Abstract

In contrast to static objects, liquid structures such as drops, blobs, as well as waves and ripples on water surfaces are challenging to image in 3D due to two main reasons: first, the transient nature of those phenomena requires snapshot imaging that is fast enough to freeze the motion of the liquid. Second, the transparency of liquids and the specular reflections from their surfaces induce complex image artefacts. In this article we present a novel imaging approach to reconstruct in 3D the surface of irregular liquid structures that only requires a single snapshot. The technique is named Fringe Projection - Laser Induced Fluorescence (FP-LIF) and uses a high concentration of fluorescent dye in the probed liquid. By exciting this dye with a fringe projection structured laser beam, fluorescence is generated primarily at the liquid surface and imaged at a backward angle. By analysing the deformation of the initial projected fringes using phase-demodulation image post-processing, the 3D coordinates of the liquid surface are deduced. In this article, the approach is first numerically tested by considering a simulated pending drop, in order to analyse its performance. Then, FP-LIF is applied for two experimental cases: a quasi-static pending drop as well as a transient liquid sheet. We demonstrate reconstruction RMS errors of 1.4% and 6.1% for the simulated and experimental cases respectively. The technique presented here demonstrates, for the first time, a fringe projection approach based on LIF detection to reconstruct liquid surfaces in 3D. FP-LIF is promising for the study of more complex liquid structures and is paving the way for high-speed 3D videography of liquid surfaces.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (2)

C. Popescu, B. Tajsten, T. Blanksvard, and L. Elfgren, “3D reconstruction of existing concrete bridges using optical methods,” Struct. Infrastructure Eng. 15(7), 912–924 (2019).
[Crossref]

B. R. Halls, N. Rahman, M. N. Slipchenko, J. W. James, A. McMaster, M. D. A. Ligthfoot, J. R. Gord, and T. R. Meyer, “4D spatiotemporal evolution of liquid spray using kilohertz-rate x-ray computed tomography,” Opt. Lett. 44(20), 5013–5016 (2019).
[Crossref]

2018 (3)

S. Yang, X. Shi, G. Zhang, and C. Lv, “A dual-platform laser scanner for 3D reconstruction of dental pieces,” Engineering 4(6), 796–805 (2018).
[Crossref]

F. Liu, Y. Wu, F. Wu, N. König, R. Schmitt, Y. Wan, and Y. Xu, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148 (2018).
[Crossref]

L. Fengwei, W. Yongqian, W. Fan, K. Niels, S. Robert, W. Yongjian, and X. Yan, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148–159 (2018).
[Crossref]

2016 (2)

2015 (3)

K. Zhang, T. Wei, and H. Hu, “An experimental investigation on the surface water transport process over an airfoil by using a digital image projection technique,” Exp. Fluids 56(9), 173 (2015).
[Crossref]

T. D. Fansler and S. E. Parrish, “Spray measurement technology: a review,” Meas. Sci. Technol. 26(1), 012002 (2015).
[Crossref]

L. Marchitto, D. Hampai, S. Dabagov, L. Allocca, S. Alfuso, C. Polese, and A. Liedl, “GDI spray structure analysis by polycapillary x-ray μ-tomography,” Int. J. Multiphase Flow 70, 15–21 (2015).
[Crossref]

2014 (4)

F. Coletti, M. J. Benson, A. L. Sagues, B. H. Miller, R. Fahrig, and J. K. Eaton, “Three-Dimensional Mass Fraction Distribution of a Spray Measured by X-Ray Computed Tomography,” J. Eng. Gas Turbines Power 136(5), 051508 (2014).
[Crossref]

B. R. Halls, T. J. Heindel, A. L. Kastengren, and T. R. Meyer, “Evaluation of x-ray sources for quantitative two- and three-dimensional imaging of liquid mass distribution in atomizing sprays,” Int. J. Multiphase Flow 59, 113–120 (2014).
[Crossref]

D. R. Guildenbecher, L. Engvall, J. Gao, T. W. Grasser, P. L. Reu, and J. Chen, “Digital in-line holography to quantify secondary droplets from the impact of a single drop on a thin film,” Exp. Fluids 55(3), 1670 (2014).
[Crossref]

E. Kristensson, E. Berrocal, and M. Aldén, “Two-pulse structured illumination imaging,” Opt. Lett. 39(9), 2584–2587 (2014).
[Crossref]

2013 (1)

J. Saayman, W. Nicol, J. R. V. Ommen, and R. F. Mudde, “Fast x-ray tomography for the quantification of the bubbling-, turbulent- and fast fluidization-flow regimes and void structures,” Chem. Eng. J. 234, 437–447 (2013).
[Crossref]

2012 (2)

E. Kristensson, E. Berrocal, and M. Aldén, “Quantitative 3D imaging of scattering media using structured illumination and computed tomography,” Opt. Express 20(13), 14437–14450 (2012).
[Crossref]

J. Klinner and C. Willert, “Tomographic shadowgraphy for three-dimensional reconstruction of instantaneous spray distributions,” Exp. Fluids 53(2), 531–543 (2012).
[Crossref]

2011 (2)

R. Wellander, E. Berrocal, E. Kristensson, M. Richter, and M. Aldén, “Three-dimensional measurement of the local extinction coefficient in a dense spray,” Meas. Sci. Technol. 22(12), 125303 (2011).
[Crossref]

J. Geng, “Structured-light 3D surface imaging: a, tutorial,” Adv. Opt. Photon. 3(2), 128–160 (2011).
[Crossref]

2010 (2)

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of fourier transform, windowed fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry Fringe Projection Techniques,” Opt. Lasers Eng. 48(2), 141–148 (2010).
[Crossref]

J. W. Horbach and T. Dang, “3D reconstruction of specular surfaces using a calibrated projector-camera setup,” Mach. Vis. Appl. 21(3), 331–340 (2010).
[Crossref]

2009 (1)

X. Liu, K.-S. Im, Y. Wang, J. Wang, M. W. Tate, A. Ercan, D. R. Schuette, and S. M. Gruner, “Four dimensional visualization of highly transient fuel sprays by microsecond quantitative x-ray tomography,” Appl. Phys. Lett. 94(8), 084101 (2009).
[Crossref]

2007 (1)

Q. Kemao, “Two-dimensional windowed fourier transform for fringe pattern analysis: Principles, applications and implementations Phase Measurement Techniques and their applications,” Opt. Lasers Eng. 45(2), 304–317 (2007).
[Crossref]

2006 (1)

2004 (2)

M. Servin, J. L. Marroquin, and J. A. Quiroga, “Regularized quadrature and phase tracking from a single closed-fringe interferogram,” J. Opt. Soc. Am. A 21(3), 411–419 (2004).
[Crossref]

J. Muller, V. Kebbel, and W. Juptner, “Characterization of spatial particle distributions in a spray-forming process using digital holography,” Meas. Sci. Technol. 15(4), 706–710 (2004).
[Crossref]

2003 (1)

W. Cai, C. F. Powell, Y. Yue, S. Narayanan, J. Wang, M. W. Tate, M. J. Renzi, A. Ercan, E. Fontes, and S. M. Gruner, “Quantitative analysis of highly transient fuel sprays by time-resolved x-radiography,” Appl. Phys. Lett. 83(8), 1671–1673 (2003).
[Crossref]

2001 (1)

X. Su and W. Chen, “Fourier transform profilometry:: a review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
[Crossref]

1992 (1)

H. Murase, “Surface shape reconstruction of a nonrigid transport object using refraction and motion,” IEEE Transactions on Pattern Analysis Mach. Intell. 14(10), 1045–1052 (1992).
[Crossref]

1983 (1)

Aldén, M.

Alfuso, S.

L. Marchitto, D. Hampai, S. Dabagov, L. Allocca, S. Alfuso, C. Polese, and A. Liedl, “GDI spray structure analysis by polycapillary x-ray μ-tomography,” Int. J. Multiphase Flow 70, 15–21 (2015).
[Crossref]

Allocca, L.

L. Marchitto, D. Hampai, S. Dabagov, L. Allocca, S. Alfuso, C. Polese, and A. Liedl, “GDI spray structure analysis by polycapillary x-ray μ-tomography,” Int. J. Multiphase Flow 70, 15–21 (2015).
[Crossref]

Asundi, A. K.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of fourier transform, windowed fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry Fringe Projection Techniques,” Opt. Lasers Eng. 48(2), 141–148 (2010).
[Crossref]

Benson, M. J.

F. Coletti, M. J. Benson, A. L. Sagues, B. H. Miller, R. Fahrig, and J. K. Eaton, “Three-Dimensional Mass Fraction Distribution of a Spray Measured by X-Ray Computed Tomography,” J. Eng. Gas Turbines Power 136(5), 051508 (2014).
[Crossref]

Berrocal, E.

Blanksvard, T.

C. Popescu, B. Tajsten, T. Blanksvard, and L. Elfgren, “3D reconstruction of existing concrete bridges using optical methods,” Struct. Infrastructure Eng. 15(7), 912–924 (2019).
[Crossref]

Bodenmann, A.

A. Bodenmann, B. Thornton, T. Ura, M. Sangekar, T. Nakatani, and T. Sakamaki, “Pixel based mapping using a sheet laser and camera for generation of coloured 3D seafloor reconstructions,” in OCEANS 2010 MTS/IEEE SEATTLE, (2010), pp. 1–5.

Boulogne, F.

S. van der Walt, J. L. Schönberger, J. Nunez-Iglesias, F. Boulogne, J. D. Warner, N. Yager, E. Gouillart, and T. Yu, “scikit-image: Image processing in python,” CoRR abs/1407.6245 (2014).

Burton, D. R.

D. R. Burton, “Phase Unwrapping,” in Phase estimation in optical interferometry, E. Hack and P. K. Rastogi, eds. (CTC Press, 2015).

Cai, W.

W. Cai, C. F. Powell, Y. Yue, S. Narayanan, J. Wang, M. W. Tate, M. J. Renzi, A. Ercan, E. Fontes, and S. M. Gruner, “Quantitative analysis of highly transient fuel sprays by time-resolved x-radiography,” Appl. Phys. Lett. 83(8), 1671–1673 (2003).
[Crossref]

Chen, J.

D. R. Guildenbecher, L. Engvall, J. Gao, T. W. Grasser, P. L. Reu, and J. Chen, “Digital in-line holography to quantify secondary droplets from the impact of a single drop on a thin film,” Exp. Fluids 55(3), 1670 (2014).
[Crossref]

Chen, W.

X. Su and W. Chen, “Fourier transform profilometry:: a review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
[Crossref]

Chengda, L.

Y. Yidan, L. Chengda, Z. Ruifang, and C. Kun, “Rapeseed 3D reconstruction and morphological parameter measurement based on laser point cloud,” in 2016 Fifth International Conference on Agro-Geoinformatics (Agro-Geoinformatics) (2016), pp. 1–6.x

Coletti, F.

F. Coletti, M. J. Benson, A. L. Sagues, B. H. Miller, R. Fahrig, and J. K. Eaton, “Three-Dimensional Mass Fraction Distribution of a Spray Measured by X-Ray Computed Tomography,” J. Eng. Gas Turbines Power 136(5), 051508 (2014).
[Crossref]

Dabagov, S.

L. Marchitto, D. Hampai, S. Dabagov, L. Allocca, S. Alfuso, C. Polese, and A. Liedl, “GDI spray structure analysis by polycapillary x-ray μ-tomography,” Int. J. Multiphase Flow 70, 15–21 (2015).
[Crossref]

Dang, T.

J. W. Horbach and T. Dang, “3D reconstruction of specular surfaces using a calibrated projector-camera setup,” Mach. Vis. Appl. 21(3), 331–340 (2010).
[Crossref]

Eaton, J. K.

F. Coletti, M. J. Benson, A. L. Sagues, B. H. Miller, R. Fahrig, and J. K. Eaton, “Three-Dimensional Mass Fraction Distribution of a Spray Measured by X-Ray Computed Tomography,” J. Eng. Gas Turbines Power 136(5), 051508 (2014).
[Crossref]

Elfgren, L.

C. Popescu, B. Tajsten, T. Blanksvard, and L. Elfgren, “3D reconstruction of existing concrete bridges using optical methods,” Struct. Infrastructure Eng. 15(7), 912–924 (2019).
[Crossref]

Engvall, L.

D. R. Guildenbecher, L. Engvall, J. Gao, T. W. Grasser, P. L. Reu, and J. Chen, “Digital in-line holography to quantify secondary droplets from the impact of a single drop on a thin film,” Exp. Fluids 55(3), 1670 (2014).
[Crossref]

Ercan, A.

X. Liu, K.-S. Im, Y. Wang, J. Wang, M. W. Tate, A. Ercan, D. R. Schuette, and S. M. Gruner, “Four dimensional visualization of highly transient fuel sprays by microsecond quantitative x-ray tomography,” Appl. Phys. Lett. 94(8), 084101 (2009).
[Crossref]

W. Cai, C. F. Powell, Y. Yue, S. Narayanan, J. Wang, M. W. Tate, M. J. Renzi, A. Ercan, E. Fontes, and S. M. Gruner, “Quantitative analysis of highly transient fuel sprays by time-resolved x-radiography,” Appl. Phys. Lett. 83(8), 1671–1673 (2003).
[Crossref]

Fahrig, R.

F. Coletti, M. J. Benson, A. L. Sagues, B. H. Miller, R. Fahrig, and J. K. Eaton, “Three-Dimensional Mass Fraction Distribution of a Spray Measured by X-Ray Computed Tomography,” J. Eng. Gas Turbines Power 136(5), 051508 (2014).
[Crossref]

Fan, W.

L. Fengwei, W. Yongqian, W. Fan, K. Niels, S. Robert, W. Yongjian, and X. Yan, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148–159 (2018).
[Crossref]

Fansler, T. D.

T. D. Fansler and S. E. Parrish, “Spray measurement technology: a review,” Meas. Sci. Technol. 26(1), 012002 (2015).
[Crossref]

Fengwei, L.

L. Fengwei, W. Yongqian, W. Fan, K. Niels, S. Robert, W. Yongjian, and X. Yan, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148–159 (2018).
[Crossref]

Fontes, E.

W. Cai, C. F. Powell, Y. Yue, S. Narayanan, J. Wang, M. W. Tate, M. J. Renzi, A. Ercan, E. Fontes, and S. M. Gruner, “Quantitative analysis of highly transient fuel sprays by time-resolved x-radiography,” Appl. Phys. Lett. 83(8), 1671–1673 (2003).
[Crossref]

Gao, J.

D. R. Guildenbecher, L. Engvall, J. Gao, T. W. Grasser, P. L. Reu, and J. Chen, “Digital in-line holography to quantify secondary droplets from the impact of a single drop on a thin film,” Exp. Fluids 55(3), 1670 (2014).
[Crossref]

Geng, J.

Goidluecke, B.

I. Ihrke, B. Goidluecke, and M. Magnor, “Reconstructing the geometry of flowing water,” in Tenth IEEE International Conference on Computer Vision (ICCV’05) Volume 1, vol. 2 (2005), pp. 1055–1060.x

Gong, M.

Y. Qian, M. Gong, and Y. Yang, “Stereo-based 3D reconstruction of dynamic fluid surfaces by global optimization,” in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2017), pp. 6650–6659.

Gord, J. R.

Gouillart, E.

S. van der Walt, J. L. Schönberger, J. Nunez-Iglesias, F. Boulogne, J. D. Warner, N. Yager, E. Gouillart, and T. Yu, “scikit-image: Image processing in python,” CoRR abs/1407.6245 (2014).

Grasser, T. W.

D. R. Guildenbecher, L. Engvall, J. Gao, T. W. Grasser, P. L. Reu, and J. Chen, “Digital in-line holography to quantify secondary droplets from the impact of a single drop on a thin film,” Exp. Fluids 55(3), 1670 (2014).
[Crossref]

Greene, M.

P. Lillo, M. Greene, and V. Sick, “Plenoptic single-shot 3D imaging of in-cylinder fuel spray geometry,” Zeitschrift für Physikalische Chemie 229 (2015).

Gruner, S. M.

X. Liu, K.-S. Im, Y. Wang, J. Wang, M. W. Tate, A. Ercan, D. R. Schuette, and S. M. Gruner, “Four dimensional visualization of highly transient fuel sprays by microsecond quantitative x-ray tomography,” Appl. Phys. Lett. 94(8), 084101 (2009).
[Crossref]

W. Cai, C. F. Powell, Y. Yue, S. Narayanan, J. Wang, M. W. Tate, M. J. Renzi, A. Ercan, E. Fontes, and S. M. Gruner, “Quantitative analysis of highly transient fuel sprays by time-resolved x-radiography,” Appl. Phys. Lett. 83(8), 1671–1673 (2003).
[Crossref]

Guildenbecher, D. R.

E. M. Hall, B. S. Thurow, and D. R. Guildenbecher, “Comparison of three-dimensional particle tracking and sizing using plenoptic imaging and digital in-line holography,” Appl. Opt. 55(23), 6410–6420 (2016).
[Crossref]

D. R. Guildenbecher, L. Engvall, J. Gao, T. W. Grasser, P. L. Reu, and J. Chen, “Digital in-line holography to quantify secondary droplets from the impact of a single drop on a thin film,” Exp. Fluids 55(3), 1670 (2014).
[Crossref]

Hall, E. M.

Halls, B. R.

B. R. Halls, N. Rahman, M. N. Slipchenko, J. W. James, A. McMaster, M. D. A. Ligthfoot, J. R. Gord, and T. R. Meyer, “4D spatiotemporal evolution of liquid spray using kilohertz-rate x-ray computed tomography,” Opt. Lett. 44(20), 5013–5016 (2019).
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B. R. Halls, T. J. Heindel, A. L. Kastengren, and T. R. Meyer, “Evaluation of x-ray sources for quantitative two- and three-dimensional imaging of liquid mass distribution in atomizing sprays,” Int. J. Multiphase Flow 59, 113–120 (2014).
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L. Marchitto, D. Hampai, S. Dabagov, L. Allocca, S. Alfuso, C. Polese, and A. Liedl, “GDI spray structure analysis by polycapillary x-ray μ-tomography,” Int. J. Multiphase Flow 70, 15–21 (2015).
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Heindel, T. J.

B. R. Halls, T. J. Heindel, A. L. Kastengren, and T. R. Meyer, “Evaluation of x-ray sources for quantitative two- and three-dimensional imaging of liquid mass distribution in atomizing sprays,” Int. J. Multiphase Flow 59, 113–120 (2014).
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K. Zhang, T. Wei, and H. Hu, “An experimental investigation on the surface water transport process over an airfoil by using a digital image projection technique,” Exp. Fluids 56(9), 173 (2015).
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L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of fourier transform, windowed fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry Fringe Projection Techniques,” Opt. Lasers Eng. 48(2), 141–148 (2010).
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I. Ihrke, B. Goidluecke, and M. Magnor, “Reconstructing the geometry of flowing water,” in Tenth IEEE International Conference on Computer Vision (ICCV’05) Volume 1, vol. 2 (2005), pp. 1055–1060.x

Im, K.-S.

X. Liu, K.-S. Im, Y. Wang, J. Wang, M. W. Tate, A. Ercan, D. R. Schuette, and S. M. Gruner, “Four dimensional visualization of highly transient fuel sprays by microsecond quantitative x-ray tomography,” Appl. Phys. Lett. 94(8), 084101 (2009).
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James, J. W.

Juptner, W.

J. Muller, V. Kebbel, and W. Juptner, “Characterization of spatial particle distributions in a spray-forming process using digital holography,” Meas. Sci. Technol. 15(4), 706–710 (2004).
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Kastengren, A. L.

B. R. Halls, T. J. Heindel, A. L. Kastengren, and T. R. Meyer, “Evaluation of x-ray sources for quantitative two- and three-dimensional imaging of liquid mass distribution in atomizing sprays,” Int. J. Multiphase Flow 59, 113–120 (2014).
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J. Muller, V. Kebbel, and W. Juptner, “Characterization of spatial particle distributions in a spray-forming process using digital holography,” Meas. Sci. Technol. 15(4), 706–710 (2004).
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L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of fourier transform, windowed fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry Fringe Projection Techniques,” Opt. Lasers Eng. 48(2), 141–148 (2010).
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Q. Kemao, “Two-dimensional windowed fourier transform for fringe pattern analysis: Principles, applications and implementations Phase Measurement Techniques and their applications,” Opt. Lasers Eng. 45(2), 304–317 (2007).
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J. Klinner and C. Willert, “Tomographic shadowgraphy for three-dimensional reconstruction of instantaneous spray distributions,” Exp. Fluids 53(2), 531–543 (2012).
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F. Liu, Y. Wu, F. Wu, N. König, R. Schmitt, Y. Wan, and Y. Xu, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148 (2018).
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Kristensson, E.

Kun, C.

Y. Yidan, L. Chengda, Z. Ruifang, and C. Kun, “Rapeseed 3D reconstruction and morphological parameter measurement based on laser point cloud,” in 2016 Fifth International Conference on Agro-Geoinformatics (Agro-Geoinformatics) (2016), pp. 1–6.x

Liedl, A.

L. Marchitto, D. Hampai, S. Dabagov, L. Allocca, S. Alfuso, C. Polese, and A. Liedl, “GDI spray structure analysis by polycapillary x-ray μ-tomography,” Int. J. Multiphase Flow 70, 15–21 (2015).
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Ligthfoot, M. D. A.

Lillo, P.

P. Lillo, M. Greene, and V. Sick, “Plenoptic single-shot 3D imaging of in-cylinder fuel spray geometry,” Zeitschrift für Physikalische Chemie 229 (2015).

Liu, F.

F. Liu, Y. Wu, F. Wu, N. König, R. Schmitt, Y. Wan, and Y. Xu, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148 (2018).
[Crossref]

Liu, X.

X. Liu, K.-S. Im, Y. Wang, J. Wang, M. W. Tate, A. Ercan, D. R. Schuette, and S. M. Gruner, “Four dimensional visualization of highly transient fuel sprays by microsecond quantitative x-ray tomography,” Appl. Phys. Lett. 94(8), 084101 (2009).
[Crossref]

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S. Yang, X. Shi, G. Zhang, and C. Lv, “A dual-platform laser scanner for 3D reconstruction of dental pieces,” Engineering 4(6), 796–805 (2018).
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Magnor, M.

I. Ihrke, B. Goidluecke, and M. Magnor, “Reconstructing the geometry of flowing water,” in Tenth IEEE International Conference on Computer Vision (ICCV’05) Volume 1, vol. 2 (2005), pp. 1055–1060.x

Marchitto, L.

L. Marchitto, D. Hampai, S. Dabagov, L. Allocca, S. Alfuso, C. Polese, and A. Liedl, “GDI spray structure analysis by polycapillary x-ray μ-tomography,” Int. J. Multiphase Flow 70, 15–21 (2015).
[Crossref]

Marroquin, J. L.

McMaster, A.

Meyer, T. R.

B. R. Halls, N. Rahman, M. N. Slipchenko, J. W. James, A. McMaster, M. D. A. Ligthfoot, J. R. Gord, and T. R. Meyer, “4D spatiotemporal evolution of liquid spray using kilohertz-rate x-ray computed tomography,” Opt. Lett. 44(20), 5013–5016 (2019).
[Crossref]

B. R. Halls, T. J. Heindel, A. L. Kastengren, and T. R. Meyer, “Evaluation of x-ray sources for quantitative two- and three-dimensional imaging of liquid mass distribution in atomizing sprays,” Int. J. Multiphase Flow 59, 113–120 (2014).
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Miller, B. H.

F. Coletti, M. J. Benson, A. L. Sagues, B. H. Miller, R. Fahrig, and J. K. Eaton, “Three-Dimensional Mass Fraction Distribution of a Spray Measured by X-Ray Computed Tomography,” J. Eng. Gas Turbines Power 136(5), 051508 (2014).
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Mishra, Y. N.

Mudde, R. F.

J. Saayman, W. Nicol, J. R. V. Ommen, and R. F. Mudde, “Fast x-ray tomography for the quantification of the bubbling-, turbulent- and fast fluidization-flow regimes and void structures,” Chem. Eng. J. 234, 437–447 (2013).
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Muller, J.

J. Muller, V. Kebbel, and W. Juptner, “Characterization of spatial particle distributions in a spray-forming process using digital holography,” Meas. Sci. Technol. 15(4), 706–710 (2004).
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Nada, F. A.

Nakatani, T.

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Narayanan, S.

W. Cai, C. F. Powell, Y. Yue, S. Narayanan, J. Wang, M. W. Tate, M. J. Renzi, A. Ercan, E. Fontes, and S. M. Gruner, “Quantitative analysis of highly transient fuel sprays by time-resolved x-radiography,” Appl. Phys. Lett. 83(8), 1671–1673 (2003).
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Nicol, W.

J. Saayman, W. Nicol, J. R. V. Ommen, and R. F. Mudde, “Fast x-ray tomography for the quantification of the bubbling-, turbulent- and fast fluidization-flow regimes and void structures,” Chem. Eng. J. 234, 437–447 (2013).
[Crossref]

Niels, K.

L. Fengwei, W. Yongqian, W. Fan, K. Niels, S. Robert, W. Yongjian, and X. Yan, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148–159 (2018).
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Nunez-Iglesias, J.

S. van der Walt, J. L. Schönberger, J. Nunez-Iglesias, F. Boulogne, J. D. Warner, N. Yager, E. Gouillart, and T. Yu, “scikit-image: Image processing in python,” CoRR abs/1407.6245 (2014).

Ommen, J. R. V.

J. Saayman, W. Nicol, J. R. V. Ommen, and R. F. Mudde, “Fast x-ray tomography for the quantification of the bubbling-, turbulent- and fast fluidization-flow regimes and void structures,” Chem. Eng. J. 234, 437–447 (2013).
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Padilla, M.

M. Padilla and M. Servin, “Regularized Phase Estimation Methods in Interferometry,” in Phase estimation in optical interferometry, E. Hack and P. K. Rastogi, eds. (CTC Press, 2015).

Pan, B.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of fourier transform, windowed fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry Fringe Projection Techniques,” Opt. Lasers Eng. 48(2), 141–148 (2010).
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T. D. Fansler and S. E. Parrish, “Spray measurement technology: a review,” Meas. Sci. Technol. 26(1), 012002 (2015).
[Crossref]

Polese, C.

L. Marchitto, D. Hampai, S. Dabagov, L. Allocca, S. Alfuso, C. Polese, and A. Liedl, “GDI spray structure analysis by polycapillary x-ray μ-tomography,” Int. J. Multiphase Flow 70, 15–21 (2015).
[Crossref]

Polster, S.

Popescu, C.

C. Popescu, B. Tajsten, T. Blanksvard, and L. Elfgren, “3D reconstruction of existing concrete bridges using optical methods,” Struct. Infrastructure Eng. 15(7), 912–924 (2019).
[Crossref]

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W. Cai, C. F. Powell, Y. Yue, S. Narayanan, J. Wang, M. W. Tate, M. J. Renzi, A. Ercan, E. Fontes, and S. M. Gruner, “Quantitative analysis of highly transient fuel sprays by time-resolved x-radiography,” Appl. Phys. Lett. 83(8), 1671–1673 (2003).
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Y. Qian, M. Gong, and Y. Yang, “Stereo-based 3D reconstruction of dynamic fluid surfaces by global optimization,” in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2017), pp. 6650–6659.

Quiroga, J. A.

Rahman, N.

Renzi, M. J.

W. Cai, C. F. Powell, Y. Yue, S. Narayanan, J. Wang, M. W. Tate, M. J. Renzi, A. Ercan, E. Fontes, and S. M. Gruner, “Quantitative analysis of highly transient fuel sprays by time-resolved x-radiography,” Appl. Phys. Lett. 83(8), 1671–1673 (2003).
[Crossref]

Reu, P. L.

D. R. Guildenbecher, L. Engvall, J. Gao, T. W. Grasser, P. L. Reu, and J. Chen, “Digital in-line holography to quantify secondary droplets from the impact of a single drop on a thin film,” Exp. Fluids 55(3), 1670 (2014).
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R. Wellander, E. Berrocal, E. Kristensson, M. Richter, and M. Aldén, “Three-dimensional measurement of the local extinction coefficient in a dense spray,” Meas. Sci. Technol. 22(12), 125303 (2011).
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L. Fengwei, W. Yongqian, W. Fan, K. Niels, S. Robert, W. Yongjian, and X. Yan, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148–159 (2018).
[Crossref]

Ruifang, Z.

Y. Yidan, L. Chengda, Z. Ruifang, and C. Kun, “Rapeseed 3D reconstruction and morphological parameter measurement based on laser point cloud,” in 2016 Fifth International Conference on Agro-Geoinformatics (Agro-Geoinformatics) (2016), pp. 1–6.x

Saayman, J.

J. Saayman, W. Nicol, J. R. V. Ommen, and R. F. Mudde, “Fast x-ray tomography for the quantification of the bubbling-, turbulent- and fast fluidization-flow regimes and void structures,” Chem. Eng. J. 234, 437–447 (2013).
[Crossref]

Sagues, A. L.

F. Coletti, M. J. Benson, A. L. Sagues, B. H. Miller, R. Fahrig, and J. K. Eaton, “Three-Dimensional Mass Fraction Distribution of a Spray Measured by X-Ray Computed Tomography,” J. Eng. Gas Turbines Power 136(5), 051508 (2014).
[Crossref]

Sakamaki, T.

A. Bodenmann, B. Thornton, T. Ura, M. Sangekar, T. Nakatani, and T. Sakamaki, “Pixel based mapping using a sheet laser and camera for generation of coloured 3D seafloor reconstructions,” in OCEANS 2010 MTS/IEEE SEATTLE, (2010), pp. 1–5.

Sangekar, M.

A. Bodenmann, B. Thornton, T. Ura, M. Sangekar, T. Nakatani, and T. Sakamaki, “Pixel based mapping using a sheet laser and camera for generation of coloured 3D seafloor reconstructions,” in OCEANS 2010 MTS/IEEE SEATTLE, (2010), pp. 1–5.

Schmitt, R.

F. Liu, Y. Wu, F. Wu, N. König, R. Schmitt, Y. Wan, and Y. Xu, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148 (2018).
[Crossref]

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S. van der Walt, J. L. Schönberger, J. Nunez-Iglesias, F. Boulogne, J. D. Warner, N. Yager, E. Gouillart, and T. Yu, “scikit-image: Image processing in python,” CoRR abs/1407.6245 (2014).

Schuette, D. R.

X. Liu, K.-S. Im, Y. Wang, J. Wang, M. W. Tate, A. Ercan, D. R. Schuette, and S. M. Gruner, “Four dimensional visualization of highly transient fuel sprays by microsecond quantitative x-ray tomography,” Appl. Phys. Lett. 94(8), 084101 (2009).
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M. Padilla and M. Servin, “Regularized Phase Estimation Methods in Interferometry,” in Phase estimation in optical interferometry, E. Hack and P. K. Rastogi, eds. (CTC Press, 2015).

Shi, X.

S. Yang, X. Shi, G. Zhang, and C. Lv, “A dual-platform laser scanner for 3D reconstruction of dental pieces,” Engineering 4(6), 796–805 (2018).
[Crossref]

Sick, V.

P. Lillo, M. Greene, and V. Sick, “Plenoptic single-shot 3D imaging of in-cylinder fuel spray geometry,” Zeitschrift für Physikalische Chemie 229 (2015).

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Su, X.

X. Su and W. Chen, “Fourier transform profilometry:: a review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
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Tajsten, B.

C. Popescu, B. Tajsten, T. Blanksvard, and L. Elfgren, “3D reconstruction of existing concrete bridges using optical methods,” Struct. Infrastructure Eng. 15(7), 912–924 (2019).
[Crossref]

Takeda, M.

Tate, M. W.

X. Liu, K.-S. Im, Y. Wang, J. Wang, M. W. Tate, A. Ercan, D. R. Schuette, and S. M. Gruner, “Four dimensional visualization of highly transient fuel sprays by microsecond quantitative x-ray tomography,” Appl. Phys. Lett. 94(8), 084101 (2009).
[Crossref]

W. Cai, C. F. Powell, Y. Yue, S. Narayanan, J. Wang, M. W. Tate, M. J. Renzi, A. Ercan, E. Fontes, and S. M. Gruner, “Quantitative analysis of highly transient fuel sprays by time-resolved x-radiography,” Appl. Phys. Lett. 83(8), 1671–1673 (2003).
[Crossref]

Thornton, B.

A. Bodenmann, B. Thornton, T. Ura, M. Sangekar, T. Nakatani, and T. Sakamaki, “Pixel based mapping using a sheet laser and camera for generation of coloured 3D seafloor reconstructions,” in OCEANS 2010 MTS/IEEE SEATTLE, (2010), pp. 1–5.

Thurow, B. S.

Ura, T.

A. Bodenmann, B. Thornton, T. Ura, M. Sangekar, T. Nakatani, and T. Sakamaki, “Pixel based mapping using a sheet laser and camera for generation of coloured 3D seafloor reconstructions,” in OCEANS 2010 MTS/IEEE SEATTLE, (2010), pp. 1–5.

van der Walt, S.

S. van der Walt, J. L. Schönberger, J. Nunez-Iglesias, F. Boulogne, J. D. Warner, N. Yager, E. Gouillart, and T. Yu, “scikit-image: Image processing in python,” CoRR abs/1407.6245 (2014).

Wan, Y.

F. Liu, Y. Wu, F. Wu, N. König, R. Schmitt, Y. Wan, and Y. Xu, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148 (2018).
[Crossref]

Wang, J.

X. Liu, K.-S. Im, Y. Wang, J. Wang, M. W. Tate, A. Ercan, D. R. Schuette, and S. M. Gruner, “Four dimensional visualization of highly transient fuel sprays by microsecond quantitative x-ray tomography,” Appl. Phys. Lett. 94(8), 084101 (2009).
[Crossref]

W. Cai, C. F. Powell, Y. Yue, S. Narayanan, J. Wang, M. W. Tate, M. J. Renzi, A. Ercan, E. Fontes, and S. M. Gruner, “Quantitative analysis of highly transient fuel sprays by time-resolved x-radiography,” Appl. Phys. Lett. 83(8), 1671–1673 (2003).
[Crossref]

Wang, Y.

X. Liu, K.-S. Im, Y. Wang, J. Wang, M. W. Tate, A. Ercan, D. R. Schuette, and S. M. Gruner, “Four dimensional visualization of highly transient fuel sprays by microsecond quantitative x-ray tomography,” Appl. Phys. Lett. 94(8), 084101 (2009).
[Crossref]

Warner, J. D.

S. van der Walt, J. L. Schönberger, J. Nunez-Iglesias, F. Boulogne, J. D. Warner, N. Yager, E. Gouillart, and T. Yu, “scikit-image: Image processing in python,” CoRR abs/1407.6245 (2014).

Watkins, L. R.

L. R. Watkins, “Continuous Wavelet Transforms,” in Phase estimation in optical interferometry, E. Hack and P. K. Rastogi, eds. (CTC Press, 2015).

Wei, T.

K. Zhang, T. Wei, and H. Hu, “An experimental investigation on the surface water transport process over an airfoil by using a digital image projection technique,” Exp. Fluids 56(9), 173 (2015).
[Crossref]

Wellander, R.

R. Wellander, E. Berrocal, E. Kristensson, M. Richter, and M. Aldén, “Three-dimensional measurement of the local extinction coefficient in a dense spray,” Meas. Sci. Technol. 22(12), 125303 (2011).
[Crossref]

Weng, J.

J. Zhong and J. Weng, “Windowed Fourier Transforms,” in Phase estimation in optical interferometry, E. Hack and P. K. Rastogi, eds. (CTC Press, 2015).

Willert, C.

J. Klinner and C. Willert, “Tomographic shadowgraphy for three-dimensional reconstruction of instantaneous spray distributions,” Exp. Fluids 53(2), 531–543 (2012).
[Crossref]

Wu, F.

F. Liu, Y. Wu, F. Wu, N. König, R. Schmitt, Y. Wan, and Y. Xu, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148 (2018).
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Wu, Y.

F. Liu, Y. Wu, F. Wu, N. König, R. Schmitt, Y. Wan, and Y. Xu, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148 (2018).
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Xu, Y.

F. Liu, Y. Wu, F. Wu, N. König, R. Schmitt, Y. Wan, and Y. Xu, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148 (2018).
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Yager, N.

S. van der Walt, J. L. Schönberger, J. Nunez-Iglesias, F. Boulogne, J. D. Warner, N. Yager, E. Gouillart, and T. Yu, “scikit-image: Image processing in python,” CoRR abs/1407.6245 (2014).

Yan, X.

L. Fengwei, W. Yongqian, W. Fan, K. Niels, S. Robert, W. Yongjian, and X. Yan, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148–159 (2018).
[Crossref]

Yang, S.

S. Yang, X. Shi, G. Zhang, and C. Lv, “A dual-platform laser scanner for 3D reconstruction of dental pieces,” Engineering 4(6), 796–805 (2018).
[Crossref]

Yang, Y.

Y. Qian, M. Gong, and Y. Yang, “Stereo-based 3D reconstruction of dynamic fluid surfaces by global optimization,” in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2017), pp. 6650–6659.

Yau, S.-T.

Yidan, Y.

Y. Yidan, L. Chengda, Z. Ruifang, and C. Kun, “Rapeseed 3D reconstruction and morphological parameter measurement based on laser point cloud,” in 2016 Fifth International Conference on Agro-Geoinformatics (Agro-Geoinformatics) (2016), pp. 1–6.x

Yongjian, W.

L. Fengwei, W. Yongqian, W. Fan, K. Niels, S. Robert, W. Yongjian, and X. Yan, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148–159 (2018).
[Crossref]

Yongqian, W.

L. Fengwei, W. Yongqian, W. Fan, K. Niels, S. Robert, W. Yongjian, and X. Yan, “Precise phase demodulation of single carrier-frequency interferogram by pixel-level lissajous figure and ellipse fitting,” Sci. Rep. 8(1), 148–159 (2018).
[Crossref]

Yu, T.

S. van der Walt, J. L. Schönberger, J. Nunez-Iglesias, F. Boulogne, J. D. Warner, N. Yager, E. Gouillart, and T. Yu, “scikit-image: Image processing in python,” CoRR abs/1407.6245 (2014).

Yue, Y.

W. Cai, C. F. Powell, Y. Yue, S. Narayanan, J. Wang, M. W. Tate, M. J. Renzi, A. Ercan, E. Fontes, and S. M. Gruner, “Quantitative analysis of highly transient fuel sprays by time-resolved x-radiography,” Appl. Phys. Lett. 83(8), 1671–1673 (2003).
[Crossref]

Zhang, G.

S. Yang, X. Shi, G. Zhang, and C. Lv, “A dual-platform laser scanner for 3D reconstruction of dental pieces,” Engineering 4(6), 796–805 (2018).
[Crossref]

Zhang, K.

K. Zhang, T. Wei, and H. Hu, “An experimental investigation on the surface water transport process over an airfoil by using a digital image projection technique,” Exp. Fluids 56(9), 173 (2015).
[Crossref]

Zhang, S.

S. Zhang and S.-T. Yau, “High-resolution, real-time 3D absolute coordinate measurement based on a phase-shifting method,” Opt. Express 14(7), 2644–2649 (2006).
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S. Zhang, High-speed 3D imaging with digital fringe projection techniques., Optical sciences and applications of light (CRC Press, 2016).

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J. Zhong and J. Weng, “Windowed Fourier Transforms,” in Phase estimation in optical interferometry, E. Hack and P. K. Rastogi, eds. (CTC Press, 2015).

Adv. Opt. Photon. (1)

Appl. Opt. (2)

Appl. Phys. Lett. (2)

W. Cai, C. F. Powell, Y. Yue, S. Narayanan, J. Wang, M. W. Tate, M. J. Renzi, A. Ercan, E. Fontes, and S. M. Gruner, “Quantitative analysis of highly transient fuel sprays by time-resolved x-radiography,” Appl. Phys. Lett. 83(8), 1671–1673 (2003).
[Crossref]

X. Liu, K.-S. Im, Y. Wang, J. Wang, M. W. Tate, A. Ercan, D. R. Schuette, and S. M. Gruner, “Four dimensional visualization of highly transient fuel sprays by microsecond quantitative x-ray tomography,” Appl. Phys. Lett. 94(8), 084101 (2009).
[Crossref]

Chem. Eng. J. (1)

J. Saayman, W. Nicol, J. R. V. Ommen, and R. F. Mudde, “Fast x-ray tomography for the quantification of the bubbling-, turbulent- and fast fluidization-flow regimes and void structures,” Chem. Eng. J. 234, 437–447 (2013).
[Crossref]

Engineering (1)

S. Yang, X. Shi, G. Zhang, and C. Lv, “A dual-platform laser scanner for 3D reconstruction of dental pieces,” Engineering 4(6), 796–805 (2018).
[Crossref]

Exp. Fluids (3)

J. Klinner and C. Willert, “Tomographic shadowgraphy for three-dimensional reconstruction of instantaneous spray distributions,” Exp. Fluids 53(2), 531–543 (2012).
[Crossref]

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Figures (10)

Fig. 1.
Fig. 1. General description of the FP-LIF approach: Periodic fringes are projected on a 3D liquid structure containing a fluorescing dye at high concentration. The fluorescing fringes are, then, imaged at a backward angle $\theta$ . By now analysing the phase of the modulation, from the recorded fringe pattern, a 3D surface reconstruction can be obtained. Here, (a) shows the light intensity along the green line of recorded fringe pattern while (b) shows the modulation the incident fringe projection. The change of frequency between (a) and (b) is induced by the 3D structure of the object surface. Thus, the shift of the modulation peaks between the signal (a) and the reference (b) is connected to the depth coordinate Z as shown in (c).
Fig. 2.
Fig. 2. For phase demodulation using the CWT method a wavelet is required and the one used in this work is the Morlet wavelet shown above. The real part of the complex spatial function, Eq. (2), is plotted in the left panel with its Fourier transform $\mathcal {F}$ on the right. From Eq. (2) $k_0 = 6$ , $\sigma = 0.6$ and $\varepsilon = 3$ where the parameters have been adapted to improve simulation results from initial values suggested by Watkins et al [33].
Fig. 3.
Fig. 3. Illustration of orthographic fringe projection and imaging with the FP-LIF technique. The projected light induce fluorescence close to the 3D structure surface, white stripes, connected to a linear global phase in the $X$ direction, Eq. (10). The projection is imaged from an angle $\theta$ where the orthographic camera converts the global coordinates $[X, Y, Z]^T$ into camera coordinates $[x, y]^T$ . The projection can be represented using a camera matrix, Eq. (9). From the global phase and the camera matrix, the expression in Eq. (13) is derived for extracting the third coordinate $d$ from the image phase.
Fig. 4.
Fig. 4. The simulated drop 3D structure and simulated fringe patterns. The top left image shows the drop structure where the colour represents $d(x, y$ -> $d(x, y)$ and the remaining fringe patterns images were simulated with different base period length $T$ and Signal to Noise Ratio (SNR). For a 3D visualisation of the drop go to https://3d.spray-imaging.com/simulated_drop.
Fig. 5.
Fig. 5. 3D reconstructed cross sections of the simulated drop show good accuracy compared to the true structure. The reconstructions are produced from fringe patterns similar to the ones found in Fig. 4 but as all combinations of base period lengths, $T$ , 7 and 12 pixels and SNR $\infty$ , $20$ and $2$ . Both horizontal cross sections (a) and (b) along $y_c$ , and vertical cross sections (c) and (d) along $x_c$ are shown. Here, (b) and (d) show the errors to the original structure in small detail. In (a), large errors are shown mainly close to the left edge which is connected to a high instantaneous frequency of the pattern that is hard to demodulate. If these large errors would be found in the reconstruction of an experimental image, the areas would be segmented as background.
Fig. 6.
Fig. 6. Optimization of the period length $T$ for 3D reconstruction of fringe patterns show an optimal value between $12-15$ pixels for the full structure or $7-9$ pixels if the left edge is disregarded. The three maps (full structure, left edge and remainder pixels indicated in the top right panel) show mean RMSE measures for 3D reconstructions of the simulated drop 3D structure with different combinations of noise level, SNR, and base fringe pattern period $T$ .
Fig. 7.
Fig. 7. Optical arrangement of the FP-LIF experimental setup.
Fig. 8.
Fig. 8. Experimental 3D reconstruction of a pending drop. Panel (a) shows the fringe pattern recorded using the FP-LIF technique. It has a transparent overlay of a manual foreground segmentation, only pixels inside this area are reconstructed in 3D. Panel (b) shows the 3D reconstruction from the front and a slightly rotated version is shown in (c). The 3D reconstructions are virtually rendered with the 3D computer graphics software Blender. For a 3D visualisation go to https://3d.spray-imaging.com/drop.
Fig. 9.
Fig. 9. The assumption of rotational symmetry for the pending drop is here used as validation of the experimental 3D reconstruction in Fig. 8. The blue envelope line in panel (a) is drawn on the edge of the original fringe pattern image and the distance from the centre of the envelope to the line is used as the rotational symmetric drop’s radius for different $Y$ coordinates. The illustration is similar to the one found in Fig. 5 where both a horizontal (c) and a vertical (d) cross section is compared to the rotational symmetric structure along $y_c$ and $x_c$ as indicated in panel (b). Panel (b) also show the full 3D reconstruction where the colour indicates the estimated $\hat {d}$ . Here, $x_c$ is chosen as the calibrated world coordinate $X = 0$ .
Fig. 10.
Fig. 10. Example of experimental images with different shapes. (a) shows the shape evolution of pending drops at different times ( $t_1$ , $t_2$ , $t_3$ ) until they break up. (b) Experimental 3D reconstruction of a transient liquid sheet which is similarly visualised as Fig. 9. This is a proof of concept example of using the FP-LIF technique on a more turbulent liquid surface structure with a short shutter time of 50 $\mu$ s. For a 3D visualisation go to https://3d.spray-imaging.com/onion.

Equations (17)

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I ( x , y ) = { A ( x , y ) + B ( x , y ) cos ( φ ( x , y ) ) + ϵ ( x , y ) ,   ( x , y ) fg   ϵ ( x , y ) ,   ( x , y ) bg
Ψ M o r l e t ( x , y ) = exp ( i k 0 x 1 2 σ 2 ( x 2 + y 2 ε ) ) .
Ψ a , b ( x ) = 1 a Ψ ( x b a ) ,
W f ( a , b ) = f ( x ) 1 a Ψ ( x b a ) d 2 x .
R ( b ) = argmax a ( | W f ( a , b ) | ) .
φ ^ wrapped ( x ) = arg ( W f ( R ( x ) , x ) ) .
W f ( a ) = a F 1 ( F ( f ) F ( Ψ ) ( a x ) ) ,
{ D = [ X , Y , d ( X , Y ) , 1 ] T , ( X , Y )  is on object } ,
P = K C = [ s 0 x 0 0 s y 0 0 0 1 ] [ cos ( θ ) 0 sin ( θ ) 0 0 1 0 0 0 0 0 1 ] .
φ ( X , Y , Z ) = 2 π T g X ,
[ x y 1 ] = P D = [ s ( cos ( θ ) X sin ( θ ) d ( x , y ) ) + x 0 s ( Y ) + y 0 1 ] .
φ ( x , y ) = 2 π T ( x x 0 + s sin ( θ ) d ( x , y ) ) .
d ( x , y ) = T 2 π s sin ( θ ) ( φ ( x , y ) 2 π T ( x x 0 ) ) .
[ X Y Z ] = [ x x 0 + s sin ( θ ) d ( x , y ) s cos ( θ ) y y 0 s d ( x , y ) ] , [ x , y ]  fg .
r ( Y ) = { 0.4 Y 3 0.6 Y 2 + 1 ,   Y 0 sin ( arccos ( Y ) ) ,   Y > 0
x = X y = Y d ( x , y ) = X sin ( θ ) + d ( X , Y ) cos ( θ )
R M S E = 1 N x , y fg ( d ( x , y ) d ^ ( x , y ) ) 2 ,

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