This paper investigates recovery of an undamped spectrally sparse signal and its spectral components from a set of regularly spaced samples within the framework of spectral compressed sensing and super-resolution. We show that the existing Hankel-based optimization methods suffer from the fundamental limitation that the prior of undampedness cannot be exploited. We propose a new low rank optimization model partially inspired by forward-backward processing for line spectral estimation and show its capability in restricting the spectral poles on the unit circle. We present convex relaxation approaches with the model and show their provable accuracy and robustness to bounded and sparse noise. All our results are generalized from the 1-D to arbitrary-dimensional spectral compressed sensing. Numerical simulations are provided that corroborate our analysis and show efficiency of our model and advantageous performance of our approach in improved accuracy and resolution as compared to the state-of-the-art Hankel and atomic norm methods.
Recovering a sparse signal from an undersampled set of random linear measurements is the main problem of interest in compressed sensing. In this paper, we consider the case where both the signal and the measurements are complex. We study the popular reconstruction method of $\ell_1$-regularized least squares or LASSO. While several studies have shown that the LASSO algorithm offers desirable solutions under certain conditions, the precise asymptotic performance of this algorithm in the complex setting is not yet known. In this paper, we extend the approximate message passing (AMP) algorithm to the complex signals and measurements and obtain the complex approximate message passing algorithm (CAMP). We then generalize the state evolution framework recently introduced for the analysis of AMP, to the complex setting. Using the state evolution, we derive accurate formulas for the phase transition and noise sensitivity of both LASSO and CAMP.
Compressed sensing (CS) enables the reconstruction of MR images from highly under-sampled k-space data via a constrained ℓ 1 -minimization problem. However, existing convex optimization techniques to solve such a constrained optimization problem suffer from slow convergence rate when dealing with data of a large size. On the other hand, many iterative thresholding techniques improve the convergence rate but at the cost of accuracy. In this work, we present a new iterative optimization technique to efficiently solve the constrained ℓ 1 optimization without compromising the accuracy of the solution. The key idea is to expand the sensing matrix into an orthonormal matrix, which casts the ℓ 1 constrained optimization into an equivalent convex optimization problem that can be exactly solved by the joint application of augmented Lagrange multipliers (ALM) method and alternating direction method (ADM). The proposed algorithm, dubbed as One - ℓ 1 , provides much faster convergence rate without compromising the reconstruction accuracy, when compared with commonly used optimization techniques, such as nonlinear conjugate gradient (NCG) method, as demonstrated with both phantom and in-vivo MR experiments.
This paper studies the problem of multichannel spectral super-resolution with either constant amplitude (CA) or not. We propose two optimization problems based on low-rank Hankel-Toeplitz matrix factorization. The two problems effectively leverage the multichannel and CA structures, while also enabling the design of low-complexity gradient descent algorithms for their solutions. Extensive simulations show the superior performance of the proposed algorithms.
For reconstruction of a nonnegative real valued object based on scanned data in the k-space, it is shown that the l 1 norm of the original motionless object is minimum among that of all objects with combined translational and rotational motions and is a sensitive measure of the object motions. These results provide an insight and theoretical fundamental for applying the l 1 optimization to the motion correction problem for image reconstruction in various practical applications.
Phase retrieval refers to a classical nonconvex problem of recovering a signal from its Fourier magnitude measurements. Inspired by the compressed sensing technique, signal sparsity is exploited in recent studies of phase retrieval to reduce the required number of measurements, known as compressive phase retrieval (CPR). In this paper, l1 minimization problems are formulated for CPR to exploit the signal sparsity and alternating direction algorithms are presented for problem solving. For real-valued, nonnegative image reconstruction, the image of interest is shown to be an optimal solution of the formulated l1 minimization in the noise free case. Numerical simulations demonstrate that the proposed approach is fast, accurate and robust to measurements noises.
Compressive multichannel frequency estimation refers to the process of retrieving the frequency profile shared by multiple signals from their compressive samples. A recent approach to this problem relies on atomic norm minimization which exploits joint sparsity among the channels, is solved using convex optimization, and has strong theoretical guarantees. We provide in this paper an average-case analysis for atomic norm minimization by assuming proper randomness on the amplitudes of the frequencies. We show that the sample size per channel required for exact frequency estimation from noiseless samples decreases as the number of channels increases and is on the order of K log K (1+ 1/L log N), where K is the number of frequencies, L is the number of channels, and N is a fixed parameter proportional to the sampling window size and inversely proportional to the desired resolution.
This paper focuses on the problem of source localization using time-of-arrival (TOA) measurements. Differently from the existing studies assuming that TOA measurement noises are independent and identically distributed, we deal with more practical TOA measurements suffering from heteroscedastic noises due to different physical distances between a source and multiple sensors. Consequently, the variance of the heteroscedastic noise is distance-dependent. From the information point of view, the distance-dependent variance contains certain information about the source location, and as such helps to improve localization accuracy. In this paper, we theoretically analyze the impact of the heteroscedastic noises on localization accuracy, and conclude that using the extra information in the distance-dependent variance does not significantly improve the estimation accuracy under low noise levels. Furthermore, considering the fact that TOA measurements are normally accurate, we propose a lightweight localization scheme based on the existing two-stage maximum likelihood (TSML) method. Finally, a simulation analysis confirms our theoretical study, and shows that the accuracy of the proposed localization scheme is comparable to the Cramer-Rao lower bound (CRLB) given moderate TOA measurement noises.
Direction augmentation (DA) and spatial smoothing (SS), followed by a subspace method such as ESPRIT or MUSIC, are two simple and successful approaches that enable localization of more uncorrelated sources than sensors with a proper sparse array. In this paper, we carry out nonasymptotic performance analyses of DA-ESPRIT and SS-ESPRIT in the practical finite-snapshot regime. We show that their absolute localization errors are bounded from above by $C_1\frac{\max\{\sigma^2, C_2\}}{\sqrt{L}}$ with overwhelming probability, where $L$ is the snapshot number, $\sigma^2$ is the Gaussian noise power, and $C_1,C_2$ are constants independent of $L$ and $\sigma^2$, if and only if they can do exact source localization with infinitely many snapshots. We also show that their resolution increases with the snapshot number, without a substantial limit. Numerical results corroborating our analysis are provided.
Diversity smoothing has been widely used for angle estimation with multiple input multiple output (MIMO) radar in the presence of coherent or correlated targets, and the parameter identifiability is an interesting issue. Previous works have shown sufficient conditions for some special cases, such as the coherent targets with conventional MIMO radar, and the array size are derived as a function of the target number and target structure. In this article, we further introduce the interelement spacings to build a complete parameter identifiability scheme. For monostatic MIMO radars, the antenna numbers and interelement spacings of transmit and receive arrays are derived as functions of the target number and the target structure. The optimization models are built to calculate the maximum number of detectable targets for a given target structure. Additionally, the conditions for the bistatic MIMO radar are derived from two-dimension viewpoint. It is shown that the new results improve upon previous spatial smoothing or diversity smoothing methods and recover them in special cases. Simulation results are presented that corroborate our theoretical findings.
'/%3e%3cuse xlink:href='%23a' transform='rotate(23.036 .093 25.536)'/%3e%3cuse xlink:href='%23a' transform='rotate(45.87 1.273 16.18)'/%3e%3cuse xlink:href='%23a' transform='rotate(69.945 .996 12.078)'/%3e%3cuse xlink:href='%23a' transform='rotate(20.66 -19.689 31.932)'/%3e%3c/svg%3e)
'/%3e%3cuse xlink:href='%23a' transform='translate(288.889 -214.211)'/%3e%3cuse xlink:href='%23a' transform='translate(108 342.749)'/%3e%3cuse xlink:href='%23a' transform='translate(469.189 342.749)'/%3e%3c/svg%3e)
'%3e%3cpath fill='%23012169' d='M0 0v30h60V0z'/%3e%3cpath stroke='%23fff' stroke-width='6' d='m0 0 60 30m0-30L0 30'/%3e%3cpath stroke='%23C8102E' stroke-width='4' d='m0 0 60 30m0-30L0 30' clip-path='url(%23b)'/%3e%3cpath stroke='%23fff' stroke-width='10' d='M30 0v30M0 15h60'/%3e%3cpath stroke='%23C8102E' stroke-width='6' d='M30 0v30M0 15h60'/%3e%3c/g%3e%3c/svg%3e)
'/%3e%3c/defs%3e%3cpath fill='%23012169' d='M0 0h10080v5040H0z'/%3e%3cpath stroke='%23fff' stroke-width='.6' d='m0 0 6 3m0-3L0 3' clip-path='url(%23b)' transform='scale(840)'/%3e%3cpath stroke='%23e4002b' stroke-width='.4' d='m0 0 6 3m0-3L0 3' clip-path='url(%23c)' transform='scale(840)'/%3e%3cpath stroke='%23fff' stroke-width='840' d='M2520 0v2520M0 1260h5040'/%3e%3cpath stroke='%23e4002b' stroke-width='504' d='M2520 0v2520M0 1260h5040'/%3e%3cg fill='%23fff'%3e%3cuse xlink:href='%23d' x='2520' y='3780'/%3e%3cuse xlink:href='%23a' x='7560' y='4200'/%3e%3cuse xlink:href='%23a' x='6300' y='2205'/%3e%3cuse xlink:href='%23a' x='7560' y='840'/%3e%3cuse xlink:href='%23a' x='8680' y='1869'/%3e%3cuse xlink:href='%23e' x='8064' y='2730'/%3e%3c/g%3e%3c/svg%3e)