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.
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
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 ${\boldsymbol{q}}^{(k+1)}$. |
| Wavelength (nm) | Simulation | Phantom | ||
| $\mu_a$ (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 |
| 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, $-$2.79, 0.70) | 1.537 |
| Method | Source | Reconstructed | Single CoM | Grouped CoM |
| CoM (mm) | deviation (mm) | deviation (mm) | ||
| GIST | S1 | ($-$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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