SAMV (algorithm)

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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 environments (e.g., limited number of snapshots and 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 M {\displaystyle M} {\displaystyle M}-element uniform linear array (ULA) receives K {\displaystyle K} {\displaystyle K} narrow band signals emitted from sources located at locations θ = { θ a , … , θ K } {\displaystyle \mathbf {\theta } =\{\theta _{a},\ldots ,\theta _{K}\}} {\displaystyle \mathbf {\theta } =\{\theta _{a},\ldots ,\theta _{K}\}}, respectively. The sensors in the ULA accumulates N {\displaystyle N} {\displaystyle N} snapshots over a specific time. The M × 1 {\displaystyle M\times 1} {\displaystyle M\times 1} dimensional snapshot vectors are

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

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

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

R = A P A H + σ I . {\displaystyle {\bf {R}}={\bf {A}}{\bf {P}}{\bf {A}}^{H}+\sigma {\bf {I}}.} {\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 R N = Y Y H / N {\displaystyle {\bf {R}}_{N}={\bf {Y}}{\bf {Y}}^{H}/N} {\displaystyle {\bf {R}}_{N}={\bf {Y}}{\bf {Y}}^{H}/N} where Y = [ y ( 1 ) , … , y ( N ) ] {\displaystyle {\bf {Y}}=[{\bf {y}}(1),\ldots ,{\bf {y}}(N)]} {\displaystyle {\bf {Y}}=[{\bf {y}}(1),\ldots ,{\bf {y}}(N)]}. After applying the vectorization operator to the matrix R {\displaystyle {\bf {R}}} {\displaystyle {\bf {R}}}, the obtained vector r ( p ) = vec ⁡ ( R ) {\displaystyle {\bf {r}}({\boldsymbol {\bf {p}}})=\operatorname {vec} ({\bf {R}})} {\displaystyle {\bf {r}}({\boldsymbol {\bf {p}}})=\operatorname {vec} ({\bf {R}})} is linearly related to the unknown parameter p {\displaystyle {\boldsymbol {\bf {p}}}} {\displaystyle {\boldsymbol {\bf {p}}}} as

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

where S = [ S 1 , a ¯ K + 1 ] {\displaystyle {\bf {S}}=[{\bf {S}}_{1},{\bar {\bf {a}}}_{K+1}]} {\displaystyle {\bf {S}}=[{\bf {S}}_{1},{\bar {\bf {a}}}_{K+1}]}, S 1 = [ a ¯ 1 , … , a ¯ K ] {\displaystyle {\bf {S}}_{1}=[{\bar {\bf {a}}}_{1},\ldots ,{\bar {\bf {a}}}_{K}]} {\displaystyle {\bf {S}}_{1}=[{\bar {\bf {a}}}_{1},\ldots ,{\bar {\bf {a}}}_{K}]}, a ¯ k = a k ∗ ⊗ a k {\displaystyle {\bar {\bf {a}}}_{k}={\bf {a}}_{k}^{*}\otimes {\bf {a}}_{k}} {\displaystyle {\bar {\bf {a}}}_{k}={\bf {a}}_{k}^{*}\otimes {\bf {a}}_{k}}, k = 1 , … , K {\displaystyle k=1,\ldots ,K} {\displaystyle k=1,\ldots ,K}, and let a ¯ K + 1 = vec ⁡ ( I ) {\displaystyle {\bar {\bf {a}}}_{K+1}=\operatorname {vec} ({\bf {I}})} {\displaystyle {\bar {\bf {a}}}_{K+1}=\operatorname {vec} ({\bf {I}})} where ⊗ {\displaystyle \otimes } {\displaystyle \otimes } is the Kronecker product.

SAMV algorithm

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

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

where S d = d r ( p ) / d p {\displaystyle {\bf {S}}_{d}={\rm {d}}{\bf {r}}({\boldsymbol {p}})/{\rm {d}}{\boldsymbol {p}}} {\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 p ^ {\displaystyle {\hat {\bf {p}}}} {\displaystyle {\hat {\bf {p}}}} obtained by minimizing

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

where f ( p ) = [ r N − r ( p ) ] H C r − 1 [ r N − r ( p ) ] . {\displaystyle f({\boldsymbol {p}})=[{\bf {r}}_{N}-{\bf {r}}({\boldsymbol {p}})]^{H}{\bf {C}}_{r}^{-1}[{\bf {r}}_{N}-{\bf {r}}({\boldsymbol {p}})].} {\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 p {\displaystyle {\boldsymbol {\bf {p}}}} {\displaystyle {\boldsymbol {\bf {p}}}} can be obtained iteratively.

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

p ^ k ( i + 1 ) = a k H R − 1 ( i ) R N R − 1 ( i ) a k ( a k H R − 1 ( i ) a k ) 2 + p ^ k ( i ) − 1 a k H R − 1 ( i ) a k , k = 1 , … , K {\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 {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}
σ ^ ( i + 1 ) = ( Tr ⁡ ( R − 2 ( i ) R N ) + σ ^ ( i ) Tr ⁡ ( R − 2 ( i ) ) − Tr ⁡ ( R − 1 ( i ) ) ) / Tr ⁡ ( R − 2 ( i ) ) , {\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)}})}},} {\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 R {\displaystyle {\bf {R}}} {\displaystyle {\bf {R}}} at the i {\displaystyle i} {\displaystyle i}th iteration is given by R ( i ) = A P ( i ) A H + σ ^ ( i ) I {\displaystyle {\bf {R}}^{(i)}={\bf {A}}{\bf {P}}^{(i)}{\bf {A}}^{H}+{\hat {\sigma }}^{(i)}{\bf {I}}} {\displaystyle {\bf {R}}^{(i)}={\bf {A}}{\bf {P}}^{(i)}{\bf {A}}^{H}+{\hat {\sigma }}^{(i)}{\bf {I}}} with P ( i ) = Diag ⁡ ( p ^ 1 ( i ) , … , p ^ K ( i ) ) {\displaystyle {\bf {P}}^{(i)}=\operatorname {Diag} ({\hat {p}}_{1}^{(i)},\ldots ,{\hat {p}}_{K}^{(i)})} {\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 x ( n ) {\displaystyle \mathbf {x} (n)} {\displaystyle \mathbf {x} (n)} is dependent on the distance between the adjacent element in the overcomplete dictionary A {\displaystyle {\bf {A}}} {\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 θ = ( θ 1 , … , θ K ) T {\displaystyle {\boldsymbol {\bf {\theta }}}=(\theta _{1},\ldots ,\theta _{K})^{T}} {\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 θ k {\displaystyle \theta _{k}} {\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 30 {\displaystyle 30} {\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 5 {\displaystyle 5} {\displaystyle 5} dB power and the rest six are of 25 {\displaystyle 25} {\displaystyle 25} dB power. The received signals are assumed to be contaminated with uniform white Gaussian noise of 0 {\displaystyle 0} {\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 5 {\displaystyle 5} {\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 5 {\displaystyle 5} {\displaystyle 5} dB targets.

Open source implementation

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

See also

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. S2CID 16276001.
  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. Bibcode:2014SigPr..95...15G. doi:10.1016/j.sigpro.2013.08.003. S2CID 41743051.
  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. S2CID 33008200.
  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. hdl:1721.1/87445. S2CID 6876056.
  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. S2CID 18834345.