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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

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.


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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