ISTA-Net: Interpretable Optimization-Inspired Deep Network for Image Compressive Sensing

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.

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.

Compressed sensing (CS) is a sparse signal sampling methodology for efficiently acquiring and reconstructing a signal from relatively few measurements. Recent work shows that CS is well-suited to be applied to problems in genomics, including probe design in microarrays, RNA interference (RNAi), and taxonomic assignment in metagenomics. The principle of using different CS recovery methods in these applications has thus been established, but a comprehensive study of using a wide range of CS methods has not been done. For each of these applications, we apply three hitherto unused CS methods, namely, l1-magic, CoSaMP, and l1-homotopy, in conjunction with CS measurement matrices such as randomly generated CS m matrix, Hamming matrix, and projective geometry-based matrix. We find that, in RNAi, the l1-magic (the standard package for l1 minimization) and l1-homotopy methods show significant reduction in reconstruction error compared to the baseline. In metagenomics, we find that l1-homotopy as well as CoSaMP estimate concentration with significantly reduced time when compared to the GPSR and WGSQuikr methods.

Recently, there are a lot of study of direction of arrival (DoA) estimation using compressive sensing (CS). As CS is a new paradigm in signal processing, there are many aspects of this method that can be investigated. In the case of DoA estimation in noisy measurement, it is important to correctly determine a correct threshold of CS reconstruction, particularly when CS reconstruction is implemented using L 1 -norm minimization. Too small threshold value will make the correct DoA does not lies in CS reconstruction searching area, while too large threshold value will burden CS iteration to select a solution from a large number of possible solutions. In this paper, we derived an optimal threshold value for CS reconstruction for DoA estimation mathematically and verified the result using computer simulation. Using Gaussian noise model, we obtain the chi-square distribution of euclidean distance of noisy and noiseless received vector. We introduce the thresholding index κ to scale the standard deviation of chi-square distribution to determine the CS reconstruction threshold and simulate this value for various SNR. We find that the optimal κ value 0.5 to 1 for high noise environment, and optimal κ value 1 to 2 in low noise environment.

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.

Compressive sensing (CS) can reduce the number of data transmissions and balance the traffic load throughout networks. However, the total number of data transmissions required in CS method is still large. It is observed that there are many zero elements in the measurement matrix. In each round of data transmission in CS method, the sensor nodes corresponding to the zero elements in the measurement matrix do not have their own data to transmit. To further reduce the number of data transmissions in the network, we aim to compute a data gathering tree by taking advantages of these zero elements in the measurement matrix, such that the total number of data transmissions is minimized. We formulate the problem as linear programming with boolean variables. The problem is NP-hard. We propose heuristic algorithm to compute the Minimum Transmission Tree (MTT) for data gathering in CS methods. The MTT algorithm constructs a spanning tree by iteratively including the edge whose average incremental transmission cost is minimum. The simulation results demonstrate that our algorithm can reduce the number of transmissions significantly, compared with the methods using minimum spanning tree (MST), shortest path tree and the CS method with nonzero measurement coefficient using MST.

In this paper, we address the problem of green Compressed Sensing (CS) reconstruction within Internet of Things (IoT) networks, both in terms of computing architecture and reconstruction algorithms. The approach is novel since, unlike most of the literature dealing with energy efficient gathering of the CS measurements, we focus on the energy efficiency of the signal reconstruction stage given the CS measurements. As a first novel contribution, we present an analysis of the energy consumption within the IoT network under two computing architectures. In the first one, reconstruction takes place within the IoT network and the reconstructed data are encoded and transmitted out of the IoT network; in the second one, all the CS measurements are forwarded to off-network devices for reconstruction and storage, i.e., reconstruction is off-loaded. Our analysis shows that the two architectures significantly differ in terms of consumed energy, and it outlines a theoretically motivated criterion to select a green CS reconstruction computing architecture. Specifically, we present a suitable decision function to determine which architecture outperforms the other in terms of energy efficiency. The presented decision function depends on a few IoT network features, such as the network size, the sink connectivity, and other systems’ parameters. As a second novel contribution, we show how to overcome classical performance comparison of different CS reconstruction algorithms usually carried out w.r.t. the achieved accuracy. Specifically, we consider the consumed energy and analyze the energy vs. accuracy trade-off. The herein presented approach, jointly considering signal processing and IoT network issues, is a relevant contribution for designing green compressive sampling architectures in IoT networks.

Recent advances in compressed sensing (CS) enable accurate CT image reconstruction from highly undersampled and noisy projection measurements, due to the sparsifiable feature of most CT images using total variation (TV). These novel reconstruction methods have demonstrated advantages in clinical applications where radiation dose reduction is critical, such as onboard cone-beam CT (CBCT) imaging in radiation therapy. The image reconstruction using CS is formulated as either a constrained problem to minimize the TV objective within a small and fixed data fidelity error, or an unconstrained problem to minimize the data fidelity error with TV regularization. However, the conventional solutions to the above two formulations are either computationally inefficient or involved with inconsistent regularization parameter tuning, which significantly limit the clinical use of CS-based iterative reconstruction. In this paper, we propose an optimization algorithm for CS reconstruction which overcomes the above two drawbacks. The data fidelity tolerance of CS reconstruction can be well estimated based on the measured data, as most of the projection errors are from Poisson noise after effective data correction for scatter and beam-hardening effects. We therefore adopt the TV optimization framework with a data fidelity constraint. To accelerate the convergence, we first convert such a constrained optimization using a logarithmic barrier method into a form similar to that of the conventional TV regularization based reconstruction but with an automatically adjusted penalty weight. The problem is then solved efficiently by gradient projection with an adaptive Barzilai-Borwein step-size selection scheme. The proposed algorithm is referred to as accelerated barrier optimization for CS (ABOCS), and evaluated using both digital and physical phantom studies. ABOCS directly estimates the data fidelity tolerance from the raw projection data. Therefore, as demonstrated in both digital Shepp-Logan and physical head phantom studies, consistent reconstruction performances are achieved using the same algorithm parameters on scans with different noise levels and∕or on different objects. On the contrary, the penalty weight in a TV regularization based method needs to be fine-tuned in a large range (up to seven times) to maintain the reconstructed image quality. The improvement of ABOCS on computational efficiency is demonstrated in the comparisons with adaptive-steepest-descent-projection-onto-convex-sets (ASD-POCS), an existing CS reconstruction algorithm also using constrained optimization. ASD-POCS alternatively minimizes the TV objective using adaptive steepest descent (ASD) and the data fidelity error using projection onto convex sets (POCS). For similar image qualities of the Shepp-Logan phantom, ABOCS requires less computation time than ASD-POCS in MATLAB by more than one order of magnitude. We propose ABOCS for CBCT reconstruction. As compared to other published CS-based algorithms, our method has attractive features of fast convergence and consistent parameter settings for different datasets. These advantages have been demonstrated on phantom studies.

Impulsive noise is always present in real-world image Compressive Sensing (CS) acquisition systems, where existing CS reconstruction performance may seriously deteriorate. In this article, we propose a robust CS formulation for image reconstruction to suppress outliers in the presence of impulsive noise. To address this issue, we consider a novel truncated-Cauchy loss function as the metric of residual error to elevate the reconstruction robustness. Specifically, we design a complementary priors model to incorporate nonconvex nonlocal low-rank prior and deep denoiser prior for high-accuracy image reconstruction. By means of the half-quadratic optimization theory and generalized soft-thresholding technique, we also develop an alternative optimization algorithm for solving the induced nonconvex optimization problem. Numerical simulations demonstrate the robustness and accuracy of the proposed robust CS method compared to some recent CS methods for image reconstruction in impulsive noise.

To develop a phase encoding reduction scheme based on compressed sensing (CS) for phase-encoded (1) H MR spectroscopic imaging (MRSI). Phantom and in vivo rat brain MRSI experiments were performed at 7 Tesla to examine the performance of CS approach and compare it with the full k-space acquisition. The CS undersampling was performed by acquiring a pseudorandom and density-varying subset of phase encodings. Residual water resonance was first removed from the undersampled k-space dataset before CS reconstruction. The CS reconstruction was performed by a linearized Bregman iteration procedure for efficiency in computing large scale L1 minimization. The spectral and spatial fidelity was evaluated by comparing spectral linewidths and metabolite maps and voxel-wise Bland-Altman analysis. CS preserved the spectral resolution and metabolite content levels. No spectral broadening was observed, and the estimation biases of the metabolite content levels were no more than 4%. CS clearly preserved the boundaries of metabolite maps but led to slight loss of details in metabolite maps. This study demonstrated the feasibility of the proposed CS approach to accelerate phase-encoded (1) H MRSI.

Compressive sensing (CS) is a theory about that one may achieve a nearly exact signal reconstruction from the fewer samples, if the signal is sparse or compressible under some basis. The reconstruction of signal can be obtained by solving a convex program, which is equivalent to a LASSO problem with l1-formulation. In this paper, we propose a stage-wise fast LASSO (StF-LASSO) algorithm for the image reconstruction from CS. It uses an insensitive Huber loss function to the objective function of LASSO, and iteratively builds the decision function and updates the parameters by introducing a stagewise fast learning strategy. Simulation studies in the CS reconstruction of the natural images and SAR images widely applied in practice demonstrate that the good reconstruction performance both in evaluation indexes and visual effect can be achieved by StF-LASSO with the fast recovered speed among the algorithms which have been implemented in our simulations in most of the cases. Theoretical analysis and experiments show that StF-LASSO is a CS reconstruction algorithm with the low complexity and stability.

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.

Compressed sensing (CS) enables accurate CT image reconstruction from low-dose measurements, due to the sparsifiable feature of most CT images using total variation (TV). The CS reconstruction is formulated as either a constrained problem to minimize the TV objective within a small data fidelity error, or an unconstrained problem to minimize the data fidelity error with TV regularization. However, the conventional solutions to the above two formulations are either computationally inefficient or involved with inconsistent regularization parameter tuning. In this work, we propose an optimization algorithm for cone-beam CT (CBCT) CS reconstruction which overcomes the above two drawbacks. The data tolerance is well estimated using the measured data, as most of the projection errors are from Poisson noise. We adopt the TV optimization framework with data fidelity as constraints. To accelerate the convergence, we first convert such a constrained optimization using a barrier method into a similar form to the conventional TV-regularization reconstruction but with an automatically adjusted penalty weight. The problem is then solved efficiently by gradient projection. The proposed algorithm is referred to as Accelerated Barrier Optimization for CS (ABOCS). As demonstrated on Shepp-Logan and head phantoms, ABOCS achieves consistent performances using the same parameters on scans with different datasets, while the TV-regularization method needs a large-scale tuning on the penalty weight. ABOCS also requires less computation time than ASDPOCS in Matlab by more than 10 times. ABOCS is further accelerated on GPU to reconstruct a 3D Shepp-Logan volume of 256×256×256 voxels in less than 20 mins using 10% projections, and the image quality is comparable to that of the full-view FDK reconstruction. We propose ABOCS for CBCT reconstruction. As compared to other published CS-based algorithms, our method has attractive features of fast convergence and consistent parameter setting for different datasets.

Compressed sensing (CS) algorithms exploit sparseness properties to reconstruct high spatial resolution magnetic resonance (MR) images from k-space data acquisitions significantly under sampled to reduce imaging times. CS algorithm effectiveness is frequently shown using under-sampled k-space data from NxN simulated images. These demonstration reconstructions are near perfect with quality higher than reconstructions using under-sampled NxN experimental k-space data sets. These differences are explained in terms of the interaction between the explicit transform domain sparsity requirement employed during iterative CS reconstruction and an inherent frequency domain property of the discrete Fourier transform (DFT). We report on experiments to overcome the limitations imposed by this DFT property by modifying the CS objective function to use a sparseness transform with a resolution higher that the standard transform related to the acquired NxN data matrix size. We demonstrate the relative effectiveness and limitations of standard CS and our proposed highresolution k-space extrapolation enabled (Hi-KEE) CS reconstruction on underand fully-sampled, simulated and experimental MR k-space data. (8 pages)

Compressed sensing or compressive sampling (CS) is a new framework for simultaneous data sampling and compression which was proposed by Candes, Donoho, and Tao several years ago. Ever since the advent of a single-pixel camera, one of the CS applications - compressive imaging (CI, also referred as feature-specific imaging) has aroused more interest of numerous researchers. However, it is still a challenging problem to choose a simple and efficient measurement matrix in such a hardware system, especially for large scale image. In this paper, we propose a new measurement matrix whose rows are the odd rows of N order Hadamard matrix and discuss the validity of the matrix theoretically. The advantage of the matrix is its universality and easy implementation in the optical domain owing to its integer-valued elements. In addition, we demonstrate the validity of the matrix through the reconstruction of natural images using Orthogonal Matching Pursuit (OMP) algorithm. Due to the limitation of the memory of the hardware system and personal computer which is used to simulate the process, it is impossible to create such a large matrix that is used to conduct large scale images. In order to solve the problem, the block-wise notion is introduced to conduct large scale images and the experiments results present the validity of this method.