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Beam arrangement with limited projections is a critical part of research on tunable diode laser absorption tomography reconstruction for combustion diagnosis. Based on the efforts to regularize this rank-deficient and ill-posed problem with Tikhonov regularization, a novel approach to using the regularization parameter matrix is developed for designing optical component layout and predicting the reconstruction accuracy. Objective function of beam arrangement is established by the rigorous mathematical derivation, and genetic algorithm is adopted to realize the optimization of function to overcome the difficulty associated with the multimodal nature of the problem. Nonuniform distribution properties of matrix elements in physical space relate to location and alignment of the laser/detector pairs, and form a basis for adjusting the weight between measurement and regularization to improve the reconstruction performance. A mathematical model of double Gauss distributions is established in a 10 × 10 element discrete tomography domain, and typically 20 measurement beams scanning the H2O transition at 7185.6 cm–1 are available to probe the domain of interest. The systematic comparison between optimized beam array here and four existing beam arrangements in the literature is analyzed to validate the method. The reconstruction with Tikhonov regularization parameter matrix shows obvious advantages of reducing errors especially under the condition of fewer projections. The validation of reconstruction performance of the optimized beam array is also examined by simulating the laser absorption measurement which is carried out on phantoms generated using a simulation of external flow field of an air-gasoline pulsed detonation engine. The result shows that the optimized beam array consistently outperforms other arrangements reported in complicated fluid field. A demonstration reconstruction experiment is performed on the distribution from small gas burners. Both locations and amplitudes are in good agreement with those in the actual case. This proposed design method will be valuable in broadening the scope of applications of tunable diode laser absorption tomography reconstruction for engine diagnosis and combustion efficiency improvement.
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Keywords:
- laser absorption spectroscopy /
- tomography reconstruction /
- Tikhonov regularization /
- singular value decomposition
[1] Jatana G, Geckler S, Koeberlein D, Partridge W 2017 Sens. Actuators B-Chem. 240 1197
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[5] Strand C L, Hanson R K 2015 AIAA 53 2978
Google Scholar
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Google Scholar
[7] Tsekenis S A, Polydorides, N 2017 IEEE Sens. J. 17 8072
Google Scholar
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Google Scholar
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Google Scholar
[10] Liu C, Xu L J, Chen J L, Cao Z, Lin Y Z, Cai WW 2015 Opt. Express 23 22494
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Google Scholar
[12] Xia H H, Kan R F, Xu Z Y, He Y B, Liu J G, Chen B, Yang C G, Yao L, Wei M, Zhang G L 2017 Opt. Laser Eng. 90 10
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 激光吸收光谱二维测量光路系统示意图
Figure 1. Geometry of transmitted laser beam in tomography measurement.
图 2 不同正则化参数
$\lambda $ 影响下增广矩阵$\left[ {{{A}}; \lambda {{L}}} \right]$ 奇异值变化曲线Figure 2. Singular values of the augmented matrix
$\left[ {{{A}}; \lambda {{L}}} \right]$ with various λ.
图 3 光路布置目标函数遗传算法求解过程
Figure 3. Evolution of objective function by genetic algorithm.
图 4 光路布置方式与相应的正则化参数矩阵 (a) 扇形光路布置; (b) 正则化参数矩阵元素分布图像
Figure 4. Beam configuration and corresponding regularization parameter matrix: (a) Fanned beam configuration; (b) distribution of matrix elements in the physical space.
图 5 4 × 4扇形光路布置方式下分别利用单一正则化参数与正则化参数矩阵方法对100个高斯分布模型重建结果对比
Figure 5. Comparison of reconstruction errors for 4 × 4 fanned beam arrangement and 100 Gauss phantoms calculated by single regularization parameter and regularization parameter matrix.
图 6 5种光路布置方式空间分布图与对应投影域图 (a) 2 × 10平行光路布置方式; (b) 4 × 5扇形光路布置方式; (c) 基于Mod设计的光路布置方式; (d) 基于单一正则化参数设计的光路布置方式; (e) 基于正则化参数矩阵设计的光路布置方式
Figure 6. Five example beam configuration in the physical space and in Radon space: (a) 2 × 10 parallel beams arrangement; (b) 4 × 5 fanned beams arrangement; (c) beams arrangement designed based on MOD method; (d) beams arrangement designed based on single regularization parameter; (e) beams arrangement designed based on regularization parameter matrix.
图 7 二维重建模型与不同光路布置方式重建结果 (a) 重建模型; (b) 2 × 10平行光路布置方式; (c) 4 × 5扇形光路布置方式; (d) 基于Mod设计的光路布置方式; (e) 基于单一正则化参数设计的光路布置方式; (f) 基于正则化参数矩阵设计的光路布置方式
Figure 7. Phantom and reconstruction results from different beam arrangement: (a) Phantom; (b) 2 × 10 parallel beams arrangement; (c) 4 × 5 fanned beams arrangement; (d) beams arrangement designed based on MOD method; (e) beams arrangement designed based on single regularization parameter; (f) beams arrangement designed based on regularization parameter matrix.
图 8 投影数据噪音对重建误差的影响
Figure 8. Effect of noise in projections on reconstruction error.
图 9 气液两相爆轰外流场重建模型
Figure 9. Model of simulated tomography measurement in a two-phase detonation flow.
图 11 激光吸收光谱二维重建实验示意图
Figure 11. Experiment on tomography reconstruction based on tunable diode laser absorption.
图 12 燃气炉重建结果 (a) 单炉实验结果; (b) 双炉实验结果
Figure 12. Reconstruction results of gas burners: (a) single burner; (b) double burners.
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[1] Jatana G, Geckler S, Koeberlein D, Partridge W 2017 Sens. Actuators B-Chem. 240 1197
Google Scholar
[2] Kamimoto T, Deguchi Y, Shisawa Y, Kitauchi Y, Eto Y 2016 Appl. Therm. Eng. 102 596
Google Scholar
[3] Wood M P, Ozanyan K B 2015 IEEE Sens. J. 15 545
Google Scholar
[4] Goldenstein C S, Spearrin R M, Jeffries J B, Hanson R K 2014 Appl. Phys. B 116 705
Google Scholar
[5] Strand C L, Hanson R K 2015 AIAA 53 2978
Google Scholar
[6] Tsekenis S A, Wilson D, Lengden M, Hyvonen J, Leinonen J, Shah A, Andersson O, McCann H 2017 Flow Meas. Instrum. 53 116
Google Scholar
[7] Tsekenis S A, Polydorides, N 2017 IEEE Sens. J. 17 8072
Google Scholar
[8] Xu L J, Liu C, Jing W Y, Cao Z, Xue X, Lin Y Z 2016 Rev. Sci. Instrum. 87 013101
Google Scholar
[9] Cai W W, Kaminski C F 2017 Prog. Energy Combust.Sci. 59 1
Google Scholar
[10] Liu C, Xu L J, Chen J L, Cao Z, Lin Y Z, Cai WW 2015 Opt. Express 23 22494
Google Scholar
[11] Wang F, Wu Q, Huang Q X, Zhang H D, Yan J H, Cen K F 2015 Opt. Commun. 346 53
Google Scholar
[12] Xia H H, Kan R F, Xu Z Y, He Y B, Liu J G, Chen B, Yang C G, Yao L, Wei M, Zhang G L 2017 Opt. Laser Eng. 90 10
Google Scholar
[13] Cai W W, Ewing D J, Ma L 2008 Comput. Phys. Commun. 179 250
Google Scholar
[14] Cai W W, Ewing D J, Ma L 2011 Appl.Math. Comput. 217 5754
Google Scholar
[15] Terzija N, Davidson J L, Garciastewart C A, Wright P, Ozanyan K B, Pegrum S, Litt T J, Mccann H 2009 Meas. Sci.Technol. 19 094007
Google Scholar
[16] Song J L, Hong Y J, Pan H, Wang G Y 2013 Proceedings of 5th International Symposium on Photoelectronic Detection and Imaging Beijing, China, June 25–27, 2013 p89070K
[17] Twynstra M G, Daun K J 2012 Appl. Opt. 51 7059
Google Scholar
[18] Grauer S J, Hadwin P J, Daun K J 2016 Appl. Opt. 55 5772
Google Scholar
[19] Yu T, Tian B, Cai W W 2017 Opt. Express 25 5982
Google Scholar
[20] Kang Y, Li N, Weng C S, Wang C W 2018 Chin. Phys. B 27 104703
Google Scholar
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