SAMV (algorithm)

SAMV (iterative sparse asymptotic minimum variance[1][2]) is a parameter-free superresolution algorithm for the linear inverse problem in spectral estimation, direction-of-arrival (DOA) estimation and tomographic reconstruction with applications in signal processing, medical imaging and remote sensing. The name was coined in 2013[1] 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 environment (e.g., limited number of snapshots, low signal-to-noise ratio. Applications include synthetic-aperture radar[2][3], computed tomography scan, and magnetic resonance imaging (MRI).

Definition

The formulation of the SAMV algorithm is given as an inverse problem in the context of DOA estimation. Suppose an ${\displaystyle M}$-element uniform linear array (ULA) receive ${\displaystyle K}$ narrow band signals emitted from sources located at locations ${\displaystyle \mathbf {\theta } =\{\theta _{a},\ldots ,\theta _{K}\}}$, respectively. The sensors in the ULA accumulates ${\displaystyle N}$ snapshots over a specific time. The ${\displaystyle M\times 1}$ dimensional snapshot vectors are

${\displaystyle \mathbf {y} (n)=\mathbf {A} \mathbf {x} (n)+\mathbf {e} (n),n=1,\ldots ,N}$

where ${\displaystyle \mathbf {A} =[\mathbf {a} (\theta _{1}),\ldots ,\mathbf {a} (\theta _{K})]}$ is the steering matrix, ${\displaystyle {\bf {x}}(n)=[{\bf {x}}_{1}(n),\ldots ,{\bf {x}}_{K}(n)]^{T}}$ contains the source waveforms, and ${\displaystyle {\bf {e}}(n)}$ is the noise term. Assume that ${\displaystyle \mathbf {E} \left({\bf {e}}(n){\bf {e}}^{H}({\bar {n}})\right)=\sigma {\bf {I}}_{M}\delta _{n,{\bar {n}}}}$, where ${\displaystyle \delta _{n,{\bar {n}}}}$ is the Dirac delta and it equals to 1 only if ${\displaystyle n={\bar {n}}}$ and 0 otherwise. Also assume that ${\displaystyle {\bf {e}}(n)}$ and ${\displaystyle {\bf {x}}(n)}$ are independent, and that ${\displaystyle \mathbf {E} \left({\bf {x}}(n){\bf {x}}^{H}({\bar {n}})\right)={\bf {P}}\delta _{n,{\bar {n}}}}$, where ${\displaystyle {\bf {P}}=\operatorname {Diag} ({p_{1},\ldots ,p_{K}})}$. Let ${\displaystyle {\bf {p}}}$ be a vector containing the unknown signal powers and noise variance, ${\displaystyle {\bf {p}}=[p_{1},\ldots ,p_{K},\sigma ]^{T}}$.

The covariance matrix of ${\displaystyle {\bf {y}}(n)}$ that contains all information about ${\displaystyle {\boldsymbol {\bf {p}}}}$ is

${\displaystyle {\bf {R}}={\bf {A}}{\bf {P}}{\bf {A}}^{H}+\sigma {\bf {I}}.}$

This covariance matrix can be traditionally estimated by the sample covariance matrix ${\displaystyle {\bf {R}}_{N}={\bf {Y}}{\bf {Y}}^{H}/N}$ where ${\displaystyle {\bf {Y}}=[{\bf {y}}(1),\ldots ,{\bf {y}}(N)]}$. After applying the vectorization operator to the matrix ${\displaystyle {\bf {R}}}$, the obtained vector ${\displaystyle {\bf {r}}({\boldsymbol {\bf {p}}})=\operatorname {vec} ({\bf {R}})}$ is linearly related to the unknown parameter ${\displaystyle {\boldsymbol {\bf {p}}}}$ as

${\displaystyle {\bf {r}}({\boldsymbol {\bf {p}}})=\operatorname {vec} ({\bf {R}})={\bf {S}}{\boldsymbol {\bf {p}}}}$,

where ${\displaystyle {\bf {S}}=[{\bf {S}}_{1},{\bar {\bf {a}}}_{K+1}]}$, ${\displaystyle {\bf {S}}_{1}=[{\bar {\bf {a}}}_{1},\ldots ,{\bar {\bf {a}}}_{K}]}$, ${\displaystyle {\bar {\bf {a}}}_{k}={\bf {a}}_{k}^{*}\otimes {\bf {a}}_{k}}$, ${\displaystyle k=1,\ldots ,K}$, and let ${\displaystyle {\bar {\bf {a}}}_{K+1}=\operatorname {vec} ({\bf {I}})}$.

SAMV algorithm

To estimate the parameter ${\displaystyle {\boldsymbol {\bf {p}}}}$ from the statistic ${\displaystyle {\bf {r}}_{N}}$, we develop a series of iterative SAMV approaches based on the asymptotically minimum variance criterion. From [1], the covariance matrix ${\displaystyle \operatorname {Cov} _{\boldsymbol {p}}^{\operatorname {Alg} }}$ of an arbitrary consistent estimator of ${\displaystyle {\boldsymbol {p}}}$ based on the second-order statistic ${\displaystyle {\bf {r}}_{N}}$ is bounded by the real symmetric positive definite matrix

${\displaystyle \operatorname {Cov} _{\boldsymbol {p}}^{\operatorname {Alg} }\geq [{\bf {S}}_{d}^{H}{\bf {C}}_{r}^{-1}{\bf {S}}_{d}]^{-1},}$

where ${\displaystyle {\bf {S}}_{d}={\rm {d}}{\bf {r}}({\boldsymbol {p}})/{\rm {d}}{\boldsymbol {p}}}$. In addition, this lower bound is attained by the covariance matrix of the asymptotic distribution of ${\displaystyle {\hat {\bf {p}}}}$ obtained by minimizing,

${\displaystyle {\hat {\boldsymbol {p}}}=\arg \min _{\boldsymbol {p}}f({\boldsymbol {p}}),}$

where ${\displaystyle f({\boldsymbol {p}})=[{\bf {r}}_{N}-{\bf {r}}({\boldsymbol {p}})]^{H}{\bf {C}}_{r}^{-1}[{\bf {r}}_{N}-{\bf {r}}({\boldsymbol {p}})].}$

Therefore, the estimate of ${\displaystyle {\boldsymbol {\bf {p}}}}$ can be obtained iteratively.

The ${\displaystyle \{{\hat {p}}_{k}\}_{k=1}^{K}}$ and ${\displaystyle {\hat {\sigma }}}$ that minimize ${\displaystyle f({\boldsymbol {p}})}$ can be computed as follows. Assume ${\displaystyle {\hat {p}}_{k}^{(i)}}$ and ${\displaystyle {\hat {\sigma }}^{(i)}}$ have been approximated to a certain degree in the ${\displaystyle i}$th iteration, they can be refined at the ${\displaystyle (i+1)}$th iteration by,

${\displaystyle {\hat {p}}_{k}^{(i+1)}={\frac {{\bf {a}}_{k}^{H}{\bf {R}}^{-1{(i)}}{\bf {R}}_{N}{\bf {R}}^{-1{(i)}}{\bf {a}}_{k}}{({\bf {a}}_{k}^{H}{\bf {R}}^{-1{(i)}}{\bf {a}}_{k})^{2}}}+{\hat {p}}_{k}^{(i)}-{\frac {1}{{\bf {a}}_{k}^{H}{\bf {R}}^{-1{(i)}}{\bf {a}}_{k}}},\quad k=1,\ldots ,K}$

${\displaystyle {\hat {\sigma }}^{(i+1)}=\left(\operatorname {Tr} ({\bf {R}}^{-2^{(i)}}{\bf {R}}_{N})+{\hat {\sigma }}^{(i)}\operatorname {Tr} ({\bf {R}}^{-2^{(i)}})-\operatorname {Tr} ({\bf {R}}^{-1^{(i)}})\right)/{\operatorname {Tr} {({\bf {R}}^{-2^{(i)}})}},}$

where the estimate of ${\displaystyle {\bf {R}}}$ at the ${\displaystyle i}$th iteration is given by ${\displaystyle {\bf {R}}^{(i)}={\bf {A}}{\bf {P}}^{(i)}{\bf {A}}^{H}+{\hat {\sigma }}^{(i)}{\bf {I}}}$ with ${\displaystyle {\bf {P}}^{(i)}=\operatorname {Diag} ({\hat {p}}_{1}^{(i)},\ldots ,{\hat {p}}_{K}^{(i)})}$.

Beyond scanning grid accuracy

The resolution of most compressed sensing based source localization techniques is limited by the fineness of the direction grid that covers the location parameter space.[4] In the sparse signal recovery model, the sparsity of the truth signal ${\displaystyle \mathbf {x} (n)}$ is dependent on the distance between the adjacent element in the overcomplete dictionary ${\displaystyle {\bf {A}}}$, 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[1], which refine the location estimates ${\displaystyle {\boldsymbol {\bf {\theta }}}=(\theta _{1},\ldots ,\theta _{K})^{T}}$ by iteratively minimizing a stochastic maximum likelihood cost function with respect to a single scalar parameter ${\displaystyle \theta _{k}}$.

Application to range-Doppler imaging

SISO range Doppler imaging results comparison with three 5 dB and six 25 dB targets. (a) ground truth, (b) matched filter (MF), (c) IAA algorithm, (d) SAMV-0 algorithm. All power levels are in dB. Both MF and IAA methods are limited in resolution with respect to the doppler axis. SAMV-0 offers superior resolution in terms of both range and doppler. [1]

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[5], and a variant of the SAMV algorithm (SAMV-0). The simulation conditions are identical to [5]: A ${\displaystyle 30}$-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 ${\displaystyle 5}$ dB power and the rest six are of ${\displaystyle 25}$ dB power. The received signals are assumed to be contaminated with uniform white Gaussian noise of ${\displaystyle 0}$ 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 ${\displaystyle 5}$ 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 ${\displaystyle 5}$ dB targets.

Open source implementation

An open source MATLAB implementation of SAMV algorithm could be downloaded here.

See also

• Array processing
• Matched filter
• Periodogram
• Filtered backprojection (Radon transform)
• MUltiple SIgnal Classification (MUSIC), a popular parametric superresolution method
• Pulse-Doppler radar
• Super-resolution imaging
• Compressed sensing
• Inverse problem
• Tomographic reconstruction

References

1. Abeida, Habti; Zhang, Qilin; Li, Jian; Merabtine, Nadjim (2013). "Iterative Sparse Asymptotic Minimum Variance Based Approaches for Array Processing" (PDF). IEEE Transactions on Signal Processing. 61 (4): 933–944. arXiv:1802.03070. Bibcode:2013ITSP...61..933A. doi:10.1109/tsp.2012.2231676. ISSN 1053-587X.
2. Glentis, George-Othon; Zhao, Kexin; Jakobsson, Andreas; Abeida, Habti; Li, Jian (2014). "SAR imaging via efficient implementations of sparse ML approaches" (PDF). Signal Processing. 95: 15–26. doi:10.1016/j.sigpro.2013.08.003.
3. Yang, Xuemin; Li, Guangjun; Zheng, Zhi (2015-02-03). "DOA Estimation of Noncircular Signal Based on Sparse Representation". Wireless Personal Communications. 82 (4): 2363–2375. doi:10.1007/s11277-015-2352-z.
4. Malioutov, D.; Cetin, M.; Willsky, A.S. (2005). "A sparse signal reconstruction perspective for source localization with sensor arrays". IEEE Transactions on Signal Processing. 53 (8): 3010–3022. Bibcode:2005ITSP...53.3010M. doi:10.1109/tsp.2005.850882.
5. Yardibi, Tarik; Li, Jian; Stoica, Petre; Xue, Ming; Baggeroer, Arthur B. (2010). "Source Localization and Sensing: A Nonparametric Iterative Adaptive Approach Based on Weighted Least Squares". IEEE Transactions on Aerospace and Electronic Systems. 46 (1): 425–443. Bibcode:2010ITAES..46..425Y. doi:10.1109/taes.2010.5417172. hdl:1721.1/59588.