Abstract

Interferometric tomography can reconstruct 3D refractive-index distributions through phase-shift measurements for different beam angles. To reconstruct a complex refractive-index distribution, many projections along different directions are required. For the purpose of increasing the number of the projections, we earlier proposed a beam-angle-controllable interferometer with mechanical stages; however, the quality of reconstructed distribution by conventional algorithms was poor because the background fringes cannot be precisely controlled. To improve the quality, we propose a weighted reconstruction algorithm that can consider projection errors. We demonstrate the validity of the weighted reconstruction through simulations and a reconstruction from experimental data for three candle flames.

© 2017 Optical Society of America

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References

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2016 (1)

A. Iriarte, R. Marabini, S. Matej, C. O. S. Sorzano, and R. M. Lewitt, “System models for pet statistical iterative reconstruction: a review,” Comput. Med. Imaging Graph. 48, 30–48 (2016).
[Crossref]

2015 (1)

S. Tomioka, S. Nishiyama, S. Heshmat, Y. Hashimoto, and K. Kurita, “Three-dimensional gas temperature measurements by computed tomography with incident angle variable interferometer,” Proc. SPIE 9401, 94010J (2015).
[Crossref]

2014 (2)

S. Tomioka, S. Nishiyama, and S. Heshmat, “Carrier peak isolation from single interferogram using spectrum shift technique,” Appl. Opt. 53, 5620–5631 (2014).
[Crossref]

S. Heshmat, S. Tomioka, and S. Nishiyama, “Performance evaluation of phase unwrapping algorithms for noisy phase measurements,” Int. J. Optomechatron. 8, 260–274 (2014).
[Crossref]

2012 (2)

S. Tomioka and S. Nishiyama, “Phase unwrapping for noisy phase map using localized compensator,” Appl. Opt. 51, 4984–4994 (2012).
[Crossref]

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in x-ray ct,” Physica Medica 28, 94–108 (2012).
[Crossref]

2011 (1)

2010 (1)

2007 (1)

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45, 3240–3251 (2007).
[Crossref]

2003 (1)

D. Naylor, “Recent developments in the measurement of convective heat transfer rates by laser interferometry,” Int. J. Heat Fluid Flow 24, 345–355 (2003).
[Crossref]

2002 (1)

1999 (1)

C. Shakher and A. K. Nirala, “A review on refractive index and temperature profile measurements using laser-based interferometric techniques,” Opt. Lasers Eng. 31, 455–491 (1999).
[Crossref]

1998 (1)

1997 (3)

T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692–2701 (1997).
[Crossref]

J. M. Anderson, B. A. Mair, M. Rao, and C.-H. Wu, “Weighted least-squares reconstruction methods for positron emission tomography,” IEEE Trans. Med. Imaging 16, 159–165 (1997).
[Crossref]

H. El-Ghandoor, “Tomographic investigation of the refractive index profiling using speckle photography technique,” Opt. Commun. 133, 33–38 (1997).
[Crossref]

1994 (3)

H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13, 601–609 (1994).
[Crossref]

J. A. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imaging 13, 290–300 (1994).
[Crossref]

D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
[Crossref]

1989 (1)

G. N. Blinkov, N. A. Fomin, M. N. Rolin, R. I. Soloukhin, D. E. Vitkin, and N. L. Yadrevskaya, “Speckle tomography of a gas flame,” Exp. Fluids 8, 72–76 (1989).
[Crossref]

1988 (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

1987 (1)

1986 (1)

1985 (1)

H. M. Hertz, “Experimental determination of 2-d flame temperature fields by interferometric tomography,” Opt. Commun. 54, 131–136 (1985).
[Crossref]

1984 (2)

K. Lange and R. Carson, “Em reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr. 8, 306–316 (1984).

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (sart): a superior implementation of the art algorithm,” Ultrason. Imag. 6, 81–94 (1984).
[Crossref]

1982 (1)

L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[Crossref]

1975 (1)

L. Pera and B. Gebhart, “Laminar plume interactions,” J. Fluid Mech. 68, 259–271 (1975).
[Crossref]

1974 (1)

L. A. Shepp and B. F. Logan, “The fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. 21, 21–43 (1974).
[Crossref]

1972 (1)

P. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,” J. Theoret. Biol. 36, 105–117 (1972).
[Crossref]

1970 (1)

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (art) for three-dimensional electron microscopy and x-ray photography,” J. Theoret. Biol. 29, 471–481 (1970).
[Crossref]

1964 (1)

A. Savitzky and M. J. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[Crossref]

Andersen, A. H.

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (sart): a superior implementation of the art algorithm,” Ultrason. Imag. 6, 81–94 (1984).
[Crossref]

Anderson, J. M.

J. M. Anderson, B. A. Mair, M. Rao, and C.-H. Wu, “Weighted least-squares reconstruction methods for positron emission tomography,” IEEE Trans. Med. Imaging 16, 159–165 (1997).
[Crossref]

Bachor, H.-A.

Beister, M.

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in x-ray ct,” Physica Medica 28, 94–108 (2012).
[Crossref]

Bender, R.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (art) for three-dimensional electron microscopy and x-ray photography,” J. Theoret. Biol. 29, 471–481 (1970).
[Crossref]

Blinkov, G. N.

G. N. Blinkov, N. A. Fomin, M. N. Rolin, R. I. Soloukhin, D. E. Vitkin, and N. L. Yadrevskaya, “Speckle tomography of a gas flame,” Exp. Fluids 8, 72–76 (1989).
[Crossref]

Bone, D. J.

Burton, D. R.

Byer, R. L.

Carson, R.

K. Lange and R. Carson, “Em reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr. 8, 306–316 (1984).

Cha, S. S.

El-Ghandoor, H.

H. El-Ghandoor, “Tomographic investigation of the refractive index profiling using speckle photography technique,” Opt. Commun. 133, 33–38 (1997).
[Crossref]

Faris, G. W.

Fessler, J. A.

J. A. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imaging 13, 290–300 (1994).
[Crossref]

Flynn, T. J.

Fomin, N. A.

G. N. Blinkov, N. A. Fomin, M. N. Rolin, R. I. Soloukhin, D. E. Vitkin, and N. L. Yadrevskaya, “Speckle tomography of a gas flame,” Exp. Fluids 8, 72–76 (1989).
[Crossref]

Gdeisat, M. A.

Gebhart, B.

L. Pera and B. Gebhart, “Laminar plume interactions,” J. Fluid Mech. 68, 259–271 (1975).
[Crossref]

Gerzen, T.

T. Gerzen and D. Minkwitz, “Simultaneous multiplicative column normalized method (smart) for the 3d ionosphere tomography in comparison with other algebraic methods,” in Annales Geophysicae (European Geosciences Union, 2016), Vol. 34, pp. 97–115.

Ghiglia, D. C.

Gilbert, P.

P. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,” J. Theoret. Biol. 36, 105–117 (1972).
[Crossref]

Golay, M. J.

A. Savitzky and M. J. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[Crossref]

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Gordon, R.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (art) for three-dimensional electron microscopy and x-ray photography,” J. Theoret. Biol. 29, 471–481 (1970).
[Crossref]

Hashimoto, Y.

S. Tomioka, S. Nishiyama, S. Heshmat, Y. Hashimoto, and K. Kurita, “Three-dimensional gas temperature measurements by computed tomography with incident angle variable interferometer,” Proc. SPIE 9401, 94010J (2015).
[Crossref]

Herman, G. T.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (art) for three-dimensional electron microscopy and x-ray photography,” J. Theoret. Biol. 29, 471–481 (1970).
[Crossref]

Herráez, M. A.

Hertz, H. M.

H. M. Hertz, “Experimental determination of 2-d flame temperature fields by interferometric tomography,” Opt. Commun. 54, 131–136 (1985).
[Crossref]

Heshmat, S.

S. Tomioka, S. Nishiyama, S. Heshmat, Y. Hashimoto, and K. Kurita, “Three-dimensional gas temperature measurements by computed tomography with incident angle variable interferometer,” Proc. SPIE 9401, 94010J (2015).
[Crossref]

S. Heshmat, S. Tomioka, and S. Nishiyama, “Performance evaluation of phase unwrapping algorithms for noisy phase measurements,” Int. J. Optomechatron. 8, 260–274 (2014).
[Crossref]

S. Tomioka, S. Nishiyama, and S. Heshmat, “Carrier peak isolation from single interferogram using spectrum shift technique,” Appl. Opt. 53, 5620–5631 (2014).
[Crossref]

S. Heshmat, S. Tomioka, and S. Nishiyama, “Reliable phase unwrapping algorithm based on rotational and direct compensators,” Appl. Opt. 50, 6225–6233 (2011).
[Crossref]

S. Tomioka, S. Heshmat, N. Miyamoto, and S. Nishiyama, “Phase unwrapping for noisy phase maps using rotational compensator with virtual singular points,” Appl. Opt. 49, 4735–4745 (2010).
[Crossref]

Hirose, A.

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45, 3240–3251 (2007).
[Crossref]

Hudson, H. M.

H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13, 601–609 (1994).
[Crossref]

Iriarte, A.

A. Iriarte, R. Marabini, S. Matej, C. O. S. Sorzano, and R. M. Lewitt, “System models for pet statistical iterative reconstruction: a review,” Comput. Med. Imaging Graph. 48, 30–48 (2016).
[Crossref]

Kak, A. C.

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (sart): a superior implementation of the art algorithm,” Ultrason. Imag. 6, 81–94 (1984).
[Crossref]

Kalender, W. A.

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in x-ray ct,” Physica Medica 28, 94–108 (2012).
[Crossref]

Kolditz, D.

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in x-ray ct,” Physica Medica 28, 94–108 (2012).
[Crossref]

Kurita, K.

S. Tomioka, S. Nishiyama, S. Heshmat, Y. Hashimoto, and K. Kurita, “Three-dimensional gas temperature measurements by computed tomography with incident angle variable interferometer,” Proc. SPIE 9401, 94010J (2015).
[Crossref]

Lalor, M. J.

Lange, K.

K. Lange and R. Carson, “Em reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr. 8, 306–316 (1984).

Larkin, R. S.

H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13, 601–609 (1994).
[Crossref]

Lewitt, R. M.

A. Iriarte, R. Marabini, S. Matej, C. O. S. Sorzano, and R. M. Lewitt, “System models for pet statistical iterative reconstruction: a review,” Comput. Med. Imaging Graph. 48, 30–48 (2016).
[Crossref]

Logan, B. F.

L. A. Shepp and B. F. Logan, “The fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. 21, 21–43 (1974).
[Crossref]

Mair, B. A.

J. M. Anderson, B. A. Mair, M. Rao, and C.-H. Wu, “Weighted least-squares reconstruction methods for positron emission tomography,” IEEE Trans. Med. Imaging 16, 159–165 (1997).
[Crossref]

Marabini, R.

A. Iriarte, R. Marabini, S. Matej, C. O. S. Sorzano, and R. M. Lewitt, “System models for pet statistical iterative reconstruction: a review,” Comput. Med. Imaging Graph. 48, 30–48 (2016).
[Crossref]

Matej, S.

A. Iriarte, R. Marabini, S. Matej, C. O. S. Sorzano, and R. M. Lewitt, “System models for pet statistical iterative reconstruction: a review,” Comput. Med. Imaging Graph. 48, 30–48 (2016).
[Crossref]

Minkwitz, D.

T. Gerzen and D. Minkwitz, “Simultaneous multiplicative column normalized method (smart) for the 3d ionosphere tomography in comparison with other algebraic methods,” in Annales Geophysicae (European Geosciences Union, 2016), Vol. 34, pp. 97–115.

Miyamoto, N.

Naylor, D.

D. Naylor, “Recent developments in the measurement of convective heat transfer rates by laser interferometry,” Int. J. Heat Fluid Flow 24, 345–355 (2003).
[Crossref]

Nirala, A. K.

C. Shakher and A. K. Nirala, “A review on refractive index and temperature profile measurements using laser-based interferometric techniques,” Opt. Lasers Eng. 31, 455–491 (1999).
[Crossref]

Nishiyama, S.

Pera, L.

L. Pera and B. Gebhart, “Laminar plume interactions,” J. Fluid Mech. 68, 259–271 (1975).
[Crossref]

Rao, M.

J. M. Anderson, B. A. Mair, M. Rao, and C.-H. Wu, “Weighted least-squares reconstruction methods for positron emission tomography,” IEEE Trans. Med. Imaging 16, 159–165 (1997).
[Crossref]

Rolin, M. N.

G. N. Blinkov, N. A. Fomin, M. N. Rolin, R. I. Soloukhin, D. E. Vitkin, and N. L. Yadrevskaya, “Speckle tomography of a gas flame,” Exp. Fluids 8, 72–76 (1989).
[Crossref]

Romero, L. A.

Sandeman, R. J.

Savitzky, A.

A. Savitzky and M. J. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[Crossref]

Shakher, C.

C. Shakher and A. K. Nirala, “A review on refractive index and temperature profile measurements using laser-based interferometric techniques,” Opt. Lasers Eng. 31, 455–491 (1999).
[Crossref]

Shepp, L. A.

L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[Crossref]

L. A. Shepp and B. F. Logan, “The fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. 21, 21–43 (1974).
[Crossref]

Soloukhin, R. I.

G. N. Blinkov, N. A. Fomin, M. N. Rolin, R. I. Soloukhin, D. E. Vitkin, and N. L. Yadrevskaya, “Speckle tomography of a gas flame,” Exp. Fluids 8, 72–76 (1989).
[Crossref]

Sorzano, C. O. S.

A. Iriarte, R. Marabini, S. Matej, C. O. S. Sorzano, and R. M. Lewitt, “System models for pet statistical iterative reconstruction: a review,” Comput. Med. Imaging Graph. 48, 30–48 (2016).
[Crossref]

Tomioka, S.

Vardi, Y.

L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[Crossref]

Vitkin, D. E.

G. N. Blinkov, N. A. Fomin, M. N. Rolin, R. I. Soloukhin, D. E. Vitkin, and N. L. Yadrevskaya, “Speckle tomography of a gas flame,” Exp. Fluids 8, 72–76 (1989).
[Crossref]

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Wu, C.-H.

J. M. Anderson, B. A. Mair, M. Rao, and C.-H. Wu, “Weighted least-squares reconstruction methods for positron emission tomography,” IEEE Trans. Med. Imaging 16, 159–165 (1997).
[Crossref]

Yadrevskaya, N. L.

G. N. Blinkov, N. A. Fomin, M. N. Rolin, R. I. Soloukhin, D. E. Vitkin, and N. L. Yadrevskaya, “Speckle tomography of a gas flame,” Exp. Fluids 8, 72–76 (1989).
[Crossref]

Yamaki, R.

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45, 3240–3251 (2007).
[Crossref]

Yan, D.

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Anal. Chem. (1)

A. Savitzky and M. J. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[Crossref]

Appl. Opt. (7)

Comput. Med. Imaging Graph. (1)

A. Iriarte, R. Marabini, S. Matej, C. O. S. Sorzano, and R. M. Lewitt, “System models for pet statistical iterative reconstruction: a review,” Comput. Med. Imaging Graph. 48, 30–48 (2016).
[Crossref]

Exp. Fluids (1)

G. N. Blinkov, N. A. Fomin, M. N. Rolin, R. I. Soloukhin, D. E. Vitkin, and N. L. Yadrevskaya, “Speckle tomography of a gas flame,” Exp. Fluids 8, 72–76 (1989).
[Crossref]

IEEE Trans. Geosci. Remote Sens. (1)

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45, 3240–3251 (2007).
[Crossref]

IEEE Trans. Med. Imaging (4)

L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[Crossref]

H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13, 601–609 (1994).
[Crossref]

J. A. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imaging 13, 290–300 (1994).
[Crossref]

J. M. Anderson, B. A. Mair, M. Rao, and C.-H. Wu, “Weighted least-squares reconstruction methods for positron emission tomography,” IEEE Trans. Med. Imaging 16, 159–165 (1997).
[Crossref]

IEEE Trans. Nucl. Sci. (1)

L. A. Shepp and B. F. Logan, “The fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. 21, 21–43 (1974).
[Crossref]

Int. J. Heat Fluid Flow (1)

D. Naylor, “Recent developments in the measurement of convective heat transfer rates by laser interferometry,” Int. J. Heat Fluid Flow 24, 345–355 (2003).
[Crossref]

Int. J. Optomechatron. (1)

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Figures (9)

Fig. 1.
Fig. 1.

Coordinate system and contribution from internal pixel to projection at a slice corresponding to z = const .

Fig. 2.
Fig. 2.

Improvement of the reconstructed density via exclusion of an incorrect projection. (a) Projections for a square-shaped object along eight directions, where arrows denote the incident directions and the acquired projection data are indicated by the rows of pixels at the end of the arrows. The object for θ 4 is replaced by a rectangle instead of the square, i.e., p k , 4 M is incorrect. (b) Sinogram, p k , l M , and reconstructed density, f i , j ( n ) , at n = 100 when p k , 4 M denoted by the arrow is included. (c) Same as (b) but with p k , 4 M excluded.

Fig. 3.
Fig. 3.

Simulation for limited projections: (a) True density with three Gaussian functions. (b)–(d) Conditions of the incident beams (top radial chart), and reconstructed densities at n = 10 (middle), and those at n = 50 (bottom). The projection beam angles, θ l , in (b) correspond to θ l [ 30 ° , 30 ° ] with 2° intervals. In both (c) and (d), a horizontal projection (90°) is added to the projections in (b). The weights, W l , for (b) and (c) are uniform, while W l ( 90 ° ) = 5 W l ( 90 ° ) in (d).

Fig. 4.
Fig. 4.

Comparison of moving average for projection data with speckle noise. The true density and the condition of the projection beams are the same as those corresponding to Figs. 3(a) and 3(d), respectively. The speckle noise included in p k l M obeys the normal distribution, with the standard deviation being 10% of the maximum of the true projection data.

Fig. 5.
Fig. 5.

Comparison of unweighted and weighted reconstructions for projection data with line error. The true density and the conditions of the projection beams are identical to the corresponding ones of Figs. 3(a) and 3(d), respectively. The random number, α l , to determine p k l M in Eq. (15), obeys the normal distribution with a standard deviation of 1. Parameter σ W in Eq. (16) is determined as max { p k l True } / σ W = 0.1 .

Fig. 6.
Fig. 6.

Experimental arrangement. FM1-5, full-mirrors; HM1-5: half-mirrors; X1 and X2, linear stages; LS, He–Ne laser (wavelength: 633 nm); L1 and L2, lenses (expanded beam diameter: 50 mm); OBJ, object; S , half-transparent screen. FM1, FM2, HM1, and HM2 are mounted on rotatable stages, which in turn are mounted on X1 and X2.

Fig. 7.
Fig. 7.

Process of phase evaluation from an interferogram. (a) Observed interferogram ( θ = 20.0 ° ). (b) Modified interferogram obtained by both normalization and the Wiener filter. (c)–(f) Spectral intensities in the spectral domain limit k [ k max / 2 , + k max / 2 ] , where k max denotes the Nyquist frequency; I ^ twin ( k ) = I ^ 1 ϕ ( k k ) + [ I ^ 1 ϕ ( k κ ) ] * in (e). (g) Wrapped phase with range [ π , + π ] . (h) Grouped tree of singular points to determine compensated local area for phase unwrapping. (i) Unwrapped phase with contour lines with interval 2 π . (j) Wrapped difference between rewrapped phase of (i) and wrapped phase in (g). (k) Additional phase shift to satisfy equi-phase condition on both sides of the interferogram. (l) Corrected unwrapped phase. Image sizes of (a) and (b) are 128 × 128 pixels, and those of (g)–(l) are 512 × 512 pixels.

Fig. 8.
Fig. 8.

Reconstructed density from experimental data with different operators for a slice ( z = 10    mm ). (a) Object with three different-sized flames on a candle. (b) Frequency histogram of SD l [ ε k l ] and weight function, W l , for weighted reconstruction. (c) Sinogram, p k l ; the horizontal and vertical axes represent k and l , respectively. (d) Weight, W l ; the horizontal axis represents W l within [0,1] and the vertical axis is the same as that in (c). (e) Reconstructed densities obtained with application of different operators in the backward-projection procedure; WGT denotes weighted projection and M.A. moving average. Intervals between contour lines are 2 × 10 5 for Δ n .

Fig. 9.
Fig. 9.

Reconstructed 3D temperature distribution of gas around candle flames.

Equations (34)

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p k l ( n ) = F k l { f i j ( n ) } ,
f i j ( n + 1 ) = B i j { p k l M p k l ( n ) } f i j ( n ) ,
p k l = 1 Δ s ξ k Δ s / 2 ξ k + Δ s / 2 η l f ( x ( ξ , η ) , y ( ξ , η ) ) d η d ξ .
F k l { f i j } = i , j c i j k l f i j .
A = k C k l , C k l = i , j c i j k l , L k l = C k l Δ s , Δ s 2 = k c i j k l ,
B i j { p k l } = 1 N l Δ s 2 k , l c i j k l p k l L k l ,
B i j { p ^ k l } = B i j { min ( p ^ k l , p ^ lim ) } ,
f i j ( n + 1 ) = B i j { p k l M p k l ( n ) } f i j ( n ) ,
f i j ( n + 1 ) = M { B i j { W l p k l M p k l ( n ) } f i j ( n ) } ,
l W l = 1 .
p k l M = p k l True + N k l , p k l True = F k l { f i j True } ,
Δ ^ { g p q ( n ) , ref } = | | g p q ( n ) g p q ref | | 2 | | g p q ref | | 2 , | | g p q | | 2 = g p q 2 p q ,
p k l M = p k l True + p k l δ .
f i j ( ) = F i j 1 { p k l True } + F i j 1 { p k l δ } = f i j True + F i j 1 { p k l δ } ,
p k l M = ( 1 + α l ) p k l True .
W l exp ( ( p k l M p k l True ) 2 k / σ W 2 ) ,
I ( r ) = I 0 ( r ) + 2 I 1 ( r ) cos ( Δ ϕ ( r ) ) ,
Δ ϕ ( r ) = ϕ obj ( r ) ϕ ref ( r ) ,
ϕ obj ( r ) = η k 0 obj n obj ( r ) · d r ,
ϕ ref ( r ) = η k 0 ref n ref · d r ,
Δ ϕ ( r ) = ϕ ( r ) + κ · r ,
ϕ ( r ) = η k 0 obj Δ n ( r ) · d r = k 0 η Δ n ( r ) d η ,
Δ n ( r ) = n obj ( r ) n ref ,
κ = ( k 0 obj k 0 ref ) n ref .
I ^ ( k ) = I ^ 0 ( k ) + I ^ 1 ϕ ( k κ ) + [ I ^ 1 ϕ ( k κ ) ] * ,
I ^ 1 ϕ ( k ) = F { I 1 ( r ) exp ( j ϕ ( r ) ) } ,
ϕ w ( r ) = W { ϕ ( r ) } = arg ( I 1 ( r ) exp ( j ϕ ( r ) ) ) ,
ϕ ( r ) = U { ϕ w ( r ) } .
ε ( r ) = W { ϕ ( r ) ϕ w ( r ) } .
W l ( θ 90 ° ) = exp ( SD l [ ε k l ] 2 σ W 2 ) ,
W l ( θ = 90 ° ) = W Fix ,
W l = W l / l W l ,
Δ n i j = f i j k 0 Δ s .
T = n 0 1 n 1 T 0 , n = n 0 + Δ n < n 0 ,