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SAMV (iterative sparse asymptotic minimum variance) is a parameter-free
superresolution Super-resolution imaging (SR) is a class of techniques that enhance (increase) the resolution of an imaging system. In optical SR the diffraction limit of systems is transcended, while in geometrical SR the resolution of digital imaging sensors i ...
algorithm for the linear
inverse problem An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the ...
in
spectral estimation In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the power spectral density) of a signal from a sequence of time samples of the signa ...
, direction-of-arrival (DOA) estimation and
tomographic reconstruction Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections. The mathematical basis for tomographic imaging was laid down by Johann ...
with applications in
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, d ...
,
medical imaging Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to rev ...
and
remote sensing Remote sensing is the acquisition of information about an object or phenomenon without making physical contact with the object, in contrast to in situ or on-site observation. The term is applied especially to acquiring information about Earth ...
. The name was coined in 2013 to emphasize its basis on the asymptotically minimum variance (AMV) criterion. It is a powerful tool for the recovery of both the amplitude and frequency characteristics of multiple highly correlated sources in challenging environments (e.g., limited number of snapshots and low
signal-to-noise ratio Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in de ...
). Applications include synthetic-aperture radar, computed tomography scan, and magnetic resonance imaging (MRI).


Definition

The formulation of the SAMV algorithm is given as an
inverse problem An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the ...
in the context of DOA estimation. Suppose an M-element uniform linear array (ULA) receive K narrow band signals emitted from sources located at locations \mathbf = \, respectively. The sensors in the ULA accumulates N snapshots over a specific time. The M \times 1 dimensional snapshot vectors are : \mathbf(n) = \mathbf \mathbf(n) + \mathbf(n), n = 1, \ldots, N where \mathbf = \mathbf(\theta_1), \ldots, \mathbf(\theta_K) /math> is the steering matrix, (n)= 1(n), \ldots, _K(n)T contains the source waveforms, and (n) is the noise term. Assume that \mathbf\left((n)^H(\bar)\right)= \sigma_M\delta_, where \delta_ is the Dirac delta and it equals to 1 only if n=\bar and 0 otherwise. Also assume that (n) and (n) are independent, and that \mathbf\left((n)^H(\bar)\right)=\delta_, where = \operatorname( ). Let be a vector containing the unknown signal powers and noise variance, = _1,\ldots,p_K, \sigmaT. The
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square Matrix (mathematics), matrix giving the covariance between ea ...
of (n) that contains all information about \boldsymbol is : = ^H+\sigma. This covariance matrix can be traditionally estimated by the sample covariance matrix _ = ^H/N where = 1), \ldots,(N)/math>. After applying the vectorization operator to the matrix , the obtained vector (\boldsymbol) = \operatorname() is linearly related to the unknown parameter \boldsymbol as (\boldsymbol) = \operatorname()=\boldsymbol, where = 1,\bar_/math>, _1 = bar_1,\ldots,\bar_K/math>, \bar_k = ^_k \otimes_k, k=1,\ldots, K, and let \bar_ = \operatorname() where \otimes is the Kronecker product.


SAMV algorithm

To estimate the parameter \boldsymbol from the statistic _N, we develop a series of iterative SAMV approaches based on the asymptotically minimum variance criterion. From, the covariance matrix \operatorname^\operatorname_ of an arbitrary consistent estimator of \boldsymbol based on the second-order statistic _N is bounded by the real symmetric positive definite matrix : \operatorname^\operatorname_\geq H_d ^_r_d, where _d = (\boldsymbol)/ \boldsymbol. In addition, this lower bound is attained by the covariance matrix of the asymptotic distribution of \hat obtained by minimizing, : \hat =\arg \min_ f(\boldsymbol), where f(\boldsymbol) = N-(\boldsymbol)H _r^ N-(\boldsymbol) Therefore, the estimate of \boldsymbol can be obtained iteratively. The \_^K and \hat that minimize f(\boldsymbol) can be computed as follows. Assume \hat^_k and \hat^ have been approximated to a certain degree in the ith iteration, they can be refined at the (i+1)th iteration by, : \hat^_k = \frac+\hat^_k-\frac, \quad k=1, \ldots,K : \hat^ = \left(\operatorname(^_N) + \hat^\operatorname(^) -\operatorname(^)\right)/, where the estimate of at the ith iteration is given by ^=^^H+\hat^ with ^=\operatorname(\hat^_1, \ldots, \hat^_K).


Beyond scanning grid accuracy

The resolution of most
compressed sensing Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems. This ...
based source localization techniques is limited by the fineness of the direction grid that covers the location parameter space. In the sparse signal recovery model, the sparsity of the truth signal \mathbf(n) is dependent on the distance between the adjacent element in the overcomplete dictionary , therefore, the difficulty of choosing the optimum overcomplete dictionary arises. The computational complexity is directly proportional to the fineness of the direction grid, a highly dense grid is not computational practical. To overcome this resolution limitation imposed by the grid, the grid-free SAMV-SML (iterative Sparse Asymptotic Minimum Variance - Stochastic Maximum Likelihood) is proposed, which refine the location estimates \boldsymbol=(\theta_1,\ldots,\theta_K)^T by iteratively minimizing a stochastic
maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statis ...
cost function with respect to a single scalar parameter \theta_k.


Application to range-Doppler imaging

A typical application with the SAMV algorithm in SISO radar/ sonar range-Doppler imaging problem. This imaging problem is a single-snapshot application, and algorithms compatible with single-snapshot estimation are included, i.e., matched filter (MF, similar to the periodogram or backprojection, which is often efficiently implemented as fast Fourier transform (FFT)), IAA, and a variant of the SAMV algorithm (SAMV-0). The simulation conditions are identical to: A 30-element polyphase pulse compression P3 code is employed as the transmitted pulse, and a total of nine moving targets are simulated. Of all the moving targets, three are of 5 dB power and the rest six are of 25 dB power. The received signals are assumed to be contaminated with uniform white Gaussian noise of 0 dB power. The matched filter detection result suffers from severe smearing and leakage effects both in the Doppler and range domain, hence it is impossible to distinguish the 5 dB targets. On contrary, the IAA algorithm offers enhanced imaging results with observable target range estimates and Doppler frequencies. The SAMV-0 approach provides highly sparse result and eliminates the smearing effects completely, but it misses the weak 5 dB targets.


Open source implementation

An open source
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementatio ...
implementation of SAMV algorithm could be downloade
here


See also

*
Array processing Array processing is a wide area of research in the field of signal processing that extends from the simplest form of 1 dimensional line arrays to 2 and 3 dimensional array geometries. Array structure can be defined as a set of sensors that are sp ...
* Matched filter * Periodogram *
Filtered backprojection In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the ...
(Radon transform) * MUltiple SIgnal Classification (MUSIC), a popular parametric
superresolution Super-resolution imaging (SR) is a class of techniques that enhance (increase) the resolution of an imaging system. In optical SR the diffraction limit of systems is transcended, while in geometrical SR the resolution of digital imaging sensors i ...
method * Pulse-Doppler radar * Super-resolution imaging *
Compressed sensing Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems. This ...
*
Inverse problem An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the ...
*
Tomographic reconstruction Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections. The mathematical basis for tomographic imaging was laid down by Johann ...


References

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