Laser Absorption Spectroscopy Tomography Based on Cartoon-Texture Model

Jingjing Si; Lü Dongcan; Rui Zhang; Yinbo Cheng; Chang Liu

doi:10.3788/CJL231024

Journals >Chinese Journal of Lasers >Volume 51 >Issue 6 >Page 0611001 > Article

Chinese Journal of Lasers
Vol. 51, Issue 6, 0611001 (2024)

Laser Absorption Spectroscopy Tomography Based on Cartoon-Texture Model

Jingjing Si^1、4, Lü Dongcan¹, Rui Zhang¹, Yinbo Cheng^2、*, and Chang Liu³

Author Affiliations

¹School of Information Science and Engineering, Yanshan University, Qinhuangdao 066004, Hebei , China

²Ocean College, Hebei Agricultural University, Qinhuangdao 066003, Hebei , China

³School of Engineering, the University of Edinburgh, Edinburgh EH93JL, UK

⁴Hebei Key Laboratory of Information Transmission and Signal Processing, Qinhuangdao 066004, Hebei , China

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DOI: 10.3788/CJL231024 Cite this Article Set citation alerts

Jingjing Si, Lü Dongcan, Rui Zhang, Yinbo Cheng, Chang Liu. Laser Absorption Spectroscopy Tomography Based on Cartoon-Texture Model[J]. Chinese Journal of Lasers, 2024, 51(6): 0611001 Copy Citation Text

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Reconstruction of absorbance density image for a simulated phantom. (a) Original absorbance density image; (b) image reconstructed with the Landweber algorithm; (c) residual image

Fig. 1. Reconstruction of absorbance density image for a simulated phantom. (a) Original absorbance density image; (b) image reconstructed with the Landweber algorithm; (c) residual image

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Fig. 2. Overall architecture of the proposed TRACT algorithm

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Fig. 3. Network block implementing the nonlinear operation in the k-th layer of ISTA-mNet

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Fig. 4. Optical layout of the TDLAT experimental system

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Fig. 5. Dependence of NMSE on SNRs for five different reconstruction algorithms

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Fig. 6. Temperature distribution images reconstructed with five different algorithms. (a) Original temperature distribution image; (b)‒(f) temperature distribution images reconstructed with Landweber, ART-TV, HCNN, Landweber-TV, and TRACT, respectively; (g)‒(k) residual images correspond to the images reconstructed with Landweber, ART-TV, HCNN, Landweber-TV, and TRACT, respectively

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Fig. 7. Temperature distribution images reconstructed for the RoI in real TDLAT experimental system. (a) Top view of the measurement field of the TDLAT experimental system; (b)‒(f) temperature distribution images of the RoI reconstructed by Landweber, ART-TV, HCNN, Landweber-TV, and TRACT, respectively

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Algorithm 1: Landweber-TV algorithm

Input: path integrated absorbance vector $A \in R^{G \times 1}$ , chord length matrix $L \in R^{G \times M}$ , regularization parameter $λ_{c}$ , and the number of iterations $N$

Initialize: $a^{0} = 0^{M \times 1}$

Iterations:

For $n = 1, 2, \dots, N$ do

$a^{n} = a^{n - 1} + λ_{c} L^{T} (A - L a^{n - 1})$

$a^{n} = m a x (a^{n}, 0)$

${\bar{a}}^{n} = V e c t o r 2 M a t r i x (a^{n})$

$\bar{V} = \frac{\partial T V ({\bar{a}}^{n})}{\partial {\bar{a}}^{n}}$

$V = M a t r i x 2 V e c t o r (\bar{V})$

$d^{n} = {‖a^{n} - a^{n - 1}‖}_{2}$

$a^{n} = a^{n - 1} + λ_{c} d^{n} \frac{V}{{‖V‖}_{2}}$

$a^{n} = m a x (a^{n}, 0)$

End for

${\hat{a}}_{c} = a^{N}$

Output: ${\hat{a}}_{c} \in R^{M \times 1}$

Table 1. Detailed steps of the Landweber-TV algorithm

Algorithm 2: ISTA-mNet

Input: path integrated absorbance vector $A_{t} \in R^{G \times 1}$ , chord length matrix $L \in R^{G \times M}$ , the number of layers K, initial step size $ρ^{0}$ , and initial threshold $θ^{0}$

Initialize: $a^{0} = L^{T} A_{t}$

Iterations:

For $k = 1, 2, \dots, K$ do

Linear operation:

$r^{k} = a^{k - 1} + ρ^{k - 1} L^{T} (A_{t} - L a^{k - 1})$

Nonlinear operation:

${\bar{r}}^{k} = V e c t o r 2 M a t r i x (r^{k})$

${\bar{a}}^{k} = y_{N e t}^{k} ({\bar{r}}^{k}) = {\tilde{f}}_{n e t}^{k} \{s o f t [f_{n e t}^{k} ({\bar{r}}^{k}); θ^{k - 1}]\}$

$a^{k} = M a t r i x 2 V e c t o r ({\bar{a}}^{k})$

End for

${\hat{a}}_{t} = a^{K}$

Output: ${\hat{a}}_{t} \in R^{M \times 1}$

Table 2. Implementation process of ISTA-mNet

Algorithm 3: TRACT

Input: path integrated absorbance vectors $A_{v_{1}} \in R^{G \times 1}$ and $A_{v_{2}} \in R^{G \times 1}$ , chord length matrix $L \in R^{G \times M}$ , regularization parameter $λ_{c}$ , the number of iterations N in Landweber-TV, the number of layers K in ISTA-mNet, initial step size $ρ^{0}$ , and initial threshold $θ^{0}$

Initialize: $a_{v_{1}}^{0} = 0^{M \times 1}$ , $a_{v_{2}}^{0} = 0^{M \times 1}$

Iterations:

For $v = v_{1}, v_{2}$ do

Cartoon component reconstruction stage:

${\hat{a}}_{c, v} = L a n d w e b e r - T V \{A_{v}, L, λ_{c}, N\}$

Texture component reconstruction stage:

$A_{t, v} = A_{v} - L {\hat{a}}_{c, v}$

${\hat{a}}_{t, v} = I S T A - m N e t \{A_{t, v}, L, K, ρ^{0}, θ^{0}\}$

Reconstructed absorbance density vector:

${\hat{a}}_{v} = {\hat{a}}_{c, v} + {\hat{a}}_{t, v}$

End for

For m=1, 2, …, M do

${\hat{T}}_{m} = \frac{(E_{2}^{″} - E_{1}^{″}) \frac{h c}{k}}{l n \frac{{\hat{a}}_{v_{1}, m}}{{\hat{a}}_{v_{2}, m}} + l n \frac{S_{v_{2}} (T_{0})}{S_{v_{1}} (T_{0})} + (E_{2}^{″} - E_{1}^{″}) \frac{h c}{k T_{0}}}$

End for

Output: temperature distribution vector $\hat{T} = {[{\hat{T}}_{1}, {\hat{T}}_{2}, \dots, {\hat{T}}_{M}]}^{T}$

Table 3. Implementation process of TRACT

Jingjing Si, Lü Dongcan, Rui Zhang, Yinbo Cheng, Chang Liu. Laser Absorption Spectroscopy Tomography Based on Cartoon-Texture Model[J]. Chinese Journal of Lasers, 2024, 51(6): 0611001

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