SCIENTIA SINICA Informationis, Volume 49, Issue 6: 726(2019) https://doi.org/10.1360/N112017-00195

Multispectral bioluminescence tomography-based general iterative shrinkage and threshold algorithm

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  • ReceivedJan 20, 2018
  • AcceptedFeb 5, 2018
  • PublishedMay 16, 2019


Bioluminescence tomography (BLT) is a noninvasive optical molecular imaging modality with high sensitivity. The complexity of near-infrared light transmission in biological tissues and the limitation of measurable information place a higher demand on BLT source reconstruction algorithms. In this paper, we present a reconstruction algorithm based on general iterative shrinkage and threshold (GIST), which uses a non-convex smoothly clipped absolute deviation function as the penalty term, and solves a proximal operator problem that has a closed-form solution for the penalty. In addition, we utilize multispectral measurements and an iteratively shrinking permissible region strategy to address the ill-posedness of the BLT inverse problem. To investigate the source location and multi-source resolution abilities of the proposed method, we perform comparisons between three typical sparse reconstruction algorithms based on several groups of simulations and phantom experiments. The reconstruction results demonstrate great advantages of the proposed GIST algorithm in terms of source location accuracy in all considered source settings with different source depths and separations.

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    Algorithm 1 GIST

    Choose parameter $\eta$, ${t_{\min~}}$, and ${t_{\max~}}$, which satisfy $\eta~>~1$, and $0~<~{t_{\min~}}~<~{t_{\max~}}$, (in this paper, $\eta~{\rm{~=~2}}$, ${t_{\min~}}~=$ 1E $-$ 20, ${t_{\max~}}~=~1{\rm~E}+20$);

    Initialize iteration number $k~=~0,{\rm{~count}}~=~0$ and the initial solution ${{\boldsymbol{q}}^{(0)}}={\boldsymbol~0}$;

    Choose an step size ${t^{\left(~k~\right)}}~\in~\left[~{{t_{\min~}},{t_{\max~}}}~\right]$ according to (11), and for $k=0$, ${t^{\left(~0~\right)}}{\rm{~=~}}1$;

    Update ${{\boldsymbol{q}}^{\left(~{k~+~1}~\right)}}~\leftarrow~\mathop~{\arg~\min~}\limits_{\boldsymbol{q}}~{\rm{~}}l({{\boldsymbol{q}}^{\left(~k~\right)}})~+~\left\langle~{\nabla~l({{\boldsymbol{q}}^{\left(~k~\right)}}),{\boldsymbol{q}}~-~{{\boldsymbol{q}}^{\left(~k~\right)}}}~\right\rangle~+~\frac{{{t^{\left(~k~\right)}}}}{2}{\left\|~{{\boldsymbol{q}}~-~{{\boldsymbol{q}}^{\left(~k~\right)}}}~\right\|^2}~+~r({\boldsymbol{q}})$ according to (14)–(16);

    Update step size ${t^{\left(~k~\right)}}~\leftarrow~\eta~{t^{\left(~k~\right)}}$;

    If $f({{\boldsymbol{q}}^{\left(~{k~+~1}~\right)}})~\le~\mathop~{\max~}\limits_{i~=~\max~(0,k~-~m~+~1),~\ldots~,k}~f({{\boldsymbol{q}}^{\left(~i~\right)}})~-~\frac{\sigma~}{2}{t^{\left(~k~\right)}}{\left\|~{{{\boldsymbol{q}}^{\left(~{k~+~1}~\right)}}~-~{{\boldsymbol{q}}^{\left(~k~\right)}}}~\right\|^2}$, $\sigma\in(0,1)$, go to step 4; else go to step 7;

    If $|~{\frac{{f({\boldsymbol{q}}^k)~-~f({\boldsymbol{q}}^{(k~+~1))}}}{{f({\boldsymbol{q}}^{(k~+~1)})}}}~|~<~\tau$, count $\leftarrow$ count + 1, (we set $\tau~=~1{\rm~E}~-~3$ in this paper);

    If count $<$ 5 and $k<20$, $k~\leftarrow~k~+~1$, and go to step 3; else output the solution ${\boldsymbol{q}}^{(k+1)}$.

  • Table 1   Optical properties for simulation and phantom experiments
    Wavelength (nm)Simulation [33]Phantom [34]
    $\mu_a$ (mm$^{-1}$)$\mu&apos;_s$ (mm$^{-1}$)$\mu_a$ (mm$^{-1}$)$\mu&apos;_s$ (mm$^{-1}$)
  • Table 2   Reconstruction results of phantom experiment Case 1
    MethodSource Reconstructed Single CoM Grouped CoM
    CoM (mm)deviation (mm)deviation (mm)
    GISTS1($-$0.44, 2.62, 0.49)0.9930.671
    S2($-$0.73, $-$1.22, 0.38) 0.684
    FISTAS1($-$0.54, 2.59, 0.60)1.0650.975
    S2($-$1.03, $-$1.21, 0.65)0.922
    IRW-L$_{1/2}$S1($-$0.81, 1.34, 0.36)1.8820.577
    S2($-$0.37, $-$1.96, $-$0.01)0.697
    IVTCGS1($-$0.61, 1.96, 0.57)1.449 0.941
    S2($-$0.79, $-$2.79, 0.70)1.537
  • Table 3   Reconstruction results of phantom experiment Case 2
    MethodSource Reconstructed Single CoM Grouped CoM
    CoM (mm)deviation (mm)deviation (mm)
    GISTS1($-$0.11, $-$3.13, $-$0.76)1.4550.732
    S2(6.19, $-$2.67, $-$0.21) 0.834
    FISTAS1(0.59, $-$3.07, $-$2.34)2.9841.503
    S2(5.87, $-$2.79, 0.29)0.750
    IRW-L$_{1/2}$S1(1.25, $-$2.87, $-$2.48)3.5231.651
    S2(5.89, $-$2.81, 0.48)0.911
    IVTCGS1(0.58, $-$3.07, $-$2.35)2.987 1.509
    S2(5.87, $-$2.78, 0.29)0.749

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