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

• AcceptedFeb 5, 2018
• PublishedMay 16, 2019
Share
Rating

### 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

•

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

Citations

• #### 0

Altmetric

Copyright 2019 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有