Compressive Sensing Algorithm Research Articles

Hybrid Reconstruction Approach for Polychromatic Computed Tomography in Highly Limited-Data Scenarios.

Conventional strategies aimed at mitigating beam-hardening artifacts in computed tomography (CT) can be categorized into two main approaches: (1) postprocessing following conventional reconstruction and (2) iterative reconstruction incorporating a beam-hardening model. While the former fails in low-dose and/or limited-data cases, the latter substantially increases computational cost. Although deep learning-based methods have been proposed for several cases of limited-data CT, few works in the literature have dealt with beam-hardening artifacts, and none have addressed the problems caused by randomly selected projections and a highly limited span. We propose the deep learning-based prior image constrained (PICDL) framework, a hybrid method used to yield CT images free from beam-hardening artifacts in different limited-data scenarios based on the combination of a modified version of the Prior Image Constrained Compressed Sensing (PICCS) algorithm that incorporates the L2 norm (L2-PICCS) with a prior image generated from a preliminary FDK reconstruction with a deep learning (DL) algorithm. The model is based on a modification of the U-Net architecture, incorporating ResNet-34 as a replacement of the original encoder. Evaluation with rodent head studies in a small-animal CT scanner showed that the proposed method was able to correct beam-hardening artifacts, recover patient contours, and compensate streak and deformation artifacts in scenarios with a limited span and a limited number of projections randomly selected. Hallucinations present in the prior image caused by the deep learning model were eliminated, while the target information was effectively recovered by the L2-PICCS algorithm.

A 3D fast MR elastography sequence with interleaved multislab acquisition and Hadamard encoding.

To enhance the functional capability of MRI, this study aims to develop a novel MR elastography (MRE) sequence that achieves rapid acquisition without distortion artifacts. A displacement-encoded stimulated echo (DENSE) with multiphase acquisition scheme was used to capture wave images. A center-out golden-angle stack-of-stars sampling pattern was introduced for improved SNR and data incoherence. A combination of Hadamard encoding and interleaved multislab acquisition schemes was used to increase the acquisition efficiency of MRE data with multiple directions and phase offsets. A generalized parallel-imaging and compressed-sensing method was further applied to accelerate the acquisition process. The imaging results of the proposed sequence were compared with those from six gradient echo (GRE)/EPI/DENSE-based MRE sequences via phantom and brain acquisitions. The proposed sequence achieved a 6-fold acceleration compared with GRE MRE. With the application of a conventional parallel-imaging and compressed-sensing algorithm, the scanning speed was further accelerated by 8-fold, matching the speed of EPI-based MRE. Phantom tests revealed small variances in stiffness measurements across the seven sequences (< 9.23%). The proposed sequence exhibited a higher contrast-to-noise ratio (1.38) than the two EPI-based sequences (0.61/0.76) and similar to GRE-based sequences (1.34/1.22/1.58). Brain imaging validated the effectiveness of the proposed sequence in accurate stiffness estimation and distortion artifact avoidance. A rapid DENSE-based MRE sequence with interleaved multislab acquisition and Hadamard encoding was developed at a speed matching EPI-based sequences, without compromising SNR or introducing distortion artifacts.

A Lightweight Recurrent Learning Network for Sustainable Compressed Sensing

Recently, deep learning-based compressed sensing (CS) has achieved great success in reducing the sampling and computational cost of sensing systems and improving the reconstruction quality. These approaches, however, largely overlook the issue of the computational cost; they rely on complex structures and task-specific operator designs, resulting in extensive storage and high energy consumption in CS imaging systems. In this article, we propose a lightweight but effective deep neural network based on recurrent learning to achieve a sustainable CS system; it requires a smaller number of parameters but obtains high-quality reconstructions. Specifically, our proposed network consists of an initial reconstruction sub-network and a residual reconstruction sub-network. While the initial reconstruction sub-network has a hierarchical structure to progressively recover the image, reducing the number of parameters, the residual reconstruction sub-network facilitates recurrent residual feature extraction via recurrent learning to perform both feature fusion and deep reconstructions across different scales. In addition, we also demonstrate that, after the initial reconstruction, feature maps with reduced sizes are sufficient to recover the residual information, and thus we achieved a significant reduction in the amount of memory required. Extensive experiments illustrate that our proposed model can achieve a better reconstruction quality than existing state-of-the-art CS algorithms, and it also has a smaller number of network parameters than these algorithms. Our source codes are available at: <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/C66YU/CSRN</uri> .

A Highly Efficient Compressive Sensing Algorithm Based on Root-Sparse Bayesian Learning for RFPA Radar

Off-grid issues and high computational complexity are two major challenges faced by sparse Bayesian learning (SBL)-based compressive sensing (CS) algorithms used for random frequency pulse interval agile (RFPA) radar. Therefore, this paper proposes an off-grid CS algorithm for RFPA radar based on Root-SBL to address these issues. To effectively cope with off-grid issues, this paper derives a root-solving formula inspired by the Root-SBL algorithm for velocity parameters applicable to RFPA radar, thus enabling the proposed algorithm to directly solve the velocity parameters of targets during the fine search stage. Meanwhile, to ensure computational feasibility, the proposed algorithm utilizes a simple single-level hierarchical prior distribution model and employs the derived root-solving formula to avoid the refinement of velocity grids. Moreover, during the fine search stage, the proposed algorithm combines the fixed-point strategy with the Expectation-Maximization algorithm to update the hyperparameters, further reducing computational complexity. In terms of implementation, the proposed algorithm updates hyperparameters based on the single-level prior distribution to approximate values for the range and velocity parameters during the coarse search stage. Subsequently, in the fine search stage, the proposed algorithm performs a grid search only in the range dimension and uses the derived root-solving formula to directly solve for the target velocity parameters. Simulation results demonstrate that the proposed algorithm maintains low computational complexity while exhibiting stable performance for parameter estimation in various multi-target off-grid scenarios.

A dictionary learning based unsupervised neural network for single image compressed sensing

In the field of Compressed Sensing (CS), the sparse representation of signals and the advancement of reconstruction algorithms are two critical challenges. However, conventional CS algorithms often fail to sufficiently exploit the structured sparsity present in images and suffer from poor reconstruction quality. Most deep learning-based CS methods are typically trained on large-scale datasets. Obtaining a sufficient number of training sets is challenging in many practical applications and there may be no training sets available at all in some cases. In this paper, a novel deep Dictionary Learning (DL) based unsupervised neural network for single image CS (dubbed DL-CSNet) is proposed. It is an effective trainless neural network that consists of three components and their corresponding loss functions: 1) a DL layer that consists of multi-layer perceptron (MLP) and convolution neural networks (CNN) for latent sparse features extraction with the L1-norm sparsity loss function; 2) an image smoothing layer with the Total Variation (TV) like image smoothing loss function; and 3) a CS acquisition layer for image compression, with the Mean Square Error (MSE) loss function between the original image compression and the reconstructed image compression. In particular, the proposed DL-CSNet is a lightweight and fast model that does not require datasets for training and exhibits a fast convergence speed, making it suitable for deployment in resource-constrained environments. Experiments have demonstrated that the proposed DL-CSNet achieves superior performance compared to traditional CS methods and other unsupervised state-of-the-art deep learning-based CS methods.

High Resolution TOF-MRA Using Compressed Sensing-based Deep Learning Image Reconstruction for the Visualization of Lenticulostriate Arteries: A Preliminary Study.

To investigate the visibility of the lenticulostriate arteries (LSAs) in time-of-flight (TOF)-MR angiography (MRA) using compressed sensing (CS)-based deep learning (DL) image reconstruction by comparing its image quality with that obtained by the conventional CS algorithm. Five healthy volunteers were included. High-resolution TOF-MRA images with the reduction (R)-factor of 1 were acquired as full-sampling data. Images with R-factors of 2, 4, and 6 were then reconstructed using CS-DL and conventional CS (the combination of CS and sensitivity conceding; CS-SENSE) reconstruction, respectively. In the quantitative assessment, the number of visible LSAs (identified by two radiologists), length of each depicted LSA (evaluated by one radiological technologist), and normalized mean squared error (NMSE) value were assessed. In the qualitative assessment, the overall image quality and the visibility of the peripheral LSA were visually evaluated by two radiologists. In the quantitative assessment of the DL-CS images, the number of visible LSAs was significantly higher than those obtained with CS-SENSE in the R-factors of 4 and 6 (Reader 1) and in the R-factor of 6 (Reader 2). The length of the depicted LSAs in the DL-CS images was significantly longer in the R-factor 6 compared to the CS-SENSE result. The NMSE value in CS-DL was significantly lower than in CS-SENSE for R-factors of 4 and 6. In the qualitative assessment of DL-CS images, the overall image quality was significantly higher than that obtained with CS-SENSE in the R-factors 4 and 6 (Reader 1) and in the R-factor 4 (Reader 2). The visibility of the peripheral LSA was significantly higher than that shown by CS-SENSE in all R-factors (Reader 1) and in the R-factors 2 and 4 (Reader 2). CS-DL reconstruction demonstrated preserved image quality for the depiction of LSAs compared to the conventional CS-SENSE when the R-factor is elevated.

A signal-adaptive measurement matrix construction algorithm for compressed sensing of sEMG data

Abstract Surface electromyography (sEMG), serving as a pivotal wearable technology, is a promising tool to assess and monitor muscle function. Yet, the efficacy of a sEMG system faces inevitable constraints, primarily stemming from the challenges of transmission and energy consumption induced by big data. Compressed sensing (CS) is a promising data acquisition solution that takes advantage of the signal sparseness in a particular basis to significantly reduce the number of samples. Current CS methods usually employ random or deterministic measurement matrix to compress sEMG signal. However, these measurement matrices do not integrate the signal feature, which limits the performance of these CS methods. To address this problem, this paper proposes an improved CS method for sEMG data compression. This proposed method introduces a measurement matrix construction algorithm to produce a deterministic matrix tailored for processing sEMG signals. The deterministic measurement matrix integrates the characteristics of the magnitudes of sEMG signals. The sEMG signals acquired from the upper limb muscles of the stroke survivors were applied to evaluate the proposed CS method, with results showing that it achieves better reconstruction accuracy and robustness than the CS methods with other measurement matrices. The proposed method employing basis pursuit (BP) in the signal reconstruction presents better performance than that employing orthogonal matching pursuit (OMP). Hence, we can conclude that the proposed CS algorithm is of key importance for the popularization of sEMG in the wearable health monitoring devices.&amp;#xD;

AdaCS: Adaptive Compressive Sensing With Restricted Isometry Property-Based Error-Clamping.

Scene-dependent adaptive compressive sensing (CS) has been a long pursuing goal that has huge potential to significantly improve the performance of CS. However, with no access to the ground truth, how to design the scene-dependent adaptive strategy is still an open problem. In this paper, a restricted isometry property (RIP) condition-based error-clamping is proposed, which could directly predict the reconstruction error, i.e., the difference between the current-stage reconstructed image and the ground truth image, and adaptively allocate more samples to regions with larger reconstruction error at the next sampling stage. Furthermore, we propose a CS reconstruction network composed of Progressively inverse transform and Alternating Bi-directional Multi-grid Network, named PiABM-Net, that could efficiently utilize the multi-scale information for reconstructing the target image. The effectiveness of the proposed adaptive and cascaded CS method is demonstrated with extensive quantitative and qualitative experiments, compared with the state-of-the-art CS algorithms.

Diagnosis of Arrhythmia from Compressively Sensed ECG Signals Using Machine Learning Algorithms

Cardiovascular diseases (CVDs) represent a significant health concern in the present era, with Electrocardiogram (ECG) serving as a crucial bio-signal for their detection. Efficient health monitoring necessitates rapid and precise diagnosis, thereby mandating the utilization of Compressive Sensing (CS) alongside Machine Learning (ML) algorithms. CS functions as a sensing methodology that reduces sample numbers by capturing sparse or compressible representations, simplifying and expediting the acquisition process. In this proposed study, ECG signals are compressively sensed and preprocessed using CS reconstruction algorithms, followed by the application of various ML algorithms for diagnostic purposes. The assessment of the reconstructed ECG signal entails the assessment of Peak Signal-to-Noise Ratio (PSNR) values and Percentage Root-mean-square Difference (PRD). Concurrently, ML algorithms are evaluated based on metrics including accuracy, specificity, and sensitivity. This work demonstrates exceptional performance in terms of acquisition time and computational complexity through the application of CS technology. Comparative analysis with the existing methodologies for CVD diagnosis reveals the proposed approach’s remarkable efficacy. Notably, the reduction in data volume and hardware complexity serves as a significant advantage over conventional methods. The integration of CS and ML algorithms in the proposed methodology proves highly effective in diagnosing CVDs, achieving a classification accuracy of 94.7%. These results underscore the methodology’s ability to deliver both speed and accuracy in diagnosis, positioning it as a promising approach for health monitoring.

Operational Support Estimator Networks.

In this work, we propose a novel approach called Operational Support Estimator Networks (OSENs) for the support estimation task. Support Estimation (SE) is defined as finding the locations of non-zero elements in sparse signals. By its very nature, the mapping between the measurement and sparse signal is a non-linear operation. Traditional support estimators rely on computationally expensive iterative signal recovery techniques to achieve such non-linearity. Contrary to the convolutional layers, the proposed OSEN approach consists of operational layers that can learn such complex non-linearities without the need for deep networks. In this way, the performance of non-iterative support estimation is greatly improved. Moreover, the operational layers comprise so-called generative super neurons with non-local kernels. The kernel location for each neuron/feature map is optimized jointly for the SE task during training. We evaluate the OSENs in three different applications: i. support estimation from Compressive Sensing (CS) measurements, ii. representation-based classification, and iii. learning-aided CS reconstruction where the output of OSENs is used as prior knowledge to the CS algorithm for enhanced reconstruction. Experimental results show that the proposed approach achieves computational efficiency and outperforms competing methods, especially at low measurement rates by significant margins. The software implementation is shared at https://github.com/meteahishali/OSEN.

Efficient gridless 2D DOA estimation based on generalized matrix‐form atomic norm minimization

AbstractTwo‐dimensional (2D) direction‐of‐arrival (DOA) estimation is crucial in array signal processing. Compressed sensing (CS) provides a superior alternative to spatial spectrum estimation algorithms by enabling 2D DOA estimation of correlated sources from single snapshot data. However, the grid mismatch effect inherent in grid‐based CS algorithms impacts estimation accuracy. Despite recent advancements, the state‐of‐the‐art gridless CS algorithm, decoupled atomic norm minimization, is limited to specific 2D array geometries, such as uniform rectangular arrays. This letter presents an efficient gridless 2D DOA estimation algorithm for generalized rectangular arrays, including both uniform and sparse arrays. The proposed algorithm achieves high accuracy through a novel approach called generalized matrix‐form atomic norm minimization and provides a fast solution using the alternating direction method of multipliers. Validation through computer simulations and practical experiments underscores its efficacy.

Iterative Adaptive Based Multi-Polarimetric SAR Tomography of the Forested Areas

Synthetic aperture radar tomography (TomoSAR) is an extension of synthetic aperture radar (SAR) imaging. It introduces the synthetic aperture principle into the elevation direction to achieve three-dimensional (3-D) reconstruction of the observed target. Compressive sensing (CS) is a favorable technology for sparse elevation recovery. However, for the non-sparse elevation distribution of the forested areas, if CS is selected to reconstruct it, it is necessary to utilize some orthogonal bases to first represent the elevation reflectivity sparsely. The iterative adaptive approach (IAA) is a non-parametric algorithm that enables super-resolution reconstruction with minimal snapshots, eliminates the need for hyperparameter optimization, and requires fewer iterations. This paper introduces IAA to tomographicinversion of the forested areas and proposes a novel multi-polarimetric-channel joint 3-D imaging method. The proposed method relies on the characteristics of the consistent support of the elevation distribution of different polarimetric channels and uses the L2-norm to constrain the IAA-based 3-D reconstruction of each polarimetric channel. Compared with typical spectral estimation (SE)-based algorithms, the proposed method suppresses the elevation sidelobes and ambiguity and, hence, improves the quality of the recovered 3-D image. Compared with the wavelet-based CS algorithm, it reduces computational cost and avoids the influence of orthogonal basis selection. In addition, in comparison to the IAA, it demonstrates greater accuracy in identifying the support of the elevation distribution in forested areas. Experimental results based on BioSAR 2008 data are used to validate the proposed method.

Weighted Compressive Sensing Applied to Seismic Interferometry: Wavefield Reconstruction Using Prior Information

Abstract Seismic interferometry is widely used for passive subsurface investigation using seismic noise. The technique requires much storage for long noise records to suppress interferometric noise, which consists of spurious arrivals that do not correspond to the inter-receiver surface waves. Such long recordings may not be available in practice. Compressive sensing (CS), which is a wavefield reconstruction technique operating on incomplete data, may increase the availability, and reduce storage limitations of long noise time series. Using a numerical example of a linear array surrounded by sources and the Fourier basis for a sparse transform, we show that inter-receiver wavefields can be recovered at the locations where seismometers are unavailable, reducing the storage required for interferometry. We propose and develop a weighted CS algorithm that helps suppress the spurious arrivals by incorporating a priori information about the arrivals of surface waves that can be expected.

Fresnel incoherent compressive holography toward 3D videography via dual-channel simultaneous phase-shifting interferometry.

Fresnel incoherent correlation holography (FINCH) enables high-resolution 3D imaging of objects from several 2D holograms under incoherent light and has many attractive applications in motionless 3D fluorescence imaging. However, FINCH has difficulty implementing 3D imaging of dynamic scenes since multiple phase-shifting holograms need to be recorded for removing the bias term and twin image in the reconstructed scene, which requires the object to remain static during this progress. Here, we propose a dual-channel Fresnel noncoherent compressive holography method. First, a pair of holograms with π phase shifts obtained in a single shot are used for removing the bias term noise. Then, a physic-driven compressive sensing (CS) algorithm is used to achieve twin-image-free reconstruction. In addition, we analyze the reconstruction effect and suitability of the CS algorithm and two-step phase-shift filtering algorithm for objects with different complexities. The experimental results show that the proposed method can record hologram videos of 3D dynamic objects and scenes without sacrificing the imaging field of view or resolution. Moreover, the system refocuses images at arbitrary depth positions via computation, hence providing a new method for fast high-throughput incoherent 3D imaging.

An Off-Grid Compressive Sensing Algorithm Based on Sparse Bayesian Learning for RFPA Radar

In the application of Compressive Sensing (CS) theory for sidelobe suppression in Random Frequency and Pulse Repetition Interval Agile (RFPA) radar, the off−grid issues affect the performance of target parameter estimation in RFPA radar. Therefore, to address this issue, this paper presents an off−grid CS algorithm named Refinement and Generalized Double Pareto (GDP) distribution based on Sparse Bayesian Learning (RGDP−SBL) for RFPA radar that utilizes a coarse−to−fine grid refinement approach, allowing precise and cost−effective signal recovery while mitigating the impact of off−grid issues on target parameter estimation. To obtain a high-precision signal recovery, especially in scenarios involving closely spaced targets, the RGDP−SBL algorithm makes use of a three−level hierarchical prior model. Furthermore, the RGDP−SBL algorithm efficiently utilizes diagonal elements during the coarse search and exploits the convexity of the grid energy curve during the fine search, therefore significantly reducing computational complexity. Simulation results demonstrate that the RGDP−SBL algorithm significantly improves signal recovery performance while maintaining low computational complexity in multiple scenarios for RFPA radar.

A fast threshold OMP based on self-learning dictionary for propeller signal reconstruction

The acquisition of propeller signals can be applied to localization and fault diagnosis, etc. However, recording a large amount of propeller signals results in higher costs. Compressive sensing (CS) is an effective approach to processing large data. CS algorithms work well for sparse signals, but propeller signals are non-sparse in both the time and frequency domains. Therefore, the traditional fixed dictionary, such as discrete cosine transform (DCT), has a limited sparse representation of propeller signals. We propose a self-learning dictionary in this study, the self-learning dictionary uses the propeller signal as input to train the dictionary, and the trained dictionary can be stored and used for sparse representation of propeller signals. Based on the self-learning dictionary, we propose a threshold orthogonal matching pursuit based on QR decomposition (TOMP-QR) algorithm. We use the ratio of two energy thresholds as the stopping condition of the TOMP-QR algorithm, which is more robust under arbitrary noise. Then, we use QR decomposition to reduce the complexity of the algorithm. The simulation results verify the effectiveness of the proposed method in signal reconstruction accuracy and running time. Based on DCT, discrete wavelet transform (DWT) and self-learning dictionary, we compare the TOMP-QR algorithm with TOMP, OMP, sparsity adaptive matching pursuit (SAMP) and stage wise OMP (StOMP) algorithm to reconstruct propeller signals in the experiment, the results show that the effectiveness and superiority of the self-learning dictionary and TOMP-QR algorithm.

Orthogonalization of the Sensing Matrix Through Dominant Columns in Compressive Sensing for Speech Enhancement

This paper introduces a novel speech enhancement approach called dominant columns group orthogonalization of the sensing matrix (DCGOSM) in compressive sensing (CS). DCGOSM optimizes the sensing matrix using particle swarm optimization (PSO), ensuring separate basis vectors for speech and noise signals. By utilizing an orthogonal matching pursuit (OMP) based CS signal reconstruction with this optimized matrix, noise components are effectively avoided, resulting in lower noise in the reconstructed signal. The reconstruction process is accelerated by iterating only through the known speech-contributing columns. DCGOSM is evaluated against various noise types using speech quality measures such as SNR, SSNR, STOI, and PESQ. Compared to other OMP-based CS algorithms and deep neural network (DNN)-based speech enhancement techniques, DCGOSM demonstrates significant improvements, with maximum enhancements of 42.54%, 62.97%, 27.48%, and 8.72% for SNR, SSNR, PESQ, and STOI, respectively. Additionally, DCGOSM outperforms DNN-based techniques by 20.32% for PESQ and 8.29% for STOI. Furthermore, it reduces recovery time by at least 13.2% compared to other OMP-based CS algorithms.

Sparse Signal Recovery through Long Short-Term Memory Networks for Compressive Sensing-Based Speech Enhancement

This paper presents a novel speech enhancement approach based on compressive sensing (CS) which uses long short-term memory (LSTM) networks for the simultaneous recovery and enhancement of the compressed speech signals. The advantage of this algorithm is that it does not require an iterative process to recover the compressed signals, which makes the recovery process fast and straight forward. Furthermore, the proposed approach does not require prior knowledge of signal and noise statistical properties for sensing matrix optimization because the used LSTM can directly extract and learn the required information from the training data. The proposed technique is evaluated against white, babble, and f-16 noises. To validate the effectiveness of the proposed approach, perceptual evaluation of speech quality (PESQ), short-time objective intelligibility (STOI), and signal-to-distortion ratio (SDR) were compared to other variants of OMP-based CS algorithms The experimental outcomes show that the proposed approach achieves the maximum improvements of 50.06%, 43.65%, and 374.16% for PESQ, STOI, and SDR respectively, over the different variants of OMP-based CS algorithms.

FECG compressed sensing mode based on joint block sparsity

The development of medical Internet of Things technology has enabled the rise of telemedicine. For medical signals, real-time transmission, accuracy of signals and portability of equipment are required to be strict. The telemedicine system based on Compressed Sensing (CS) technology can avoid sampling a large number of redundant information, and reduce the complexity of data processing and computing time. However, the performance of existing Compressed Sensing algorithms in processing and reconstructing medical signals is poor. It is difficult to meet the strict requirements. In this paper, the sparse representation method which is most suitable for Fetal Electrocardiogram (FECG) signals is obtained by comparison experiment. And a Logistics-Tent Sine Bernoulli Measurement Matrix (LTSBM) construction algorithm is proposed for more convenient hardware applications. Based on the characteristics of FECG signals, a Joint Block Multiple Orthogonal Least Squares (JBMOLS) algorithm is proposed to improve the reconstruction performance in this paper. Combined with the above algorithms, the FECG compressed sensing mode is constituted, which is used to improve the compression, sampling and reconstruction performance of FECG signals. The experimental results show that, compared with traditional compressed sensing technology, the FECG compressed sensing mode improves the overall signal reconstruction performance, the running time is greatly reduced, and the practicability is stronger.

Multi-beam X-ray optical system for high-speed tomography using a σ-polarization diffraction geometry

A multi-beam X-ray optical system using a σ-polarization diffraction geometry is proposed and its potential for high-speed tomography using synchrotron radiation is experimentally evaluated. Projection images of a sample are obtained simultaneously from different directions with X-ray beams generated by diffraction of a white synchrotron radiation beam at silicon single crystals. This makes it possible to record a tomographic dataset without rotation of the sample or X-ray source. Data sets of two samples obtained in a proof-of-principle experiment with an exposure time of 1 ms were successfully reconstructed using an advanced compressed-sensing algorithm.