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SCIENCE CHINA Information Sciences, Volume 63 , Issue 11 : 219301(2020) https://doi.org/10.1007/s11432-020-2994-4

An improved iterative thresholding algorithm for $L_1$-norm regularization based sparse SAR imaging

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  • ReceivedApr 10, 2020
  • AcceptedJul 20, 2020
  • PublishedOct 21, 2020

Abstract

There is no abstract available for this article.


Acknowledgment

This work was partially supported by Fundamental Research Funds for the Central Universities (Grant No. NE2020004), National Natural Science Foundation of China (Grant No. 61901213), Natural Science Foundation of Jiangsu Province (Grant No. BK20190397), Aeronautical Science Foundation of China (Grant No. 201920052001), and Young Science and Technology Talent Support Project of Jiangsu Science and Technology Association.


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References

[1] Patel V M, Easley G R, Healy D M. Compressed Synthetic Aperture Radar. IEEE J Sel Top Signal Process, 2010, 4: 244-254 CrossRef ADS Google Scholar

[2] Zhang B C, Hong W, Wu Y R. Sparse microwave imaging: Principles and applications. Sci China Inf Sci, 2012, 55: 1722-1754 CrossRef Google Scholar

[3] Daubechies I, Defrise M, De Mol C. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm Pure Appl Math, 2004, 57: 1413-1457 CrossRef Google Scholar

[4] Tropp J A, Gilbert A C. Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit. IEEE Trans Inform Theor, 2007, 53: 4655-4666 CrossRef Google Scholar

[5] Bi H, Zhang B, Zhu X X. $L_{1}$ -Regularization-Based SAR Imaging and CFAR Detection via Complex Approximated Message Passing. IEEE Trans Geosci Remote Sens, 2017, 55: 3426-3440 CrossRef ADS Google Scholar

[6] Jian Fang , Zongben Xu , Bingchen Zhang . Fast Compressed Sensing SAR Imaging Based on Approximated Observation. IEEE J Sel Top Appl Earth Observations Remote Sens, 2014, 7: 352-363 CrossRef ADS arXiv Google Scholar

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    Algorithm 1 BiIST for $L_1$-norm regularization sparse imaging

    Require:Echo data ${{{\boldsymbol{y}}}}$, measurement matrix ${\bf{\Phi}}$;

    Initial: $\displaystyle{{{{\hat~{\boldsymbol~x}}}^{\left(~0~\right)}}~=~\bf{0}}~$,$\displaystyle{{{{\hat~{\boldsymbol~x}}}^{\left(~1~\right)}}~=~\bf{0}}~$,$\mu$, $\varepsilon$, maximum iterative step ${{T}_{\max~}}$;

    while $1~\le~t~\le~{{\rm~T}_{\max~}}$ and ${\rm{Resi~>~}}\varepsilon$ do

    $\displaystyle{~~~{\Delta{\boldsymbol{x}}^{\left(~{t~}~\right)}}~~~= \mu{\bf{\Phi}}^{\rm~H} (~~{\boldsymbol~y}~-~{\kern~1pt}~~\bf{\Phi}{{{\hat~{\boldsymbol~x}}}^{\left(~t-1~\right)}})}~$;

    $\displaystyle{~~~{{{{\tilde~{\boldsymbol~x}}}}^{\left(~{t}~\right)}}~= {\Delta{\boldsymbol{x}}^{\left(~{t~}~\right)}}~+~{{{{\hat~{\boldsymbol~x}}}}^{\left(~t~\right)}}}$;

    $\displaystyle{~~~{\beta^{\left(t\right)}}~~=~{{{{|{{{\tilde {\boldsymbol~x}}}}^{\left(~{t}\right)}~|}_{k~+~1}}}~/\mu~}}$;

    $\displaystyle{~~~{{{{\hat~{\boldsymbol~x}}}}^{\left(~{t~+~1}~\right)}}~= {\rm~sgn}~~({{\tilde~{\boldsymbol~x}}}^{\left(~{t}~\right)}~)\cdot~{\rm~max} (~|{{\tilde~{\boldsymbol~x}}}^{\left(~{t}~\right)}~|-\mu~\beta^{\left(t~\right)}, 0~)}$;

    $\displaystyle{~~~{\rm{Resi}}~=~{\|~{{{{{\hat~{\boldsymbol~x}}}}^{\left( {t~+~1}~\right)}}~-~{{{{\hat~{\boldsymbol~x}}}}^{\left(~t~\right)}}}~\|_2}}$;

    $~~~t~=~t+1$;

    end while

    Output:Recovered sparse image ${{\hat~{\boldsymbol~x}}}~=~{{{{\hat~{\boldsymbol~x}}}}^{\left(~{t}~\right)}}$; recovered non-sparse image ${{\tilde~{\boldsymbol~x}}}~=~{{\bf{\tilde~x}}^{\left(~{t}~\right)}}$.

  •   

    Algorithm 2 BiIST for $L_1$-decouple based sparse imaging

    Require:Echo data ${{{\boldsymbol{Y}}}}$, ${\bf{\Xi}}_a$, ${\bf{\Xi}}_r$;

    Initial: $\displaystyle{{{{\hat~{\boldsymbol~X}}}^{\left(~0~\right)}}~=~\bf{0}}$, $\displaystyle{{{{\hat~{\boldsymbol~X}}}^{\left(~1~\right)}}~=~\bf{0}}~$,$\mu$, $\varepsilon$, maximum iterative step ${{T}_{\max~}}$;

    while $1~\le~t~\le~{{\rm~T}_{\max~}}$ and ${\rm{Resi~>~}}\varepsilon$ do

    $\displaystyle{~~~{{\Delta~{\boldsymbol~X}}^{\left(~{t}~\right)}}~~~= \mu{\cal~R}({\boldsymbol{Y}}~-{\kern~1pt}~{\bf{\Xi}}_a~\circ~{\cal~M}({{ {{\hat~{\boldsymbol~X}}}}^{\left(~t-1~\right)}})\circ~{\bf{\Xi}}_r~)}~~$;

    $\displaystyle{~~~{{{{\tilde~{\boldsymbol~X}}}}^{\left(~{t}~\right)}}~= {{\Delta~{\boldsymbol~X}}^{\left(~{t}~\right)}}~~+~{{\hat~{\boldsymbol~X}}}^{\left(~t~\right)}}~$;

    $\displaystyle{~~~{\beta^{\left(t\right)}}~~=~~{{{{|{{{\tilde {\boldsymbol~X}}}}^{\left(~{t}\right)}~|}_{k~+~1}}}~/~\mu~}}$;

    $\displaystyle{~~~{{{{\hat~{\boldsymbol~X}}}}^{\left(~{t~+~1}~\right)}}= {\rm~sgn}~~({{\tilde~{\boldsymbol~X}}}^{\left(~{t}~\right)}~)\cdot~{\rm~max} (~~|{{\tilde~{\boldsymbol~X}}}^{\left(~{t}~\right)}~|-\mu~\beta^{\left(t~\right)}, 0~)}$;

    $\displaystyle{~~~{\rm{Resi}}~=~{\| {{{{{\hat~{\boldsymbol~X}}}}^{\left(~{t~+~1}~\right)}}~-~{{{{\hat~{\boldsymbol~X}}}} ^{\left(~t~\right)}}}~\|_F}}$;

    $~~~t~=~t+1$;

    end while

    Output:Recovered sparse image ${{\hat~{\boldsymbol~X}}}~=~{{{{\hat~{\boldsymbol~X}}}}^{\left(~{t}~\right)}}$; recovered non-sparse image ${{\tilde~{\boldsymbol~X}}}~=~{{{\tilde~{\boldsymbol~X}}}^{\left(~{t}~\right)}}$.