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SCIENTIA SINICA Informationis, Volume 49 , Issue 6 : 726-738(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

Abstract


Funded by

国家自然科学基金(61401264,11571012)

中央高校基本科研业务费专项资金(Gk201603025)

  • Figure 1

    Reconstruction results of single source under different depths

  • Figure 2

    Reconstruction results of double sources. (a) Deviation of grouped CoM for double sources at depth of 3 mm with different separation; (b) deviation of grouped CoM for double sources at different depths with 10 mm separation

  • Figure 3

    (Color online) Top views for reconstruction results of double sources (0.5 mm radius and 3 mm depth) with different separations. (a1)$\sim$(a3) GIST; (b1)$\sim$(b3), (c1)$\sim$(c3) and (d1)$\sim$(d3) show the corresponding reconstruction results of FISTA, IRW-L$_{1/2}$ and IVTCG, respectively

  • Figure 4

    (Color online) Top views for reconstruction results of double sources (0.5 mm radius and 10 mm separations) with different depths. (a1)$\sim$(a3) GIST; (b1)$\sim$(b3), (c1)$\sim$(c3) and (d1)$\sim$(d3) show the corresponding reconstruction results of FISTA, IRW-L$_{1/2}$ and IVTCG, respectively

  • Figure 5

    (Color online) Source settings in phantom experiment and photon distribution on the surface at 610 nm wave length. (a) Source setting in Case 1; (b) photon distribution in Case 1; (c) source setting in Case 2; (d) photon distribution in Case 2

  • Figure 6

    (Color online) The $X$-$Y$ and $X$-$Z$ plane views of the reconstruction results in phantom experiment Case 1. (a)$\sim$(d) show the $X$-$Y$ plane view of the reconstruction result of GIST, FISTA, IRW-L$_{1/2}$ and IVTCG, respectively; (e)$\sim$(f) show the $X$-$Z$ plane view of the reconstruction result of GIST, FISTA, IRW-L$_{1/2}$ and IVTCG, respectively

  • Figure 7

    (Color online) The $X$-$Y$ and $X$-$Z$ plane views of the reconstruction results in phantom experiment Case 2. (a)$\sim$(d) show the $X$-$Y$ plane view of the reconstruction result of GIST, FISTA, IRW-L$_{1/2}$ and IVTCG, respectively; (e)$\sim$(f) show the $X$-$Z$ plane view of the reconstruction result of GIST, FISTA, IRW-L$_{1/2}$ and IVTCG, respectively

  •   

    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$^{-1}$) $\mu'_s$ (mm$^{-1}$) $\mu_a$ (mm$^{-1}$) $\mu'_s$ (mm$^{-1}$)
    590 0.1283 1.35 0.0138 0.816
    610 0.0396 1.29 0.0094 0.756
    630 0.0214 1.24 0.0081 0.733
    650 0.0156 1.19 0.0077 0.725
  • Table 2   Reconstruction results of phantom experiment Case 1
    Method Source Reconstructed Single CoM Grouped CoM
    CoM (mm) deviation (mm) deviation (mm)
    GIST S1 ($-$0.44, 2.62, 0.49) 0.993 0.671
    S2 ($-$0.73, $-$1.22, 0.38) 0.684
    FISTA S1 ($-$0.54, 2.59, 0.60) 1.065 0.975
    S2 ($-$1.03, $-$1.21, 0.65) 0.922
    IRW-L$_{1/2}$ S1 ($-$0.81, 1.34, 0.36) 1.882 0.577
    S2 ($-$0.37, $-$1.96, $-$0.01) 0.697
    IVTCG S1 ($-$0.61, 1.96, 0.57) 1.449 0.941
    S2 ($-$0.79, $-$2.79, 0.70) 1.537
  • Table 3   Reconstruction results of phantom experiment Case 2
    Method Source Reconstructed Single CoM Grouped CoM
    CoM (mm) deviation (mm) deviation (mm)
    GIST S1 ($-$0.11, $-$3.13, $-$0.76) 1.455 0.732
    S2 (6.19, $-$2.67, $-$0.21) 0.834
    FISTA S1 (0.59, $-$3.07, $-$2.34) 2.984 1.503
    S2 (5.87, $-$2.79, 0.29) 0.750
    IRW-L$_{1/2}$ S1 (1.25, $-$2.87, $-$2.48) 3.523 1.651
    S2 (5.89, $-$2.81, 0.48) 0.911
    IVTCG S1 (0.58, $-$3.07, $-$2.35) 2.987 1.509
    S2 (5.87, $-$2.78, 0.29) 0.749
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