Learning Optimal Nonlinearities for Iterative Thresholding Algorithms

Conventional sparse spike deconvolution algorithms that are based on the iterative shrinkage-thresholding algorithm (ISTA) are widely used. The aim of this type of algorithm is to obtain accurate seismic wavelets. When this is not fulfilled, the processing stops being optimum. Using a recurrent neural network (RNN) as deep learning method and applying backpropagation to ISTA, we have developed an RNN-like ISTA as an alternative sparse spike deconvolution algorithm. The algorithm is tested with both synthetic and real seismic data. The algorithm first builds a training dataset from existing well-logs seismic data and then extracts wavelets from those seismic data for further processing. Based on the extracted wavelets, the new method uses ISTA to calculate the reflection coefficients. Next, inspired by the backpropagation through time (BPTT) algorithm, backward error correction is performed on the wavelets while using the errors between the calculated reflection coefficients and the reflection coefficients corresponding to the training dataset. Finally, after performing backward correction over multiple iterations, a set of acceptable seismic wavelets is obtained, which is then used to deduce the sequence of reflection coefficients of the real data. The new algorithm improves the accuracy of the deconvolution results by reducing the effect of wrong seismic wavelets that are given by conventional ISTA. In this study, we account for the mechanism and the derivation of the proposed algorithm, and verify its effectiveness through experimentation using theoretical and real data.

Compressive sensing (CS) is a technique that enables the recovery of sparse signals using fewer measurements than traditional sampling methods. To address the computational challenges of CS reconstruction, our objective is to develop an interpretable and concise neural network model for reconstructing natural images using CS. We achieve this by mapping one step of the iterative shrinkage thresholding algorithm (ISTA) to a deep network block, representing one iteration of ISTA. To enhance learning ability and incorporate structural diversity, we integrate aggregated residual transformations (ResNeXt) and squeeze‐and‐excitation mechanisms into the ISTA block. This block serves as a deep equilibrium layer connected to a semi‐tensor product network for convenient sampling and providing an initial reconstruction. The resulting model, called MsDC‐DEQ‐Net, exhibits competitive performance compared to state‐of‐the‐art network‐based methods. It significantly reduces storage requirements compared to deep unrolling methods, using only one iteration block instead of multiple iterations. Unlike deep unrolling models, MsDC‐DEQ‐Net can be iteratively used, gradually improving reconstruction accuracy while considering computation tradeoffs. Additionally, the model benefits from multiscale dilated convolutions, further enhancing performance.

Scanning radar can be used to obtain images of targets in forward-looking area, and has attracted much attention in many fields, such as ocean monitoring, air-to-ground attack, navigation, and so on. However, its azimuth resolution is extremely poor due to the limitation of the antenna size. In order to break through the limitation, many superresolution algorithms have been proposed, and iterative shrinkage-thresholding algorithm (ISTA) is one of the most famous methods because of its antinoise ability and simplicity. In the meantime, the slow convergence of iterative shrinkage-thresholding algorithm is also known to all. In this article, a second-order accelerated ISTA for scanning radar forward-looking superresolution imaging is proposed. In this algorithm, a prediction vector is constructed before each iteration by using the first and the second-order difference information of iteration vectors to reduce the number of iterations and get a faster convergence speed. In the end, simulations and experimental results are given to illustrate the effectiveness of the accelerated imaging algorithm.

Iterative shrinkage-thresholding algorithms provide simple methods to recover sparse signals from compressed measurements. In this paper, we propose a new class of iterative shrinkage-thresholding algorithms which preserve the computational simplicity and improve iterative estimation by incorporating a soft support detection. Indeed, at each iteration, by learning the components that are likely to be nonzero from the current signal estimation using Bayesian techniques, the shrinkage-thresholding step is adaptively tuned and optimized. Unlike other adaptive methods, we are able to prove, under suitable conditions, the convergence of the proposed methods. Moreover, we show through numerical experiments that the proposed methods outperform classical shrinkage-thresholding in terms of rate of convergence and of sparsity-undersampling tradeoff.

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.

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.

TwIST sparse regularization method using cubic B-spline dual scaling functions for impact force identification

In this paper, we propose a novel dictionary learning model based on the idea of deep unrolling to deal with the synthetic aperture radar (SAR) image change detection problem. Deep unrolling aims at unrolling the iterative algorithm into a trainable neural network. In our proposed method, the idea of unrolling is applied to the Iterative Shrinkage Threshold Algorithm (ISTA), which is one of classic algorithms for dictionary learning. Then, the proposed Unrolling Iterative Shrinkage Threshold Algorithm (UR-ISTA), is utilized to obtain the sparse codes of the difference results. Finally, the change map is computed by k-means clustering algorithm. The advantage of UR-ISTA method is relatively low time cost, which makes it possible to add dictionary updating step to calculate specific feature vectors. Experimental results show that the proposed approach has superior accuracy and precision compared to several well-known change detection techniques. The proposed UR-ISTA algorithm shows more robustness than another sparse representation algorithm.

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.

Convolution has been the core operation of modern deep neural networks. It is well known that convolutions can be implemented in the Fourier Transform domain. In this article, we propose to use binary block Walsh–Hadamard transform (WHT) instead of the Fourier transform. We use WHT-based binary layers to replace some of the regular convolution layers in deep neural networks. We utilize both one-dimensional (1D) and 2D binary WHTs in this article. In both 1D and 2D layers, we compute the binary WHT of the input feature map and denoise the WHT domain coefficients using a nonlinearity that is obtained by combining soft-thresholding with the tanh function. After denoising, we compute the inverse WHT. We use 1D-WHT to replace the 1 × 1 convolutional layers, and 2D-WHT layers can replace the 3 × 3 convolution layers and Squeeze-and-Excite layers. 2D-WHT layers with trainable weights can be also inserted before the Global Average Pooling layers to assist the dense layers. In this way, we can reduce the number of trainable parameters significantly with a slight decrease in trainable parameters. In this article, we implement the WHT layers into MobileNet-V2, MobileNet-V3-Large, and ResNet to reduce the number of parameters significantly with negligible accuracy loss. Moreover, according to our speed test, the 2D-FWHT layer runs about 24 times as fast as the regular 3 × 3 convolution with 19.51% less RAM usage in an NVIDIA Jetson Nano experiment.

Simply feeding of a last hidden layer of the deep neural network (DNN) back to the input layer recently found to be effective for noise robust acoustic modeling. Such high level feature strengthens the robustness of DNN based acoustic model while paying approximately twice the computational cost. In this paper, we proposed to feed such high level feature iteratively back to lower layers, which is referred as multi-scale feedback connection. With this intention, we firstly extract the high level feature at the last hidden layer of DNN. Second, this high level feature feed back to a lower scale features, they then generates a subsequent prediction as well as a subsequent high level feature. This subsequent high level feature is further feed down to a lower layers. We evaluated the proposed approach on both TIMIT and a large scale internal dataset. The large scale internal dataset includes voice search and far field dataset. Our finding is two aspects. First, at equivalent computational costs, the multiscale feedback connection outperforms the DNN, the DNN with skip connection and the DNN with feedback connection. The improvement is larger on the far field dataset. Second, pair layers-wise pretraining helps the proposed approach to converge better.

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.

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 &#x2113;₁ 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.

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.

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.