Abstract

An imaging technique capable of measuring the extinction coefficient in 3D is presented and demonstrated on various scattering media. The approach is able to suppress unwanted effects due to both multiple scattering and light extinction, which, in turbid situations, seriously hampers the performance of conventional imaging techniques. The main concept consists in illuminating the sample of interest with a light source that is spatially modulated in both the vertical and horizontal direction and to measure, using Structured Illumination, the correct transmission in 2D at several viewing angles. The sample is then reconstructed in 3D by means of a standard Computed Tomography algorithm. To create the adequate illumination, a novel “crossed” structured illumination approach is implemented. In this article, the accuracy and limitation of the method is first evaluated by probing several homogeneous milk solutions at various levels of turbidity. The unique possibility of visualizing an object hidden within such solutions is also demonstrated. Finally the method is applied on two different inhomogeneous scattering spray systems; one transient and one quasi-steady state.

© 2012 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  10. C. F. Powell, Y. Yue, R. Poolab, and J. Wanga, “Time-resolved measurements of supersonic fuel sprays using synchrotron x-rays,” J. Synchrotron Radiat. 7, 356–360 (2000).
    [Crossref]
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    [Crossref]

2011 (3)

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, 855–861 (2011).
[Crossref]

E. Kristensson, L. Araneo, E. Berrocal, J. Manin, M. Richter, M. Aldén, and M. Linne, “Analysis of multiple scattering suppression using structured laser illumination planar imaging in scattering and fluorescing media,” Opt. Express 19, 13674–13663 (2011).
[Crossref]

N. Ducros, A. Bassi, G. Valentini, M. Schweiger, S. Arridge, and C. D’Andrea, “Muliple-view fluorescence optical tomography reconstruction using compression of experimental data,” Opt. Lett. 36, 1377–1379 (2011).
[Crossref] [PubMed]

2010 (2)

2008 (1)

2007 (3)

2005 (1)

2004 (1)

O. S. Ugolnikov, O. V. Postylyakov, and I. A. Maslova, “Effects of multiple scattering and atmospheric aerosol on the polarization of the twilight sky,” J. Quant. Spectrosc. Radiat. Transfer 88, 233–241 (2004).
[Crossref]

2003 (1)

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

2000 (1)

C. F. Powell, Y. Yue, R. Poolab, and J. Wanga, “Time-resolved measurements of supersonic fuel sprays using synchrotron x-rays,” J. Synchrotron Radiat. 7, 356–360 (2000).
[Crossref]

1997 (1)

1993 (1)

1990 (1)

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quant. Electron. 26, 2166–2185 (1990).
[Crossref]

Aldén, M.

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, 855–861 (2011).
[Crossref]

E. Kristensson, L. Araneo, E. Berrocal, J. Manin, M. Richter, M. Aldén, and M. Linne, “Analysis of multiple scattering suppression using structured laser illumination planar imaging in scattering and fluorescing media,” Opt. Express 19, 13674–13663 (2011).
[Crossref]

E. Berrocal, E. Kristensson, M. Richter, M. Linne, and M. Aldén, “Application of structured illumination for multiple scattering suppression in planar laser imaging of dense sprays,” Opt. Express 16, 17870–17881 (2008).
[Crossref] [PubMed]

Araneo, L.

E. Kristensson, L. Araneo, E. Berrocal, J. Manin, M. Richter, M. Aldén, and M. Linne, “Analysis of multiple scattering suppression using structured laser illumination planar imaging in scattering and fluorescing media,” Opt. Express 19, 13674–13663 (2011).
[Crossref]

Arridge, S.

Badizadegan, K.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Bassi, A.

Berrocal, E.

E. Kristensson, L. Araneo, E. Berrocal, J. Manin, M. Richter, M. Aldén, and M. Linne, “Analysis of multiple scattering suppression using structured laser illumination planar imaging in scattering and fluorescing media,” Opt. Express 19, 13674–13663 (2011).
[Crossref]

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, 855–861 (2011).
[Crossref]

E. Berrocal, E. Kristensson, M. Richter, M. Linne, and M. Aldén, “Application of structured illumination for multiple scattering suppression in planar laser imaging of dense sprays,” Opt. Express 16, 17870–17881 (2008).
[Crossref] [PubMed]

E. Berrocal, D. L. Sedarsky, M. E. Paciaroni, I. V. Meglinski, and M. A. Linne, “Laser light scattering in turbid media. Part I: Experimental and simulated results for the spatial intensity distribution,” Opt. Express 15, 10649–10665 (2007).
[Crossref] [PubMed]

M. Linne, M. Paciaroni, E. Berrocal, and D. Sedarsky, “Ballistic imaging of liquid breakup processes in dense sprays,” in Proceedings of the Combustion Institute (2009), Vol. 32, pp. 2147–2161.
[Crossref]

Bevilacqua, F.

Breuninger, T.

Bruscaglioni, P.

Cai, W.

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

Castleman, K. R.

Q. Wu, F. A. Merchant, and K. R. Castleman, Microscope Image Processing (Academic Press, 2008).

Chen, J.

Cheong, W.-F.

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quant. Electron. 26, 2166–2185 (1990).
[Crossref]

Choi, W.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Cuccia, D. J.

D’Andrea, C.

Dasari, R. R.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Ducros, N.

Durkin, A. J.

Ercan, A.

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

Fang-Yen, C.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Feld, M. S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Fontes, E.

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

Greger, K.

Grunere, S. M.

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

Gurioli, M.

Intes, X.

Juškaitis, R.

Kak, A.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Academic Press, 1999).

Koppal, S. J.

S. G. Narasimhan, S. K. Nayar, B. Sun, and S. J. Koppal, “Structured light in scattering media,” in Proceedings of the Tenth IEEE International Conference on Computer Vision (2005).

Kristensson, E.

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, 855–861 (2011).
[Crossref]

E. Kristensson, L. Araneo, E. Berrocal, J. Manin, M. Richter, M. Aldén, and M. Linne, “Analysis of multiple scattering suppression using structured laser illumination planar imaging in scattering and fluorescing media,” Opt. Express 19, 13674–13663 (2011).
[Crossref]

E. Berrocal, E. Kristensson, M. Richter, M. Linne, and M. Aldén, “Application of structured illumination for multiple scattering suppression in planar laser imaging of dense sprays,” Opt. Express 16, 17870–17881 (2008).
[Crossref] [PubMed]

Lesage, F.

Linne, M.

E. Kristensson, L. Araneo, E. Berrocal, J. Manin, M. Richter, M. Aldén, and M. Linne, “Analysis of multiple scattering suppression using structured laser illumination planar imaging in scattering and fluorescing media,” Opt. Express 19, 13674–13663 (2011).
[Crossref]

E. Berrocal, E. Kristensson, M. Richter, M. Linne, and M. Aldén, “Application of structured illumination for multiple scattering suppression in planar laser imaging of dense sprays,” Opt. Express 16, 17870–17881 (2008).
[Crossref] [PubMed]

M. Linne, M. Paciaroni, E. Berrocal, and D. Sedarsky, “Ballistic imaging of liquid breakup processes in dense sprays,” in Proceedings of the Combustion Institute (2009), Vol. 32, pp. 2147–2161.
[Crossref]

Linne, M. A.

Lue, N.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Manin, J.

E. Kristensson, L. Araneo, E. Berrocal, J. Manin, M. Richter, M. Aldén, and M. Linne, “Analysis of multiple scattering suppression using structured laser illumination planar imaging in scattering and fluorescing media,” Opt. Express 19, 13674–13663 (2011).
[Crossref]

Maslova, I. A.

O. S. Ugolnikov, O. V. Postylyakov, and I. A. Maslova, “Effects of multiple scattering and atmospheric aerosol on the polarization of the twilight sky,” J. Quant. Spectrosc. Radiat. Transfer 88, 233–241 (2004).
[Crossref]

Meglinski, I. V.

Merchant, F. A.

Q. Wu, F. A. Merchant, and K. R. Castleman, Microscope Image Processing (Academic Press, 2008).

Narasimhan, S. G.

S. G. Narasimhan, S. K. Nayar, B. Sun, and S. J. Koppal, “Structured light in scattering media,” in Proceedings of the Tenth IEEE International Conference on Computer Vision (2005).

Narayanan, S.

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

Nayar, S. K.

S. G. Narasimhan, S. K. Nayar, B. Sun, and S. J. Koppal, “Structured light in scattering media,” in Proceedings of the Tenth IEEE International Conference on Computer Vision (2005).

Neil, M. A. A.

Oh, S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Paciaroni, M.

M. Linne, M. Paciaroni, E. Berrocal, and D. Sedarsky, “Ballistic imaging of liquid breakup processes in dense sprays,” in Proceedings of the Combustion Institute (2009), Vol. 32, pp. 2147–2161.
[Crossref]

Paciaroni, M. E.

Padgett, M.

M. Padgett, “Penetrating scattering media,” Nat. Photonics 4, 741–742 (2010).
[Crossref]

Poolab, R.

C. F. Powell, Y. Yue, R. Poolab, and J. Wanga, “Time-resolved measurements of supersonic fuel sprays using synchrotron x-rays,” J. Synchrotron Radiat. 7, 356–360 (2000).
[Crossref]

Postylyakov, O. V.

O. S. Ugolnikov, O. V. Postylyakov, and I. A. Maslova, “Effects of multiple scattering and atmospheric aerosol on the polarization of the twilight sky,” J. Quant. Spectrosc. Radiat. Transfer 88, 233–241 (2004).
[Crossref]

Powell, C. F.

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

C. F. Powell, Y. Yue, R. Poolab, and J. Wanga, “Time-resolved measurements of supersonic fuel sprays using synchrotron x-rays,” J. Synchrotron Radiat. 7, 356–360 (2000).
[Crossref]

Prahl, S. A.

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quant. Electron. 26, 2166–2185 (1990).
[Crossref]

Renzi, M. J.

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

Richter, M.

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, 855–861 (2011).
[Crossref]

E. Kristensson, L. Araneo, E. Berrocal, J. Manin, M. Richter, M. Aldén, and M. Linne, “Analysis of multiple scattering suppression using structured laser illumination planar imaging in scattering and fluorescing media,” Opt. Express 19, 13674–13663 (2011).
[Crossref]

E. Berrocal, E. Kristensson, M. Richter, M. Linne, and M. Aldén, “Application of structured illumination for multiple scattering suppression in planar laser imaging of dense sprays,” Opt. Express 16, 17870–17881 (2008).
[Crossref] [PubMed]

Sansoni, P.

Schweiger, M.

Sedarsky, D.

M. Linne, M. Paciaroni, E. Berrocal, and D. Sedarsky, “Ballistic imaging of liquid breakup processes in dense sprays,” in Proceedings of the Combustion Institute (2009), Vol. 32, pp. 2147–2161.
[Crossref]

Sedarsky, D. L.

Slaney, M.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Academic Press, 1999).

Stelzer, E. H. K.

Sun, B.

S. G. Narasimhan, S. K. Nayar, B. Sun, and S. J. Koppal, “Structured light in scattering media,” in Proceedings of the Tenth IEEE International Conference on Computer Vision (2005).

Tate, M. W.

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

Tromberg, B. J.

Ugolnikov, O. S.

O. S. Ugolnikov, O. V. Postylyakov, and I. A. Maslova, “Effects of multiple scattering and atmospheric aerosol on the polarization of the twilight sky,” J. Quant. Spectrosc. Radiat. Transfer 88, 233–241 (2004).
[Crossref]

Valentini, G.

Venugopal, V.

Wanga, J.

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

C. F. Powell, Y. Yue, R. Poolab, and J. Wanga, “Time-resolved measurements of supersonic fuel sprays using synchrotron x-rays,” J. Synchrotron Radiat. 7, 356–360 (2000).
[Crossref]

Welch, A. J.

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quant. Electron. 26, 2166–2185 (1990).
[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, 855–861 (2011).
[Crossref]

Wilson, T.

Wu, Q.

Q. Wu, F. A. Merchant, and K. R. Castleman, Microscope Image Processing (Academic Press, 2008).

Yue, Y.

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

C. F. Powell, Y. Yue, R. Poolab, and J. Wanga, “Time-resolved measurements of supersonic fuel sprays using synchrotron x-rays,” J. Synchrotron Radiat. 7, 356–360 (2000).
[Crossref]

Zaccanti, G.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

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

IEEE J. Quant. Electron. (1)

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quant. Electron. 26, 2166–2185 (1990).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (1)

O. S. Ugolnikov, O. V. Postylyakov, and I. A. Maslova, “Effects of multiple scattering and atmospheric aerosol on the polarization of the twilight sky,” J. Quant. Spectrosc. Radiat. Transfer 88, 233–241 (2004).
[Crossref]

J. Synchrotron Radiat. (1)

C. F. Powell, Y. Yue, R. Poolab, and J. Wanga, “Time-resolved measurements of supersonic fuel sprays using synchrotron x-rays,” J. Synchrotron Radiat. 7, 356–360 (2000).
[Crossref]

Meas. Sci. Technol. (1)

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, 855–861 (2011).
[Crossref]

Nat. Methods (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Nat. Photonics (1)

M. Padgett, “Penetrating scattering media,” Nat. Photonics 4, 741–742 (2010).
[Crossref]

Opt. Express (3)

Opt. Lett. (5)

Other (4)

M. Linne, M. Paciaroni, E. Berrocal, and D. Sedarsky, “Ballistic imaging of liquid breakup processes in dense sprays,” in Proceedings of the Combustion Institute (2009), Vol. 32, pp. 2147–2161.
[Crossref]

S. G. Narasimhan, S. K. Nayar, B. Sun, and S. J. Koppal, “Structured light in scattering media,” in Proceedings of the Tenth IEEE International Conference on Computer Vision (2005).

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Academic Press, 1999).

Q. Wu, F. A. Merchant, and K. R. Castleman, Microscope Image Processing (Academic Press, 2008).

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Figures (8)

Fig. 1
Fig. 1 Illustration of computed tomography, showing the sample at six different viewing angles together with the corresponding transmission data. A 2D representation of the data set is also shown (sinogram). Reconstruction based on back projection leads to significant blurring which is avoided by applying a frequency filter on the transmission data (filtered-back projection).

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Fig. 2
Fig. 2 (a) Optical arrangement for crossed-SLITI. Two laser pulses are each sent through a Ronchi grating, after which the beams are spatially overlapped. Their undesired frequency components are then filtered out (frequency cutter). (b) The structured light source is guided through the spray, which is mounted on a rotational stage. The transmitted light is then imaged as it falls onto a screen. (c) An example of a single modulated transmission image with the corresponding SI-image. The dark part in the top left corner is used as a reference to evaluate the unavoidable camera noise level.

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Fig. 3
Fig. 3 Two dimensional Fourier transform of one of the nine raw data images and the final structured illumination image (logarithmic intensity scale). Top row: Two Ronchi gratings oriented 90 degrees relative each other illuminated with one beam of light. Bottom row: Two beams, each (before recombination) illuminating an individual Ronchi grating.

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Fig. 4
Fig. 4 Left graph: The ratio IT /I0 as a function of C (relative concentration). The SI approach gives rise to a single exponential decay. Right graph: Estimation of optical depth as a function of C. In contrast to the SI results, which increase linearly with C, the conventional results tends to follow a second order polynomial. The error bars indicate the standard deviation.

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Fig. 5
Fig. 5 Qualitative differences in image contrast between conventional transmission imaging and structured illumination. The images show the transmitted light through a cuvette with a homogeneous solution of water and milk wherein a solid object was inserted. Scattered light, which is not suppressed in the conventional case, blurs and conceals the object.

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Fig. 6
Fig. 6 3D and 2D images of the 6-hole water spray, obtained using crossed-SLITI tomography. The numbers in the 2D sections indicate the location of the section (origin at nozzle outlet). Notice for instance how the six isolated spray plumes are clearly resolved even near the injector.

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Fig. 7
Fig. 7 3D and 2D images of the transient GDI spray, obtained using crossed-SLITI tomography. Note that only half the spray is shown in the rightmost 3D rendition. The numbers in the 2D sections indicate the location of the section (origin at nozzle outlet). Notice how the hollow region just below the nozzle tip is clearly visible.

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Fig. 8
Fig. 8 Comparison between the acquired OD data (top left image) and an “artificial” OD map extracted from the reconstructed 3D data (top right image). The graphs shows cross sections (see dashed lines) of the OD values from both images.

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Equations (4)

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I S ( x , y ) = 2 3 ⋅ ( I 0 − I 2 π / 3 ) 2 + ( I 0 − I 4 π / 3 ) 2 + ( I 2 π / 3 − I 4 π / 3 ) 2
I C ( x , y ) = I 0 + I 2 π / 3 + I 4 π / 3 3
I S ( x , y ) = 2 9 ⋅ ∑ i = 1 8 ∑ k = i + 1 9 ( I i − I k ) 2
I T = I 0 ⋅ e − σ e ⋅ C ⋅ L = I 0 ⋅ e − μ e ⋅ L = I 0 ⋅ e − O D

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