Compressive Sensing Recovery Research Articles

Using compressive sensing to recover images from PET scanners with partial detector rings.

Most positron emission tomography/computed tomography (PET/CT) scanners consist of tightly packed discrete detector rings to improve scanner efficiency. The authors' aim was to use compressive sensing (CS) techniques in PET imaging to investigate the possibility of decreasing the number of detector elements per ring (introducing gaps) while maintaining image quality. A CS model based on a combination of gradient magnitude and wavelet domains (wavelet-TV) was developed to recover missing observations in PET data acquisition. The model was designed to minimize the total variation (TV) and L1-norm of wavelet coefficients while constrained by the partially observed data. The CS model also incorporated a Poisson noise term that modeled the observed noise while suppressing its contribution by penalizing the Poisson log likelihood function. Three experiments were performed to evaluate the proposed CS recovery algorithm: a simulation study, a phantom study, and six patient studies. The simulation dataset comprised six disks of various sizes in a uniform background with an activity concentration of 5:1. The simulated image was multiplied by the system matrix to obtain the corresponding sinogram and then Poisson noise was added. The resultant sinogram was masked to create the effect of partial detector removal and then the proposed CS algorithm was applied to recover the missing PET data. In addition, different levels of noise were simulated to assess the performance of the proposed algorithm. For the phantom study, an IEC phantom with six internal spheres each filled with F-18 at an activity-to-background ratio of 10:1 was used. The phantom was imaged twice on a RX PET/CT scanner: once with all detectors operational (baseline) and once with four detector blocks (11%) turned off at each of 0 ˚, 90 ˚, 180 ˚, and 270° (partially sampled). The partially acquired sinograms were then recovered using the proposed algorithm. For the third test, PET images from six patient studies were investigated using the same strategy of the phantom study. The recovered images using WTV and TV as well as the partially sampled images from all three experiments were then compared with the fully sampled images (the baseline). Comparisons were done by calculating the mean error (%bias), root mean square error (RMSE), contrast recovery (CR), and SNR of activity concentration in regions of interest drawn in the background as well as the disks, spheres, and lesions. For the simulation study, the mean error, RMSE, and CR for the WTV (TV) recovered images were 0.26% (0.48%), 2.6% (2.9%), 97% (96%), respectively, when compared to baseline. For the partially sampled images, these results were 22.5%, 45.9%, and 64%, respectively. For the simulation study, the average SNR for the baseline was 41.7 while for WTV (TV), recovered image was 44.2 (44.0). The phantom study showed similar trends with 5.4% (18.2%), 15.6% (18.8%), and 78% (60%), respectively, for the WTV (TV) images and 33%, 34.3%, and 69% for the partially sampled images. For the phantom study, the average SNR for the baseline was 14.7 while for WTV (TV) recovered image was 13.7 (11.9). Finally, the average of these values for the six patient studies for the WTV-recovered, TV, and partially sampled images was 1%, 7.2%, 92% and 1.3%, 15.1%, 87%, and 27%, 25.8%, 45%, respectively. CS with WTV is capable of recovering PET images with good quantitative accuracy from partially sampled data. Such an approach can be used to potentially reduce the cost of scanners while maintaining good image quality.

Doppler Ambiguity Resolution Based on Random Sparse Probing Pulses

A novel method for solving Doppler ambiguous problem based on compressed sensing (CS) theory is proposed in this paper. A pulse train with the random and sparse transmitting time is transmitted. The received signals after matched filtering can be viewed as randomly sparse sampling from the traditional fixed-pulse repetition frequency (PRF) echo signals. The whole target echo could be reconstructed via CS recovery algorithms. Through refining the sensing matrix, which is equivalent to increase the sampling frequency of target characteristic, the Doppler unambiguous range is enlarged. In particular, Complex Approximate Message Passing (CAMP) algorithm is developed to estimate the unambiguity Doppler frequency. Cramer-Rao lower bound expressions are derived for the frequency. Numerical simulations validate the effectiveness of the proposed method. Finally, compared with traditional methods, the proposed method only requires transmitting a few sparse probing pulses to achieve a larger Doppler frequency unambiguous range and can also reduce the consumption of the radar time resources.

Weighted overlapped recovery for blocking artefacts reduction in block‐based compressive sensing of images

In compressive sensing (CS) of images, a block-based framework is preferred to avoid the huge memory and computation required for a frame-based approach. However, the recovered image suffers from blocking artefacts due to independent block processing, especially at a low subrate. As a result of this reported work the artifacts are reduced by weighted averaging adopting two techniques: overlapped CS recovery and adaptive weighting. Simulation results show its improvement in both subjective and objective qualities.

Energy-efficient multi-mode compressed sensing system for implantable neural recordings.

Widely utilized in the field of Neuroscience, implantable neural recording devices could capture neuron activities with an acquisition rate on the order of megabytes per second. In order to efficiently transmit neural signals through wireless channels, these devices require compression methods that reduce power consumption. Although recent Compressed Sensing (CS) approaches have successfully demonstrated their power, their full potential is yet to be explored. Built upon our previous on-chip CS implementation, we propose an energy efficient multi-mode CS framework that focuses on improving the off-chip components, including (i) a two-stage sensing strategy, (ii) a sparsifying dictionary directly using data, (iii) enhanced compression performance from Full Signal CS mode and Spike Restoration mode to Spike CS + Restoration mode and; (iv) extension of our framework to the Tetrode CS recovery using joint sparsity. This new framework achieves energy efficiency, implementation simplicity and system flexibility simultaneously. Extensive experiments are performed on simulation and real datasets. For our Spike CS + Restoration mode, we achieve a compression ratio of 6% with a reconstruction SNDR > 10 dB and a classification accuracy > 95% for synthetic datasets. For real datasets, we get a 10% compression ratio with ∼ 10 dB for Spike CS + Restoration mode.

Antenna without Phase Center for DOA Estimation in Compressive Array Processing

Recently, Compressive Sensing (CS) has been applied to array signal processing. In theory, Direction-of-Arrival (DOA) estimation based on CS recovery can work well in correlated environments. However, a large number of sensors (i.e., linear measurements) are still needed for CS recovery. To improve on this, we propose a new CS-based DOA estimation method with a recently designed antenna structure called the Asymmetric Antenna without Phase Center (Asym-AWPC). The best reconstruction is achieved by solving the l1-norm optimization problem, which is cast as an l1-regularized least-squares program. Simulated results indicate the effectiveness of the proposed CS-based Asym-AWPC DOA estimator in a multipath environment over a recent Asym-AWPC DOA estimator but using the Multiple Signal Classification (MUSIC) rather than CS. Further improvement on the resolution can be achieved by tuning the degree of asymmetry in designing the Asym-AWPC.

In-situ Soil Moisture Sensing

We consider the problem of monitoring soil moisture evolution using a wireless network of in-situ underground sensors. To reduce cost and prolong lifetime, it is highly desirable to rely on fewer measurements and estimate with higher accuracy the original signal (the temporal evolution of soil moisture). In this article, we explore the use of results from the theory of sparse sampling, including Compressive Sensing (CS) and Matrix Completion (MC), in this application context. We first consider the problem of reconstructing the soil moisture process at a single location using CS. Our physical constraint leads to very sparse measurement matrices, which makes finding a suitable representation basis very challenging: it needs to make the underlying signal sufficiently sparse while at the same time being sufficiently incoherent with the measurement matrix, two common preconditions for CS techniques to work well. We construct a representation basis by exploiting unique features of soil moisture evolution and show that this basis attains a very good tradeoff between its ability to sparsify the signal and its incoherence with measurement matrices that are consistent with our physical constraints. We next consider the problem of jointly reconstructing soil moisture processes at multiple locations, assuming sparse measurements can be taken at each location. We show that the spatial soil moisture process enjoys a low-rank property, a priority for MC. Accordingly, we introduce a spatiotemporal measurement matrix and apply the MC framework to reconstruct the soil moisture field. Extensive numerical evaluation is performed on both real, high-resolution soil moisture data and simulated data and through comparison with a closed-loop scheduling approach. Our results demonstrate that, for a single location, a uniform measurement scheduling followed by CS recovery results in a very nice tradeoff between estimation accuracy, sampling rate, flexibility, and feasibility in implementation. When multiple locations are available, our results show that joint reconstruction using MC in general produces better estimation accuracy than using a single location alone, but it requires the use of independent and random measurement schedules across locations. We also show that these sparse sampling techniques can be augmented so as to be robust against sporadic data outliers/corruption caused by, for example, intermittent sensor faults.

Generalized compressive detection of stochastic signals using Neyman–Pearson theorem

Compressive sensing (CS) enables reconstructing a sparse signal from fewer samples than those required by the classic Nyquist sampling theorem. In general, CS signal recovery algorithms have high computational complexity. However, several signal processing problems such as signal detection and classification can be tackled directly in the compressive measurement domain. This makes recovering the original signal from its compressive measurements not necessary in these applications. We consider in this paper detecting stochastic signals with known probability density function from their compressive measurements. We refer to it as the compressive detection problem to highlight that the detection task can be achieved via directly exploring the compressive measurements. The Neyman–Pearson (NP) theorem is applied to derive the NP detectors for Gaussian and non-Gaussian signals. Our work is more general over many existing literature in the sense that we do not require the orthonormality of the measurement matrix, and the compressive detection problem for stochastic signals is generalized from the case of Gaussian signals to the case of non-Gaussian signals. Theoretical performance results of the proposed NP detectors in terms of their detection probability and the false alarm rate averaged over the random measurement matrix are established. They are verified via extensive computer simulations.

Compressed Sensing Electron tomography using adaptive dictionaries: a simulation study

Electron tomography (ET) is an increasingly important technique for examining the three-dimensional morphologies of nanostructures. ET involves the acquisition of a set of 2D projection images to be reconstructed into a volumetric image by solving an inverse problem. However, due to limitations in the acquisition process this inverse problem is considered ill-posed (i.e., no unique solution exists). Furthermore reconstruction usually suffers from missing wedge artifacts (e.g., star, fan, blurring, and elongation artifacts). Compressed sensing (CS) has recently been applied to ET and showed promising results for reducing missing wedge artifacts caused by limited angle sampling. CS uses a nonlinear reconstruction algorithm that employs image sparsity as a priori knowledge to improve the accuracy of density reconstruction from a relatively small number of projections compared to other reconstruction techniques. However, The performance of CS recovery depends heavily on the degree of sparsity of the reconstructed image in the selected transform domain. Prespecified transformations such as spatial gradients provide sparse image representation, while synthesising the sparsifying transform based on the properties of the particular specimen may give even sparser results and can extend the application of CS to specimens that can not be sparsely represented with other transforms such as Total variation (TV). In this work, we show that CS reconstruction in ET can be significantly improved by tailoring the sparsity representation using a sparse dictionary learning principle.

Construction of Incoherent Unit Norm Tight Frames With Application to Compressed Sensing

Despite the important properties of unit norm tight frames (UNTFs) and equiangular tight frames (ETFs), their construction has been proven extremely difficult. The few known techniques produce only a small number of such frames while imposing certain restrictions on frame dimensions. Motivated by the application of incoherent tight frames in compressed sensing (CS), we propose a methodology to construct incoherent UNTFs. When frame redundancy is not very high, the achieved maximal column correlation becomes close to the lowest possible bound. The proposed methodology may construct frames of any dimensions. The obtained frames are employed in CS to produce optimized projection matrices. Experimental results show that the proposed optimization technique improves CS signal recovery, increasing the reconstruction accuracy. Considering that the UNTFs and ETFs are important in sparse representations, channel coding, and communications, we expect that the proposed construction will be useful in other applications, besides the CS.

Separable data hiding in encrypted image based on compressive sensing

A novel scheme for separable data hiding in an encrypted image based on compressive sensing (CS) is presented. In the first phase, the content owner encrypts the original image with an encryption key and leaves some space for the data-hider to embed data. Therefore the image is enlarged. The data-hider embeds additional data to the image for a specific purpose in a fixed location. As the image is enlarged, the data-hider can compress the image to the original size using CS with the data-hiding key. Thus, the encrypted image with embedded data is generated. The way to obtain image content and additional data is separable. If a receiver has the encryption key, he can decrypt the unintelligent data to obtain an image similar to the original one, but cannot extract the additional data. Similarly, if a receiver has the data-hiding key, he can extract the watermark without seeing the content of the image. If the receiver has both the encryption key and the data-hiding key, he can extract the additional data and recover the original image by exploiting CS recovery algorithms.

Sub-Nyquist Radar via Doppler Focusing

We investigate the problem of a monostatic pulse-Doppler radar transceiver trying to detect targets sparsely populated in the radar's unambiguous time-frequency region. Several past works employ compressed sensing (CS) algorithms to this type of problem but either do not address sample rate reduction, impose constraints on the radar transmitter, propose CS recovery methods with prohibitive dictionary size, or perform poorly in noisy conditions. Here, we describe a sub-Nyquist sampling and recovery approach called Doppler focusing, which addresses all of these problems: it performs low rate sampling and digital processing, imposes no restrictions on the transmitter, and uses a CS dictionary with size, which does not increase with increasing number of pulses P. Furthermore, in the presence of noise, Doppler focusing enjoys a signal-to-noise ratio (SNR) improvement, which scales linearly with P, obtaining good detection performance even at SNR as low as - 25 dB. The recovery is based on the Xampling framework, which allows reduction of the number of samples needed to accurately represent the signal, directly in the analog-to-digital conversion process. After sampling, the entire digital recovery process is performed on the low rate samples without having to return to the Nyquist rate. Finally, our approach can be implemented in hardware using a previously suggested Xampling radar prototype.

Measurement Coding for Compressive Sensing of Color Images

From the perspective of reducing the sampling cost of color images at high resolution, block-based compressive sensing (CS) has attracted considerable attention as a promising alternative to conventional Nyquist/Shannon sampling. On the other hand, for storing/transmitting applications, CS requires a very efficient way of representing the measurement data in terms of data volume. This paper addresses this problem by developing a measurement-coding method with the proposed customized Huffman coding. In addition, by noting the difference in visual importance between the luma and chroma channels, this paper proposes measurement coding in YCbCr space rather than in conventional RGB color space for better rate allocation. Furthermore, as the proper use of the image property in pursuing smoothness improves the CS recovery, this paper proposes the integration of a low pass filter to the CS recovery of color images, which is the block-based l<SUB>20 </SUB>-norm minimization. The proposed coding scheme shows considerable gain compared to conventional measurement coding.

Multi-Resolution Kronecker Compressive Sensing

From the perspective of reducing the sampling cost of color images at high resolution, block-based compressive sensing (CS) has attracted considerable attention as a promising alternative to conventional Nyquist/Shannon sampling. On the other hand, for storing/transmitting applications, CS requires a very efficient way of representing the measurement data in terms of data volume. This paper addresses this problem by developing a measurement-coding method with the proposed customized Huffman coding. In addition, by noting the difference in visual importance between the luma and chroma channels, this paper proposes measurement coding in YCbCr space rather than in conventional RGB color space for better rate allocation. Furthermore, as the proper use of the image property in pursuing smoothness improves the CS recovery, this paper proposes the integration of a low pass filter to the CS recovery of color images, which is the block-based -norm minimization. The proposed coding scheme l<SUB>20</SUB> shows considerable gain compared to conventional measurement coding.

Compressed Sensing Methods for Electron Tomography of Cellular Structure

Optimization-based image recovery algorithms exploiting sparse representations, such as compressed sensing (CS), are emerging as powerful tools in image denoising and dosage reduction for biomedical imaging. The efficacy of CS recovery for any imaging application hinges on the structure of the images to be reconstructed, the structure of the measurement process, and the structure of image noise. In particular, the utility of CS-ET for undersampled tomogram reconstruction relies on the image satisfying constraints of an a priori image structural model, a dependency, which hints at the theoretical connections between CS and regularization techniques for image recovery. We examine these issues in the context of electron tomography (ET) for membranous cellular organelles, analyzing the ultrastructural sparsity in the datasets, and addressing the structure of the shot noise. Using numerical simulations, we compare reconstructions from the standard tilt series acquisition technique and a randomized tilt-angle variant, against a benchmark CS measurement scheme. To assess reconstruction quality, CS reconstructions were compared with weighted backprojections of two classes of phantom: (1) derived from experimental data, and (2) derived from a generative model that approximates the structure of membrane-bound organelles. We are investigating the application of the CS approach for decreasing acquisition times and electron dose, and for improving the reconstruction quality of electron tomograms recorded from micrometer-thick specimens of neuronal tissues in the scanning transmission electron microscope.This work was supported in part by the Intramural Research Program of NIBIB, NIH.

Optimal arrays for compressed sensing in snapshot-mode radio interferometry

Radio interferometry has always faced the problem of incomplete sampling of the Fourier plane. A possible remedy can be found in the promising new theory of compressed sensing (CS), which allows for the accurate recovery of sparse signals from sub-Nyquist sampling given certain measurement conditions. We provide an introductory assessment of optimal arrays for CS in snapshot-mode radio interferometry, using orthogonal matching pursuit (OMP), a widely used CS recovery algorithm similar in some respects to CLEAN. We focus on centrally condensed (specifically, Gaussian) arrays versus uniform arrays, and the principle of randomization versus deterministic arrays such as the VLA. The theory of CS is grounded in $a)$ sparse representation of signals and $b)$ measurement matrices of low coherence. We calculate a related quantity, mutual coherence (MC), as a theoretical indicator of arrays' suitability for OMP based on the recovery error bounds in (Donoho et al. 2006). OMP reconstructions of both point and extended objects are also run from simulated incomplete data. Optimal arrays are considered for object recovery through 1) the natural pixel representation and 2) the representation by the block discrete cosine transform (BDCT). We find that reconstructions of the pixel representation perform best with the uniform random array, while reconstructions of the BDCT representation perform best with normal random arrays. Slight randomization to the VLA also improves it hugely for CS with the pixel basis. In the pixel basis, array design for CS reflects known principles of array design for small numbers of antennas, namely of randomness and uniform distribution. Differing results with the BDCT, however, emphasize the importance of studying how sparsifying bases affect array design before CS can be optimized.

Study on Compressed Sensing Recovery Algorithms

Abastruct. Compressive sensing is a novel signal sampling theory under the condition that the signalis sparse or compressible.In this case,the small amount of signal values can be reconstructed when signal is sparse or compressible.This paper has reviewed the idea of OMP,GBP and SP,given algorithms and analyzed the experiment results,suggested some improvements.

Wireless sensor networks data recovery algorithm based on quadratic programming

For improving the real-time performance of recovery algorithm in Compressed Sensing(CS) of Wireless Sensor Networks(WSN) data,a quadratic programming based network data recovery algorithm was proposed in this paper.The CS recovery was transformed to bound-constrained quadratic programming,and then the network data was recovered by solving the quadratic programming problem based on the Armijo rule.The analysis and experimental results demonstrate that the proposed algorithm can significantly reduce the complexity and ensure the accuracy of recovery,thus improving the real-time performance of data recovery in WSN.

Image compressive sensing recovery using adaptively learned sparsifying basis via L0 minimization

From many fewer acquired measurements than suggested by the Nyquist sampling theory, compressive sensing (CS) theory demonstrates that, a signal can be reconstructed with high probability when it exhibits sparsity in some domain. Most of the conventional CS recovery approaches, however, exploited a set of fixed bases (e.g. DCT, wavelet and gradient domain) for the entirety of a signal, which are irrespective of the non-stationarity of natural signals and cannot achieve high enough degree of sparsity, thus resulting in poor CS recovery performance. In this paper, we propose a new framework for image compressive sensing recovery using adaptively learned sparsifying basis via L0 minimization. The intrinsic sparsity of natural images is enforced substantially by sparsely representing overlapped image patches using the adaptively learned sparsifying basis in the form of L0 norm, greatly reducing blocking artifacts and confining the CS solution space. To make our proposed scheme tractable and robust, a split Bregman iteration based technique is developed to solve the non-convex L0 minimization problem efficiently. Experimental results on a wide range of natural images for CS recovery have shown that our proposed algorithm achieves significant performance improvements over many current state-of-the-art schemes and exhibits good convergence property.

Energy-Efficient Distributed Data Storage for Wireless Sensor Networks Based on Compressed Sensing and Network Coding

Recently, distributed data storage (DDS) for Wireless Sensor Networks (WSNs) has attracted great attention, especially in catastrophic scenarios. Since power consumption is one of the most critical factors that affect the lifetime of WSNs, the energy efficiency of DDS in WSNs is investigated in this paper. Based on Compressed Sensing (CS) and network coding theories, we propose a Compressed Network Coding based Distributed data Storage (CNCDS) scheme by exploiting the correlation of sensor readings. The CNCDS scheme achieves high energy efficiency by reducing the total number of transmissions Nttot and receptions Nrtot during the data dissemination process. Theoretical analysis proves that the CNCDS scheme guarantees good CS recovery performance. In order to theoretically verify the efficiency of the CNCDS scheme, the expressions for Nttot and Nrtot are derived based on random geometric graphs (RGG) theory. Furthermore, based on the derived expressions, an adaptive CNCDS scheme is proposed to further reduce Nttot and Nrtot. Simulation results validate that, compared with the conventional ICStorage scheme, the proposed CNCDS scheme reduces Nttot, Nrtot, and the CS recovery mean squared error (MSE) by up to 55%, 74%, and 76% respectively. In addition, compared with the CNCDS scheme, the adaptive CNCDS scheme further reduces Nttot and Nrtot by up to 63% and 32% respectively.

Data Pre-Processing in Compressive Sensing Based Indoor Fingerprinting Positioning

Presented is a novel compressive sensing (CS) based indoor positioning approach, which uses the signal strength differentials (SSDs) as location fingerprints (LFs). By using certain kernel-based transformation basis, the 2-D target location is represented as an unknown sparse location vector in the discrete spatial domain. Then it just takes a little number of online noisy SSD measurements for the exact recovery of the sparse location vector by solving an l1-minimization program. In order to effectively apply CS theory for high precision indoor positioning, we further import some data pre-processing algorithms in the LF space. Firstly, to mitigate the influence of large measurements noise on the recovery accuracy, a LF space denosing algorithm is designed to discriminate the unequal localization contribution rate of every SSD measurement in each LF. The basic idea of the denosing algorithm is to transform the original LF space into a robust and decorrelated LF space. Moreover, in order to lower the high computational complexity of the CS recovery algorithm, several LF space filtering algorithms are also exploited to remove certain percentage of useless LFs in the radio map according to the real-time RSS observations. The performance of these denosing and filtering algorithms are investigated and compared in real-world WLAN experiment test. Both experimental results and simulations demonstrate that we achieve remarkable improvements on the positioning performance of the CS based localization by using the proposed algorithms.