed algorithm approaches the Cramer Rao lower bound (CRLB) with a computational efficiency more than 100 times better than the conventional solver.

In this paper, an image deconvolution problem using triangular wavelets is addressed. A simple blurred image is generated by the Gaussian-based blur kernel and additive Gaussian white noise. Two-dimensional orthogonal Haar wavelets on a triangular lattice, which gives sparse and isotropie representations of images, are selected as a dictionary. Using the iterative shrinkage threshold algorithm, some numerical experiments are carried out for the problem of recovering the original image from the noisy blurred image. We show that the triangular wavelets result in better performance than the conventional Haar wavelet transform.

A new restoration algorithm based on Huber function for astronomy image compression was proposed in this paper. A combinatorial sensing matrix based on noiselet transform and subsample matrix was built for image acquisition. A restoration algorithm based on Huber function was introduced to reconstruct the signal. Peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) were used to evaluate the performance of the proposed algorithm. Compared with the standard compression algorithms, including JPEG and iterative shrinkage thresholding algorithm (ISTA) based reconstruction algorithm, the results obtained by the proposed algorithm are of higher structural similarity and PSNR. The proposed algorithm is suitable for the application scenarios of astronomy images with large data volume and high redundancy.

In recent years, extensive research has shown that deep learning-based compressed image reconstruction algorithms can achieve faster and better high-quality reconstruction for single-pixel imaging, and that reconstruction quality can be further improved by joint optimization of sampling and reconstruction. However, these network-based models mostly adopt end-to-end learning, and their structures are not interpretable. In this paper, we propose SRMU-Net, a sampling and reconstruction jointly optimized model unfolding network. A fully connected layer or a large convolutional layer that simulates compressed reconstruction is added to the compressed reconstruction network, which is composed of multiple cascaded iterative shrinkage thresholding algorithm (ISTA) unfolding iteration blocks. To achieve joint optimization of sampling and reconstruction, a specially designed network structure is proposed so that the sampling matrix can be input into ISTA unfolding iteration blocks as a learnable parameter. We have shown that the proposed network outperforms the existing algorithms by extensive simulations and experiments.

Iterative shrinkage/thresholding algorithm (ISTA) is a well-studied method for finding sparse solutions to ill-posed inverse problems. In this letter, we present a data-driven scheme for learning optimal thresholding functions for ISTA. The proposed scheme is obtained by relating iterations of ISTA to layers of a simple deep neural network (DNN) and developing a corresponding error backpropagation algorithm that allows to fine-tune the thresholding functions. Simulations on sparse statistical signals illustrate potential gains in estimation quality due to the proposed data adaptive ISTA.

Deep learning methods have achieved attractive performance in dynamic MR cine imaging. However, most of these methods are driven only by the sparse prior of MR images, while the important low-rank (LR) prior of dynamic MR cine images is not explored, which may limit further improvements in dynamic MR reconstruction. In this paper, a learned singular value thresholding (Learned-SVT) operator is proposed to explore low-rank priors in dynamic MR imaging to obtain improved reconstruction results. In particular, we put forward a model-based unrolling sparse and low-rank network for dynamic MR imaging, dubbed as SLR-Net. SLR-Net is defined over a deep network flow graph, which is unrolled from the iterative procedures in the iterative shrinkage-thresholding algorithm (ISTA) for optimizing a sparse and LR-based dynamic MRI model. Experimental results on a single-coil scenario show that the proposed SLR-Net can further improve the state-of-the-art compressed sensing (CS) methods and sparsity-driven deep learning-based methods with strong robustness to different undersampling patterns, both qualitatively and quantitatively. Besides, SLR-Net has been extended to a multi-coil scenario, and achieved excellent reconstruction results compared with a sparsity-driven multi-coil deep learning-based method under a high acceleration. Prospective reconstruction results on an open real-time dataset further demonstrate the capability and flexibility of the proposed method on real-time scenarios.

As an active microwave imaging technology, synthetic aperture radar (SAR) has extensive application prospects in remote sensing, surveying, and mapping. How to reconstruct SAR images from the subsampled echo data has always been a challenging issue. Compressed sensing technology attempts to use the sparse characteristics of SAR images to recover the SAR images from the compressed samples. However, in practical applications, SAR images do not always meet the characteristic of sparsity. In this paper, we propose an imaging method from sub-Nyquist sampled data based on deep priors of SAR images. First, we use a generative flow network to model the deep prior information of the images based on the existing SAR image datasets. Then, the pre-trained network modeling deep prior information of images is embedded in a typical compressed sensing method, i.e., the Iterative Shrinkage-Thresholding Algorithm (ISTA), to replace the sparse shrinkage function of it. Meanwhile, in order to improve the accuracy and convergence speed of reconstruction, we used the subsampled echo data to fine-tune the iterative parameters of the embedded ISTA method. The results of the experiments show that through the use of the deep prior information of the images, this method can accurately reconstruct non-sparse SAR images from the subsampled echoes, even if only a few echo samples are available.