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Compressed sensing: how sharp is the restricted isometry property?
, 2009
"... Compressed sensing is a recent technique by which signals can be measured at a rate proportional to their information content, combining the important task of compression directly into the measurement process. Since its introduction in 2004 there have been hundreds of manuscripts on compressed sens ..."
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Cited by 51 (7 self)
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Compressed sensing is a recent technique by which signals can be measured at a rate proportional to their information content, combining the important task of compression directly into the measurement process. Since its introduction in 2004 there have been hundreds of manuscripts on compressed sensing, a large fraction of which have focused on the design and analysis of algorithms to recover a signal from its compressed measurements. The Restricted Isometry Property (RIP) has become a ubiquitous property assumed in their analysis. We present the best known bounds on the RIP, and in the process illustrate the way in which the combinatorial nature of compressed sensing is controlled. Our quantitative bounds on the RIP allow precise statements as to how aggressively a signal can be undersampled, the essential question for practitioners.
Linear array SAR imaging via compressed sensing
- Progress In Electromagnetics Research
, 2011
"... Abstract—In recent years, various attempts have been undertaken to obtain three-dimensional (3-D) reflectivity of observed scene from synthetic aperture radar (SAR) technique. Linear array SAR (LASAR) has been demonstrated as a promising technique to achieve 3-D imaging of earth surface. The common ..."
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Cited by 16 (2 self)
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Abstract—In recent years, various attempts have been undertaken to obtain three-dimensional (3-D) reflectivity of observed scene from synthetic aperture radar (SAR) technique. Linear array SAR (LASAR) has been demonstrated as a promising technique to achieve 3-D imaging of earth surface. The common methods used for LASAR imaging are usually based on matched filter (MF) which obeys the traditional Nyquist sampling theory. However, due to limitation in the length of linear array and the “Rayleigh ” resolution, the standard MF-based methods suffer from low resolution and high sidelobes. Hence, high resolution imaging algorithms are desired. In LASAR images, dominating scatterers are always sparse compared with the total 3-D illuminated space cells. Combined with this prior knowledge of sparsity property, this paper presents a novel algorithm for LASAR imaging via compressed sensing (CS). The theory of CS indicates that sparse signal can be exactly reconstructed in high Signal-Noise-Ratio (SNR) level by solving a convex optimization problem with a very small number of samples. To overcome strong noise and clutter interference in LASAR raw echo, the new method firstly achieves range focussing by a pulse compression technique, which can greatly improve SNR level of signal in both azimuth and cross-track directions. Then, the resolution enhancement images of sparse targets are reconstructed by L1 norm regularization. High resolution properties and point localization accuracies are tested and verified by simulation and real experimental data. The results show that the CS method outperforms the conventional MF-based methods, even if very small random selected samples are used.
A novel image formation algorithm for high-resolution wide-swath spaceborne SAR using compressed sensing on azimuth displacement phase center antenna
- Progress In Electromagnetics Research
"... Abstract—High-resolution wide-swath (HRWS) imaging with space-borne synthetic aperture radar (SAR) can be achieved by using az-imuth displacement phase center antenna (DPCA) technique. How-ever, it will consequently leads to extremely high data rate on satellite downlink system. A novel sparse sampl ..."
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Cited by 10 (2 self)
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Abstract—High-resolution wide-swath (HRWS) imaging with space-borne synthetic aperture radar (SAR) can be achieved by using az-imuth displacement phase center antenna (DPCA) technique. How-ever, it will consequently leads to extremely high data rate on satellite downlink system. A novel sparse sampling scheme based on compressed sensing (CS) theory for azimuth DPCA SAR was proposed, by which only a small proportion of radar echoes are utilized for imaging to re-duce data rate. The corresponding image formation algorithm for the proposed scheme was presented in the paper. The SAR echo signal of each channel can be reconstructed with high probability by using orthogonal matching pursuit (OMP) algorithm in Doppler frequency domain. The reconstructed echo signals of each channel are jointly processed by means of spectrum reconstructing filter for compensat-ing Doppler spectrum aliasing resulting from non-uniform sampling in azimuth direction. The high quality SAR image can be obtained by us-ing chirp scaling algorithm. The effectiveness of the proposed approach was validated by computer simulations using both point targets and distributed targets. 1.
Gradient-Based Image Recovery Methods From Incomplete Fourier Measurements
"... Abstract—A major problem in imaging applications such as magnetic resonance imaging and synthetic aperture radar is the task of trying to reconstruct an image with the smallest possible set of Fourier samples, every single one of which has a potential time and/or power cost. The theory of compressiv ..."
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Cited by 5 (0 self)
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Abstract—A major problem in imaging applications such as magnetic resonance imaging and synthetic aperture radar is the task of trying to reconstruct an image with the smallest possible set of Fourier samples, every single one of which has a potential time and/or power cost. The theory of compressive sensing (CS) points to ways of exploiting inherent sparsity in such images in order to achieve accurate recovery using sub-Nyquist sampling schemes. Traditional CS approaches to this problem consist of solving total-variation (TV) minimization programs with Fourier measurement constraints or other variations thereof. This paper takes a different approach. Since the horizontal and vertical differences of a medical image are each more sparse or compressible than the corresponding TV image, CS methods will be more successful in recovering these differences individually. We develop an algorithm called GradientRec that uses a CS algorithm to recover the horizontal and vertical gradients and then estimates the original image from these gradients. We present two methods of solving the latter inverse problem, i.e., one based on least-square optimization and the other based on a generalized Poisson solver. After a thorough derivation of our complete algorithm, we present the results of various experiments that compare the effectiveness of the proposed method against other leading methods. Index Terms—Compressed sensing, Fourier transforms, image reconstruction, L1-minimization, Poisson solver, sparse recovery,
Random Steerable Arrays for Synthetic Aperture Imaging
, 2013
"... In classical spotlight-mode synthetic aperture radar (SAR), a mobile sensor array is steered to always focus on the same area (spot) as it moves, transmitting pulses to illuminate the spot and receiving and processing their reflections. The result is a high resolution image, covering a relatively sm ..."
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Cited by 3 (3 self)
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In classical spotlight-mode synthetic aperture radar (SAR), a mobile sensor array is steered to always focus on the same area (spot) as it moves, transmitting pulses to illuminate the spot and receiving and processing their reflections. The result is a high resolution image, covering a relatively small area. In this paper, we propose using a randomly steerable sensor array for synthetic aperture imaging, aiming to increase the coverage area without sacrificing the imaging resolution. This is realized by steering the beam of the array such that each transmitted pulse illuminates one of two or more spots, randomly selected with equal probability. Each of those spots has the same size as a single spot in a classical array, effectively doubling (or more) the total area illuminated. Using principles from compressive sensing (CS) we demonstrate that it is possible to reconstruct the images of all illuminated areas by exploiting the structure of the reconstructed images. Our experimental results demonstrate that our random steerable array can double coverage with almost the same imaging resolution.
ISAR imaging of nonuniform rotation targets with limited pulses via compressed sensing
- Progress In Electromagnetics Research B
"... Abstract—This research introduces compressed sensing (CS) principle into inverse synthetic aperture radar (ISAR) imaging of nonuniform rotation targets, and high azimuth resolution can be achieved with limited number of pulses. Firstly, the sparsity of the echoed signal of radar targets with non-uni ..."
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Cited by 3 (2 self)
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Abstract—This research introduces compressed sensing (CS) principle into inverse synthetic aperture radar (ISAR) imaging of nonuniform rotation targets, and high azimuth resolution can be achieved with limited number of pulses. Firstly, the sparsity of the echoed signal of radar targets with non-uniform rotation in certain matching Fourier domain is analyzed. Then the restricted isometry property (RIP) and incoherence of partial matching Fourier matrices are checked, following which an ISAR imaging method based on CS for both random sparse aperture and short aperture cases is proposed. In particular, considering the dependence of the sparse dictionary on the relative rotation parameter, a parameter estimation method by the optimal search in fractional Fourier domain is presented. Simulation experiments verify the effectiveness as well as superiority of the proposed imaging method over traditional methods in terms of imaging performance. 1.
Compressive passive millimeter wave imaging with extended depth of field,” Opt
- Eng
, 2012
"... Abstract. In this paper, we introduce a millimeter wave imaging modality with extended depthof-field that provides diffraction limited images with reduced spatial sampling. The technique uses a cubic phase element in the pupil of the system and a nonlinear recovery algorithm to produce images that a ..."
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Cited by 2 (0 self)
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Abstract. In this paper, we introduce a millimeter wave imaging modality with extended depthof-field that provides diffraction limited images with reduced spatial sampling. The technique uses a cubic phase element in the pupil of the system and a nonlinear recovery algorithm to produce images that are insensitive to object distance. We present experimental results that validate system performance and demonstrate a greater than four-fold increase in depth-of-field with a reduction in sampling requirements by a factor of at least two. c ○ 2012 Society of Photo-Optical Instrumentation Engineers. DOI: 10.0000/XXXX Subject terms: Computational imaging, millimeter wave imaging, extended depth-of-field, image
Article Compressive SAR Imaging with Joint Sparsity and Local Similarity Exploitation
"... www.mdpi.com/journal/sensors ..."
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Coprime conditions for Fourier sampling for sparse recovery
- 8th IEEE Workshop on Sensor Array and Multichannel Signal Processing
, 2014
"... Abstract-This paper considers the spark of L × N submatrices of the N × N Discrete Fourier Transform (DFT) matrix. Here a matrix has spark m if every collection of its m − 1 columns are linearly independent. The motivation comes from such applications of compressed sensing as MRI and synthetic aper ..."
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Cited by 1 (1 self)
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Abstract-This paper considers the spark of L × N submatrices of the N × N Discrete Fourier Transform (DFT) matrix. Here a matrix has spark m if every collection of its m − 1 columns are linearly independent. The motivation comes from such applications of compressed sensing as MRI and synthetic aperture radar, where device physics dictates the measurements to be Fourier samples of the signal. Consequently the observation matrix comprises certain rows of the DFT matrix. To recover an arbitrary k-sparse signal, the spark of the observation matrix must exceed 2k + 1. The technical question addressed in this paper is how to choose the rows of the DFT matrix so that its spark equals the maximum possible value L + 1. We expose certain coprimeness conditions that guarantee such a property.
