Abstract

Flame chemiluminescence tomography (FCT) plays an important role in combustion monitoring and diagnostics due to the easy implementation and non-intrusion. However, on account of the high data throughput and the inefficiency of the conventional iteration methods, the 3D reconstructions in FCT are typically conducted off-line and time-consuming. In this work, we present a 3D rapid FCT reconstruction system based on convolutional neural networks (CNN) model for practical combustion measurement, which has the ability to reconstruct 3D flame distribution rapidly after training process. First, the numerical simulation has been performed by creating three cases of phantoms which are designed to mimic the 3D conical flame. Next, after the evaluation of loss function and training time, the optimal CNN architecture has been determined and certificated quantitatively. Finally, a real time FCT system consisting of 12 color CCD cameras is realized and multispectral separation algorithm is adopted to extract CH* and C2* components. Certificated by practical measurements testing, the proposed CNN model is able to reconstruct 3D flame structure from real time captured projections with credible accuracy and structure similarity. Furthermore, compared with conventional iteration reconstruction method, the proposed CNN model shows better performance on obviously improving reconstruction speed and it is expected to achieve 3D rapid monitoring of flames.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (8)

H. Liu, J. Zhao, C. Shui, and W. Cai, “Reconstruction and analysis of non-premixed turbulent swirl flames based on kHz-rate multi-angular endoscopic volumetric tomography,” Aerosp. Sci. Technol. 91, 422–433 (2019).
[Crossref]

C. Ruan, T. Yu, F. Chen, S. Wang, W. Cai, and X. Lu, “Experimental characterization of the spatiotemporal dynamics of a turbulent flame in a gas turbine model combustor using computed tomography of chemiluminescence,” Energy 170, 744–751 (2019).
[Crossref]

A. Unterberger, A. Kempf, and K. Mohri, “3D Evolutionary Reconstruction of Scalar Fields in the Gas-Phase,” Energies 12(11), 2075 (2019).
[Crossref]

T. Liu, J. Rong, P. Gao, H. Pu, W. Zhang, X. Zhang, Z. Liang, and H. Lu, “Regularized reconstruction based on joint L1 and total variation for sparse-view cone-beam X-ray luminescence computed tomography,” Biomed. Opt. Express 10(1), 1–17 (2019).
[Crossref]

J. Huang, J. Zhao, and W. Cai, “Compressing convolutional neural networks using POD for the reconstruction of nonlinear tomographic absorption spectroscopy,” Comput. Phys. Commun. 241, 33–39 (2019).
[Crossref]

S. Colburn, Y. Chu, E. Shilzerman, and A. Majumdar, “Optical frontend for a convolutional neural network,” Appl. Opt. 58(12), 3179–3186 (2019).
[Crossref]

T. Qiu, M. Liu, G. Zhou, L. Wang, and K. Gao, “An unsupervised classification method for flame image of pulverized coal combustion based on convolutional auto-encoder and hidden Markov model,” Energies 12(13), 2585 (2019).
[Crossref]

A. J. Huang, H. Liu, and W. Cai, “Online in situ prediction of 3-D fame evolution from its history 2-D projections via deep learning,” J. Fluid Mech. 875, R2 (2019).
[Crossref]

2018 (13)

H. Wang, M. Lyu, and G. Situ, “eHoloNet: a learning-based end-to-end approach for in-line digital holographic reconstruction,” Opt. Express 26(18), 22603–22614 (2018).
[Crossref]

Z. Ren, Z. Xu, and E. Y. Lam, “Learning-based nonparametric autofocusing for digital holography,” Optica 5(4), 337–344 (2018).
[Crossref]

Y. Wu, Y. Rivenson, Y. Zhang, Z. Wei, H. Günaydin, X. Lin, and A. Ozcan, “Extended depth-of-field in holographic imaging using deep-learning-based autofocusing and phase recovery,” Optica 5(6), 704–710 (2018).
[Crossref]

J. Huang, H. Liu, J. Dai, and W. Cai, “Reconstruction for limited-data nonlinear tomographic absorption spectroscopy via deep learning,” J. Quant. Spectrosc. Radiat. Transfer 218, 187–193 (2018).
[Crossref]

T. Yu, W. Cai, and Y. Liu, “Rapid tomographic reconstruction based on machine learning for time-resolved combustion diagnostics,” Rev. Sci. Instrum. 89(4), 043101 (2018).
[Crossref]

J. Gu, Z. Wang, J. Kuen, L. Ma, A. Shahroudy, B. Shuai, T. Liu, X. Wang, G. Wang, J. Cai, and T. Chen, “Recent advances in convolutional neural networks,” Pattern Recogn. 77, 354–377 (2018).
[Crossref]

S. J. Grauer, A. Unterberger, A. Rittler, K. J. Daun, A. M. Kempf, and K. Mohri, “Instantaneous 3D flame imaging by background-oriented schlieren tomography,” Combust. Flame 196, 284–299 (2018).
[Crossref]

R. Hou, Y. Xia, Y. Bao, and X. Zhou, “Selection of regularization parameter for l1-regularized damage detection,” J. Sound Vib. 423, 141–160 (2018).
[Crossref]

R. Guo, W. Zhang, R. Liu, C. Duan, and F. Wang, “Phase unwrapping in dual-wavelength digital holographic microscopy with total variation regularization,” Opt. Lett. 43(14), 3449–3452 (2018).
[Crossref]

W. Lu, D. Lighter, and I. B. Styles, “L1-norm based nonlinear reconstruction improves quantitative accuracy of spectral diffuse optical tomography,” Biomed. Opt. Express 9(4), 1423–1444 (2018).
[Crossref]

J. Dai, T. Yu, L. Xu, and W. Cai, “On the regularization for nonlinear tomographic absorption spectroscopy,” J. Quant. Spectrosc. Radiat. Transfer 206, 233–241 (2018).
[Crossref]

T. Yu, H. Liu, J. Zhang, W. Cai, and F. Qi, “Toward real-time volumetric tomography for combustion diagnostics via dimension reduction,” Opt. Lett. 43(5), 1107–1110 (2018).
[Crossref]

A. Unterberger, M. Röder, A. Giese, A. Al-Halbouni, A. Kempf, and K. Mohri, “3D instantaneous reconstruction of turbulent industrial flames using computed tomography of chemiluminescence (CTC),” J. Combust. 2018, 1–6 (2018).
[Crossref]

2017 (12)

Y. Jin, Y. Song, X. Qu, Z. Li, Y. Ji, and A. He, “Three-dimensional dynamic measurements of CH* and C2* concentrations in flame using simultaneous chemiluminescence tomography,” Opt. Express 25(5), 4640–4654 (2017).
[Crossref]

S. M. Wiseman, M. J. Brear, R. L. Gordon, and I. Marusic, “Measurements from flame chemiluminescence tomography of forced laminar premixed propane flames,” Combust. Flame 183, 1–14 (2017).
[Crossref]

K. Wang, F. Li, H. Zeng, and X. Yu, “Three-dimensional flame measurements with large field angle,” Opt. Express 25(18), 21008–21018 (2017).
[Crossref]

T. Yu, H. Liu, and W. Cai, “On the quantification of spatial resolution for three-dimensional computed tomography of chemiluminescence,” Opt. Express 25(20), 24093–24108 (2017).
[Crossref]

J. Miller, S. Peltier, M. Slipchenko, J. Mance, T. Ombrello, J. Gord, and C. Carter, “Investigation of transient ignition processes in a model scramjet pilot cavity using simultaneous 100 kHz formaldehyde planar laser-induced fluorescence and CH* chemiluminescence imaging,” Proc. Combust. Inst. 36(2), 2865–2872 (2017).
[Crossref]

T. Yu and W. Cai, “Benchmark evaluation of inversion algorithms for tomographic absorption spectroscopy,” Appl. Opt. 56(8), 2183–2194 (2017).
[Crossref]

S. J. Grauer, P. J. Hadwin, and K. J. Daun, “Improving chemical species tomography of turbulent flows using covariance estimation,” Appl. Opt. 56(13), 3900–3912 (2017).
[Crossref]

X. Zhang, B. Javidi, and M. K. Ng, “Automatic regularization parameter selection by generalized cross-validation for total variational Poisson noise removal,” Appl. Opt. 56(9), D47–D51 (2017).
[Crossref]

H. Chen, Y. Zhang, W. Zhang, P. Liao, K. Li, J. Zhou, and G. Wang, “Low-dose CT via convolutional neural network,” Biomed. Opt. Express 8(2), 679–694 (2017).
[Crossref]

H. Chen, Y. Zhang, M. K. Kalra, F. Lin, Y. Chen, P. Liao, J. Zhou, and G. Wang, “Low-Dose CT With a residual encoder-decoder convolutional neural network,” IEEE T. Med. Imaging 36(12), 2524–2535 (2017).
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M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, “Deep-learning-based ghost imaging,” Sci. Rep. 7(1), 17865 (2017).
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M. C. Thomsen, A. Fuentes, R. Demarco, C. Volkwein, J.-L. Consalvi, and P. Reszka, “Soot measurements in candle flames,” Exp. Therm. Fluid Sci. 82, 116–123 (2017).
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2016 (5)

S. M. Ahn, “Deep learning architectures and applications,” J. Intell. Inf. Syst. 22(2), 127–142 (2016).
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G. Wang, “A Perspective on Deep Imaging,” IEEE Access 4, 8914–8924 (2016).
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K. J. Daun, S. J. Grauer, and P. J. Hadwin, “Chemical species tomography of turbulent flows: Discrete ill-posed and rank deficient problems and the use of prior information,” J. Quant. Spectrosc. Radiat. Transfer 172, 58–74 (2016).
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Y. Jin, Y. Song, X. Qu, Z. Li, Y. Ji, and A. He, “Hybrid algorithm for three-dimensional flame chemiluminescence tomography based on imaging overexposure compensation,” Appl. Opt. 55(22), 5917–5923 (2016).
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2015 (4)

X. Li and L. Ma, “Capabilities and limitations of 3D flame measurements based on computed tomography of chemiluminescence,” Combust. Flame 162(3), 642–651 (2015).
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J. Wang, Y. Song, Z. Li, A. Kempf, and A. He, “Multi-directional 3D flame chemiluminescence tomography based on lens imaging,” Opt. Lett. 40(7), 1231–1234 (2015).
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D. Sun, G. Lu, H. Zhou, Y. Yan, and S. Liu, “Quantitative assessment of flame stability through image processing and spectral analysis,” IEEE Trans. Instrum. Meas. 64(12), 3323–3333 (2015).
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Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015).
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2014 (3)

2013 (3)

W. Cai, X. Li, F. Li, and L. Ma, “Numerical and experimental validation of a three-dimensional combustion diagnostic based on tomographic chemiluminescence,” Opt. Express 21(6), 7050–7064 (2013).
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J. Sjoholm, J. Rosell, B. Li, M. Richter, Z. Li, X. Bai, and M. Aldén, “Simultaneous visualization of OH, CH, CH2O and toluene PLIF in a methane jet flame with varying degrees of turbulence,” Proc. Combust. Inst. 34(1), 1475–1482 (2013).
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A. Charogiannis and F. Beyrau, “Laser induced phosphorescence imaging for the investigation of evaporating liquid flows,” Exp. Fluids 54(5), 1518 (2013).
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2012 (4)

A. Vandersickel, M. Hartmann, K. Vogel, Y. M. Wright, M. Fikri, R. Starke, C. Schulz, and K. Boulouchos, “The autoignition of practical fuels at HCCI conditions: high-pressure shock tube experiments and phenomenological modeling,” Fuel 93, 492–501 (2012).
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P. Nau, J. Krüger, A. Lackner, M. Letzgus, and A. Brockhinke, “On the quantification of OH*, CH*, and C2* chemiluminescence in flames,” Appl. Phys. B: Lasers Opt. 107(3), 551–559 (2012).
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M. Bozkurt, M. Fikri, and C. Schulz, “Investigation of the kinetics of OH* and CH* chemiluminescence in hydrocarbon oxidation behind reflected shock waves,” Appl. Phys. B: Lasers Opt. 107(3), 515–527 (2012).
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Y. Wen and R. H. Chen, “Parameter selection for total-variation-based image restoration using discrepancy principle,” IEEE T,” Image Process. 21(4), 1770–1781 (2012).
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2011 (3)

T. D. Upton, D. D. Verhoeven, and D. E. Hudgins, “High-resolution computedtomography of a turbulent reacting flow,” Exp. Fluids 50(1), 125–134 (2011).
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J. Floyd and A. M. Kempf, “Computed Tomography of Chemiluminescence (CTC): High resolution and instantaneous 3-D measurements of a Matrix burner,” Proc. Combust. Inst. 33(1), 751–758 (2011).
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J. Floyd, P. Geipel, and A. Kempf, “Computed tomography of chemiluminescence (CTC): instantaneous 3D measurements and phantom studies of a turbulent opposed jet flame,” Combust. Flame 158(2), 376–391 (2011).
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2010 (2)

Z. Li, B. Li, Z. Sun, X. Bai, and M. Aldén, “Turbulence and combustion interaction: High resolution local flame front structure visualization using simultaneous single-shot PLIF imaging of CH, OH, and CH2O in a piloted premixed jet flame,” Combust. Flame 157(6), 1087–1096 (2010).
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L. Shi, Y. Liu, and J. Yu, “PIV measurement of separated flow over a blunt plate with different chord-to-thickness ratios,” J. Fluid. Struct. 26(4), 644–657 (2010).
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2009 (1)

V. N. Nori and J. M. Seitzman, “CH* chemiluminescence modeling for combustion diagnostics,” Proc. Combust. Inst. 32(1), 895–903 (2009).
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2008 (3)

S. S. Shy, Y. C. Chen, C. H. Yang, C. C. Liu, and C. M. Huang, “Effects of H2 or CO2 addition, equivalence ratio, and turbulent straining on turbulent burning velocities for lean premixed methane combustion,” Combust. Flame 153(4), 510–524 (2008).
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L. Ma and W. Cai, “Determination of the optimal regularization parameters in hyperspectral tomography,” Appl. Opt. 47(23), 4186–4192 (2008).
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E. O. Åkesson and K. J. Daun, “Parameter selection methods for axisymmetric flame tomography through Tikhonov regularization,” Appl. Opt. 47(3), 407–416 (2008).
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2006 (3)

K. J. Daun, K. A. Thomson, F. Liu, and G. J. Smallwood, “Deconvolution of axisymmetric flame properties using Tikhonov regularization,” Appl. Opt. 45(19), 4638–4646 (2006).
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G. E. Hinton, S. Osindero, and Y. W. Teh, “A Fast Learning Algorithm for Deep Belief Nets,” The MIT Press Journal 18(7), 1527–1554 (2006).
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Y. K. Jeong, C. H. Jeon, and Y. J. Chang, “Evaluation of the equivalence ratio of the reacting mixture using intensity ratio of chemiluminescence in laminar partially premixed CH 4-air flames,” Exp. Therm. Fluid Sci. 30(7), 663–673 (2006).
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2005 (3)

J. Kojima, Y. Ikeda, and T. Nakajima, “Basic aspects of OH(A), CH(A), and C2(d) chemiluminescence in the reaction zone of laminar methane-air premixed flames,” Combust. Flame 140(1-2), 34–45 (2005).
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S. A. Farhat, W. B. Ng, and Y. Zhang, “Chemiluminescent emission measurement of a diffusion flame jet in a loudspeaker induced standing wave,” Fuel 84(14-15), 1760–1767 (2005).
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H. Zhou, C. Lou, Q. Cheng, Z. Jiang, J. He, B. Huang, Z. Pei, and C. Lu, “Experimental investigations on visualization of three-dimensional temperature distributions in a large-scale pulverized-coal-fired boiler furnace,” Proc. Combust. Inst. 30(1), 1699–1706 (2005).
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2004 (2)

Y. Hardalupas and M. Orain, “Local measurements of the time-dependent heat release rate and equivalence ratio using chemiluminescent emission from a flame,” Combust. Flame 139(3), 188–207 (2004).
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Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
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2003 (1)

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inv. Probl. 19(6), S165–S187 (2003).
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2002 (1)

H. Zhou, S. Han, F. Sheng, and C. Zheng, “Visualization of threedimensional temperature distributions in a large-scale furnace via regularized reconstruction from radiative energy images: numerical studies,” J. Quant. Spectrosc. Radiat. Transfer 72(4), 361–383 (2002).
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J. Miller, S. Peltier, M. Slipchenko, J. Mance, T. Ombrello, J. Gord, and C. Carter, “Investigation of transient ignition processes in a model scramjet pilot cavity using simultaneous 100 kHz formaldehyde planar laser-induced fluorescence and CH* chemiluminescence imaging,” Proc. Combust. Inst. 36(2), 2865–2872 (2017).
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D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inv. Probl. 19(6), S165–S187 (2003).
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Y. K. Jeong, C. H. Jeon, and Y. J. Chang, “Evaluation of the equivalence ratio of the reacting mixture using intensity ratio of chemiluminescence in laminar partially premixed CH 4-air flames,” Exp. Therm. Fluid Sci. 30(7), 663–673 (2006).
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A. Charogiannis and F. Beyrau, “Laser induced phosphorescence imaging for the investigation of evaporating liquid flows,” Exp. Fluids 54(5), 1518 (2013).
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C. Ruan, T. Yu, F. Chen, S. Wang, W. Cai, and X. Lu, “Experimental characterization of the spatiotemporal dynamics of a turbulent flame in a gas turbine model combustor using computed tomography of chemiluminescence,” Energy 170, 744–751 (2019).
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Y. Wen and R. H. Chen, “Parameter selection for total-variation-based image restoration using discrepancy principle,” IEEE T,” Image Process. 21(4), 1770–1781 (2012).
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J. Gu, Z. Wang, J. Kuen, L. Ma, A. Shahroudy, B. Shuai, T. Liu, X. Wang, G. Wang, J. Cai, and T. Chen, “Recent advances in convolutional neural networks,” Pattern Recogn. 77, 354–377 (2018).
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J. Huang, H. Liu, J. Dai, and W. Cai, “Reconstruction for limited-data nonlinear tomographic absorption spectroscopy via deep learning,” J. Quant. Spectrosc. Radiat. Transfer 218, 187–193 (2018).
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Comput. Phys. Commun. (1)

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Supplementary Material (3)

NameDescription
» Visualization 1       3D radical concentration distribution of CH* and C2* of candle flame (Sample A) at different moments
» Visualization 2       3D radical concentration distribution of CH* and C2* of candle flame (Sample B) at different moments
» Visualization 3       3D radical concentration distribution of CH* and C2* of candle flame (Sample C) at different moments

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Figures (22)

Fig. 1.
Fig. 1. 3D projection model of reconstruction.
Fig. 2.
Fig. 2. Demonstration of convolutional natural networks (CNN).
Fig. 3.
Fig. 3. The flowchart of 3D FCT using convolutional neural networks. The blue part represents the training stage and the yellow part represents the testing stage.
Fig. 4.
Fig. 4. Locations of CCD array in numerical simulation.
Fig. 5.
Fig. 5. One example of three phantoms cases. (a)–(c) 3D distribution of three phantom cases, (d)–(f) the horizontal slices according to the yellow dash lines in (a)–(c).
Fig. 6.
Fig. 6. Architecture of CNN model for 3D FCT.
Fig. 7.
Fig. 7. Evolution of four configurations of convolution layers of CNN model. (a) the performance of loss function with different number of layers, (b) the training time with different number of layers.
Fig. 8.
Fig. 8. Evolution of four cases with different number of convolution kernels. (a) the performance of loss function with different number of convolution kernels, (b) the training time with different number of convolution kernels.
Fig. 9.
Fig. 9. Comparisons between three representative phantoms (the first row) and the corresponding prediction results via CNN model (the second row).
Fig. 10.
Fig. 10. Comparisons of three horizontal slices between three representative phantoms and the corresponding prediction results.
Fig. 11.
Fig. 11. FCT system with 12 CCD cameras.
Fig. 12.
Fig. 12. Normalized radiation intensity difference of multi-camera. (a) CH* channel, (b) C2* channel.
Fig. 13.
Fig. 13. Re-projection results and correlation coefficients of twelfth camera.
Fig. 14.
Fig. 14. 12 views instantaneous projections and averaged flame projections of Sample A. (A movie is available online. See Visualization 1)
Fig. 15.
Fig. 15. 12 views instantaneous projections and averaged flame projections of Sample B. (A movie is available online. See Visualization 2)
Fig. 16.
Fig. 16. 12 views instantaneous projections and averaged flame projections of Sample C. (A movie is available online. See Visualization 3)
Fig. 17.
Fig. 17. Projections of instantaneous candle flame from 12 directions of sample A at 40.5s. (a) chemiluminescence emission intensity images of CH*, (b) chemiluminescence emission intensity images of C2*.
Fig. 18.
Fig. 18. Projections of instantaneous candle flame from 12 directions of sample B at 60.7s. (a) chemiluminescence emission intensity images of CH*, (b) chemiluminescence emission intensity images of C2*.
Fig. 19.
Fig. 19. Projections of instantaneous candle flame from 12 directions of sample C at 64.2s. (a) chemiluminescence emission intensity images of CH*, (b) chemiluminescence emission intensity images of C2*.
Fig. 20.
Fig. 20. Comparison of 3D candle flame reconstruction results with different methods. (a) and (c) the CH* and C2* concentration reconstruction results of sample A via ART algorithm, (b) and (d) the corresponding reconstruction results of sample A via CNN model, (e) and (g) the CH* and C2* concentration reconstruction results of sample B via ART algorithm, (f) and (h) the corresponding reconstruction results of sample B via CNN model, (i) and (k) the CH* and C2* concentration reconstruction results of sample C via ART algorithm, (j) and (l) the corresponding reconstruction results of sample C via CNN model
Fig. 21.
Fig. 21. Comparison of 3D candle flame reconstruction results with different methods at 10µs. (a) and (c) the CH* and C2* concentration reconstruction results of sample A via ART algorithm, (b) and (d) the corresponding reconstruction results of sample A via CNN model, (e) and (g) the CH* and C2* concentration reconstruction results of sample B via ART algorithm, (f) and (h) the corresponding reconstruction results of sample B via CNN model, (i) and (k) the CH* and C2* concentration reconstruction results of sample C via ART algorithm, (j) and (l) the corresponding reconstruction results of sample C via CNN model
Fig. 22.
Fig. 22. Comparison of 3D candle flame reconstruction results with different methods at 20µs. (a) and (c) the CH* and C2* concentration reconstruction results of sample A via ART algorithm, (b) and (d) the corresponding reconstruction results of sample A via CNN model, (e) and (g) the CH* and C2* concentration reconstruction results of sample B via ART algorithm, (f) and (h) the corresponding reconstruction results of sample B via CNN model, (i) and (k) the CH* and C2* concentration reconstruction results of sample C via ART algorithm, (j) and (l) the corresponding reconstruction results of sample C via CNN model

Tables (6)

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Table 1. The verification results of CNN model

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Table 2. Camera parameters results

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Table 3. Evaluation results of 3D candle flames reconstruction based on CNN model

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Table 4. Evaluation results of 3D candle flames reconstruction based on CNN model at10µs

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Table 5. Evaluation results of 3D candle flames reconstruction based on CNN model at 20µs

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Table 6. Time-Consuming of 3D FCT Reconstruction via different methods

Equations (12)

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I j = i = 1 N w i j f i 1 j p × q
{ w 11 f 11 + w 21 f 21 + w 31 f 31 + + w N 1 f N 1 = I 1 w 12 f 12 + w 22 f 22 + w 32 f 32 + + w N 2 f N 2 = I 2 w 1 j f 1 j + w 2 j f 2 j + w 3 j f 3 j + + w N j f N j = I j 1 j p × q
f i ( h + 1 ) = f i ( h ) + α w i j I j i = 1 N w i j f i ( h ) i = 1 N ( w i j ) 2 1 j p × q
A = Re LU( M z + b )
Re LU(x) =  { x i f x 0 0 i f x < 0
L M S E = 1 T t = 1 T u = 1 U v = 1 V ( P ~ u , v P u , v ) 2
{ F s ( x , y , z ) = V max r x , y l 2 i f l 1 0 , l 1 z l 2 F s ( x , y , z ) = V m p r x , y l 2 i f l 1 < 0 , z l 2 x , y , z [ 1 , 50 ]
{ r x , y = ( x c x ) 2 + ( y c y ) 2 l 1 = k 1 ( z 1 ) g 1 l 2 = k 2 ( z 1 ) g 2
R M S E = [ 1 U V u = 1 U v = 1 V ( P ~ u , v P u , v ) 2 ] 1 2
S S I M = ( 2 μ P μ P ~ + c 1 ) ( 2 σ P P ~ + c 2 ) ( μ P 2 + μ P ~ 2 + c 1 ) ( σ P 2 + σ P ~ 2 + c 2 )
{ c 1 = ( k 1 L ) 2 c 2 = ( k 2 L ) 2
R ( X , Y ) = ( X Y T ) | | X | | 2 × | | Y | | 2

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