Abstract

We present a complete data-processing procedure for quantitative reconstruction of three-dimensional (3D) refractive index fields from limited multidirectional interferometric data. The proposed procedure includes two parts: (1) extraction of the projection data from limited multidirectional interferograms by a regularized phase-tracking technique and wavefront fitting and (2) reconstruction of the 3D refractive index fields by a modified polynomial approximation method. It has been shown that the procedure gives a satisfactory solution to the reconstruction issue in interferometric tomography, from the initial projection data extraction to the final image reconstruction. Computer simulation and experimental results are both presented.

© 2011 Optical Society of America

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  1. R. D. Matulka and D. J. Collins, “Determination of three-dimensional density fields from holographic interferograms,” J. Appl. Phys. 42, 1109–1119 (1971).
    [CrossRef]
  2. D. W. Sweeney and C. M. Vest, “Reconstruction of three-dimensional refractive index fields from multidirectional interferometric data,” Appl. Opt. 12, 2649–2664 (1973).
    [CrossRef] [PubMed]
  3. C. Söller, R. Wenskus, P. Middendorf, G. E. A. Meier, and F. Obermeier, “Interferometric tomography for flow visualization of density fields in supersonic jets and convective flow,” Appl. Opt. 33, 2921–2932 (1994).
    [CrossRef] [PubMed]
  4. T. Kreis, Handbook of Holographic Interferometry—Optical and Digital Methods (Wiley, 2005).
  5. 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]
  6. S. Bahl and J. A. Liburdy, “Three-dimensional image reconstruction using interferometric data from a limited field of view with noise,” Appl. Opt. 30, 4218–4226 (1991).
    [CrossRef] [PubMed]
  7. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1999).
  8. Y. Wei, D. Yan, and S. S. Cha, “Comparison of results of interferometric phase extraction algorithms for three-dimensional flow-field tomography,” Opt. Lasers Eng. 32, 147–155 (1999).
    [CrossRef]
  9. F. Becker and Y. H. Yu, “Digital fringe reduction techniques applied to the measurement of three-dimensional transonic flow fields,” Opt. Eng. 24, 429–434 (1985).
  10. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).
  11. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160(1982).
    [CrossRef]
  12. Y. Song, B. Zhang, and A. He, “Algebraic iterative algorithm for deflection tomography and its application to density flow fields in a hypersonic wind tunnel,” Appl. Opt. 45, 8092–8101(2006).
    [CrossRef] [PubMed]
  13. D. Verhoeven, “Limited-data computed tomography algorithms for the physical sciences,” Appl. Opt. 32, 3736–3754(1993).
    [CrossRef] [PubMed]
  14. S. S. Cha and H. Sun, “Tomography for reconstructing continuous fields from ill-posed multidirectional interferometric data,” Appl. Opt. 29, 251–258 (1990).
    [CrossRef] [PubMed]
  15. R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481(1970).
    [CrossRef] [PubMed]
  16. D. Mishra, J. P. Longtin, R. P. Singh, and V. Prasad, “Performance evaluation of iterative tomography algorithms for incomplete projection data,” Appl. Opt. 43, 1522–1532 (2004).
    [CrossRef] [PubMed]
  17. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18, 689–695 (2001).
    [CrossRef]
  18. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548(1997).
    [CrossRef] [PubMed]
  19. C. Tian, Y. Yang, Y. Zhuo, T. Ling, and H. Li, “Polynomial approximation method for tomographic reconstruction of three-dimensional refractive index fields with limited data,” Opt. Lasers Eng. (to be published).
  20. S. R. Deans, The Radon Transform and Some of Its Applications (Krieger, 1992).
  21. C. Tian, Y. Yang, S. Zhang, D. Liu, Y. Luo, and Y. Zhuo, “Regularized frequency-stabilizing method for single closed-fringe interferogram demodulation,” Opt. Lett. 35, 1837–1839(2010).
    [CrossRef] [PubMed]
  22. C. Tian, Y. Yang, D. Liu, Y. Luo, and Y. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49, 170–179 (2010).
    [CrossRef] [PubMed]
  23. H. X. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49, 564–569 (2011).
    [CrossRef]
  24. J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43–51 (2001).
    [CrossRef]
  25. J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  26. Y. Zhang and G. A. Ruff, “Reconstruction of refractive index fields by using relative fringe data,” Appl. Opt. 32, 2921–2926(1993).
    [CrossRef] [PubMed]
  27. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  28. G. H. Golub, Matrix Computation, 3rd ed. (Johns Hopkins University Press, 1996).

2011

H. X. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49, 564–569 (2011).
[CrossRef]

2010

2006

2004

2001

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18, 689–695 (2001).
[CrossRef]

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43–51 (2001).
[CrossRef]

1999

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]

Y. Wei, D. Yan, and S. S. Cha, “Comparison of results of interferometric phase extraction algorithms for three-dimensional flow-field tomography,” Opt. Lasers Eng. 32, 147–155 (1999).
[CrossRef]

1997

1994

1993

1991

1990

1985

F. Becker and Y. H. Yu, “Digital fringe reduction techniques applied to the measurement of three-dimensional transonic flow fields,” Opt. Eng. 24, 429–434 (1985).

1982

1980

1976

1973

1971

R. D. Matulka and D. J. Collins, “Determination of three-dimensional density fields from holographic interferograms,” J. Appl. Phys. 42, 1109–1119 (1971).
[CrossRef]

1970

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

Bahl, S.

Becker, F.

F. Becker and Y. H. Yu, “Digital fringe reduction techniques applied to the measurement of three-dimensional transonic flow fields,” Opt. Eng. 24, 429–434 (1985).

Bender, R.

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

Cha, S. S.

Y. Wei, D. Yan, and S. S. Cha, “Comparison of results of interferometric phase extraction algorithms for three-dimensional flow-field tomography,” Opt. Lasers Eng. 32, 147–155 (1999).
[CrossRef]

S. S. Cha and H. Sun, “Tomography for reconstructing continuous fields from ill-posed multidirectional interferometric data,” Appl. Opt. 29, 251–258 (1990).
[CrossRef] [PubMed]

Collins, D. J.

R. D. Matulka and D. J. Collins, “Determination of three-dimensional density fields from holographic interferograms,” J. Appl. Phys. 42, 1109–1119 (1971).
[CrossRef]

Cuevas, F. J.

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Krieger, 1992).

Garcia-Botella, A.

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43–51 (2001).
[CrossRef]

Golub, G. H.

G. H. Golub, Matrix Computation, 3rd ed. (Johns Hopkins University Press, 1996).

Gomez-Pedrero, J. Antonio

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43–51 (2001).
[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. Theor. Biol. 29, 471–481(1970).
[CrossRef] [PubMed]

He, A.

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. Theor. Biol. 29, 471–481(1970).
[CrossRef] [PubMed]

Ina, H.

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1999).

Kemao, Q.

H. X. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49, 564–569 (2011).
[CrossRef]

Kobayashi, S.

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry—Optical and Digital Methods (Wiley, 2005).

Li, H.

C. Tian, Y. Yang, Y. Zhuo, T. Ling, and H. Li, “Polynomial approximation method for tomographic reconstruction of three-dimensional refractive index fields with limited data,” Opt. Lasers Eng. (to be published).

Li, K.

H. X. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49, 564–569 (2011).
[CrossRef]

Liburdy, J. A.

Ling, T.

C. Tian, Y. Yang, Y. Zhuo, T. Ling, and H. Li, “Polynomial approximation method for tomographic reconstruction of three-dimensional refractive index fields with limited data,” Opt. Lasers Eng. (to be published).

Liu, D.

Longtin, J. P.

Luo, Y.

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

Marroquin, J. L.

Matulka, R. D.

R. D. Matulka and D. J. Collins, “Determination of three-dimensional density fields from holographic interferograms,” J. Appl. Phys. 42, 1109–1119 (1971).
[CrossRef]

Meier, G. E. A.

Middendorf, P.

Mishra, D.

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]

Noll, R. J.

Obermeier, F.

Prasad, V.

Quiroga, J. A.

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43–51 (2001).
[CrossRef]

Ruff, G. A.

Servin, M.

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]

Silva, D. E.

Singh, R. P.

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1999).

Söller, C.

Song, Y.

Sun, H.

Sweeney, D. W.

Takeda, M.

Tian, C.

Verhoeven, D.

Vest, C. M.

Wang, H. X.

H. X. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49, 564–569 (2011).
[CrossRef]

Wang, J. Y.

Wei, Y.

Y. Wei, D. Yan, and S. S. Cha, “Comparison of results of interferometric phase extraction algorithms for three-dimensional flow-field tomography,” Opt. Lasers Eng. 32, 147–155 (1999).
[CrossRef]

Wenskus, R.

Yan, D.

Y. Wei, D. Yan, and S. S. Cha, “Comparison of results of interferometric phase extraction algorithms for three-dimensional flow-field tomography,” Opt. Lasers Eng. 32, 147–155 (1999).
[CrossRef]

Yang, Y.

Yu, Y. H.

F. Becker and Y. H. Yu, “Digital fringe reduction techniques applied to the measurement of three-dimensional transonic flow fields,” Opt. Eng. 24, 429–434 (1985).

Zhang, B.

Zhang, S.

Zhang, Y.

Zhuo, Y.

Appl. Opt.

D. W. Sweeney and C. M. Vest, “Reconstruction of three-dimensional refractive index fields from multidirectional interferometric data,” Appl. Opt. 12, 2649–2664 (1973).
[CrossRef] [PubMed]

J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
[CrossRef] [PubMed]

S. S. Cha and H. Sun, “Tomography for reconstructing continuous fields from ill-posed multidirectional interferometric data,” Appl. Opt. 29, 251–258 (1990).
[CrossRef] [PubMed]

S. Bahl and J. A. Liburdy, “Three-dimensional image reconstruction using interferometric data from a limited field of view with noise,” Appl. Opt. 30, 4218–4226 (1991).
[CrossRef] [PubMed]

D. Verhoeven, “Limited-data computed tomography algorithms for the physical sciences,” Appl. Opt. 32, 3736–3754(1993).
[CrossRef] [PubMed]

Y. Zhang and G. A. Ruff, “Reconstruction of refractive index fields by using relative fringe data,” Appl. Opt. 32, 2921–2926(1993).
[CrossRef] [PubMed]

C. Söller, R. Wenskus, P. Middendorf, G. E. A. Meier, and F. Obermeier, “Interferometric tomography for flow visualization of density fields in supersonic jets and convective flow,” Appl. Opt. 33, 2921–2932 (1994).
[CrossRef] [PubMed]

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548(1997).
[CrossRef] [PubMed]

D. Mishra, J. P. Longtin, R. P. Singh, and V. Prasad, “Performance evaluation of iterative tomography algorithms for incomplete projection data,” Appl. Opt. 43, 1522–1532 (2004).
[CrossRef] [PubMed]

Y. Song, B. Zhang, and A. He, “Algebraic iterative algorithm for deflection tomography and its application to density flow fields in a hypersonic wind tunnel,” Appl. Opt. 45, 8092–8101(2006).
[CrossRef] [PubMed]

C. Tian, Y. Yang, D. Liu, Y. Luo, and Y. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49, 170–179 (2010).
[CrossRef] [PubMed]

J. Appl. Phys.

R. D. Matulka and D. J. Collins, “Determination of three-dimensional density fields from holographic interferograms,” J. Appl. Phys. 42, 1109–1119 (1971).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Theor. Biol.

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

Opt. Commun.

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43–51 (2001).
[CrossRef]

Opt. Eng.

F. Becker and Y. H. Yu, “Digital fringe reduction techniques applied to the measurement of three-dimensional transonic flow fields,” Opt. Eng. 24, 429–434 (1985).

Opt. Lasers Eng.

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]

Y. Wei, D. Yan, and S. S. Cha, “Comparison of results of interferometric phase extraction algorithms for three-dimensional flow-field tomography,” Opt. Lasers Eng. 32, 147–155 (1999).
[CrossRef]

H. X. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49, 564–569 (2011).
[CrossRef]

Opt. Lett.

Other

G. H. Golub, Matrix Computation, 3rd ed. (Johns Hopkins University Press, 1996).

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1999).

T. Kreis, Handbook of Holographic Interferometry—Optical and Digital Methods (Wiley, 2005).

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

C. Tian, Y. Yang, Y. Zhuo, T. Ling, and H. Li, “Polynomial approximation method for tomographic reconstruction of three-dimensional refractive index fields with limited data,” Opt. Lasers Eng. (to be published).

S. R. Deans, The Radon Transform and Some of Its Applications (Krieger, 1992).

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Figures (7)

Fig. 1
Fig. 1

Incomplete projection of a field f ( x , y , z i ) with an obstruction region.

Fig. 2
Fig. 2

Flow chart of the proposed procedure for interferometric tomography. K and I are the total number of projection views and slices, respectively.

Fig. 3
Fig. 3

Simulated 3D refractive index field and extracted projections by the RPT: (a) original field f ( x , y , z ) ; (b), (c), and (d) projected noisy interferograms I ( u , v ; θ k ) of (a) of views θ 1 = 0 ° , θ 6 = 90 ° and θ 10 = 162 ° ; (e), (f), and (g) extracted absolute projection data g ( u , v ; θ k ) from (b), (c), and (d) by use of the RPT and wavefront fitting techniques, respectively.

Fig. 4
Fig. 4

Reconstruction of the 3D refractive index field by the PAM and ART. First row: extracted multidirectional 1D projection data g ( ρ ; θ k ; z i ) for the sections i = 1 , 128, and 256 (from left to right). Second and third rows: reconstructed 2D fields f ( x , y , z i ) for the same three sections by the PAM and ART, respectively. Fourth row: true fields for the three sections. Last row: reconstructed 3D refractive index fields by the PAM (left) and ART (right), respectively. Note that the images from the second to the fifth rows were all plotted under the same scale.

Fig. 5
Fig. 5

Mach–Zehnder interferometer for flow diagnosis in supersonic wind tunnel.

Fig. 6
Fig. 6

Projection data extraction from an experimental interferogram: (a) upper part of an experimental fringe pattern I ( u , v ; θ ) , (b) masked image of (a), (c) recovered phase ϕ ( u , v ; θ ) of (b) by the RPT, and (d) extracted “absolute” projection data g ( u , v ; θ ) from (c).

Fig. 7
Fig. 7

Reconstruction of the 3D refractive index field f ( x , y , z ) . First row: extracted 1D projection data g ( ρ ; θ ; z i ) for the sections i = 80 , 320, and 550 (from left to right). Second, third, and fourth rows: reconstructed 2D fields f ( x , y , z i ) for the same three sections by the PAM, ART, and FBP, respectively. Fifth row: intermediate profiles of the reconstructed fields of the three sections by the three methods. Last row: reconstructed 3D refractive index fields by the PAM (left), ART (middle), and FBP (right), respectively. Note that the images in the second, third, fourth, and last rows were all plotted under the same scale.

Equations (26)

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I ( u , v ; θ k ) = a ( u , v ; θ k ) + b ( u , v ; θ k ) cos [ 2 π · ϕ ( u , v ; θ k ) / λ ] + n ( u , v ; θ k ) ,
g ( ρ ; θ k ; z i ) = l f ( x , y , z i ) d l ,
U u , v ( ϕ 0 , ω u , ω v ; θ k ) = ( ε , η ) ( N u , v L ) ( { [ I n ( ε , η ; θ k ) cos [ ϕ ( ε , η , u , v ; θ k ) ] } 2 + { [ I n ( ε , η ; θ k ) cos [ ϕ ( ε , η , u , v ; θ k ) + α ] } 2 + β [ ϕ 0 ( ε , η ; θ k ) ϕ ( ε , η , u , v ; θ k ) ] 2 m ( ε , η ; θ k ) ) ,
I n ( u , v ; θ k ) cos [ ϕ ( u , v ; θ k ) ] ,
ϕ ( ε , η , u , v ; θ k ) = ϕ 0 ( u , v ; θ k ) + [ ω u ( u , v ; θ k ) ( ε u ) + ω v ( u , v ; θ k ) ( η v ) ] ,
s k + 1 = s k τ s U u , v ,
s U u , v = [ U u , v ϕ 0 , U u , v ω u , U u , v ω v ] .
ϕ ( u , v ; θ k ) = g ( u , v ; θ k ) + A + B u + C v ,
Δ = u , v L [ ϕ ( u , v ; θ k ) A B u C v ] 2 .
A + B u + C v ϕ ( u , v ; θ k ) .
Z a = ϕ .
a = Z + Φ
f ( x , y , z i ) f ˜ ( x , y ) = j = 1 N ψ j ( x , y ) c j ,
j = 1 N Ψ j ( ρ ; θ k ; z i ) c j = g ( ρ ; θ k ; z i ) ,
ψ even j = [ 2 ( n + 1 ) ] 1 / 2 R n m ( ρ ) cos m θ ψ odd j = [ 2 ( n + 1 ) ] 1 / 2 R n m ( ρ ) sin m θ } m 0 ψ j = [ ( n + 1 ) ] 1 / 2 R n m ( ρ ) m = 0 } ,
R n m ( ρ ) = s = 0 ( n m ) / 2 ( 1 ) s ( n s ) ! s ! [ ( n + m ) / 2 s ] ! [ ( n m ) / 2 s ] ! ρ n 2 s ,
Ψ c = g ,
Ψ = [ Ψ 1 , 1 Ψ 1 , 1 Ψ 1 , N Ψ 2 , 1 Ψ 2 , 1 Ψ 2 , N Ψ M , 1 Ψ M , 1 Ψ M , N ] ,
Ψ = U Σ V T ,
c = V Σ + U T g ,
σ i + = { 1 / σ i , when     σ i > τ σ max 0 , else ,
ξ r = ψ r + s = 1 r 1 D r s ξ s ,
D r s = L ψ r ξ s / L ξ s 2 .
f ( x , y , z ) = 0.05 × { exp [ ( x + 0.8 ) 2 ( y 0.8 ) 2 ] + 0.6 exp [ 1.8 ( x + 1 ) 2 1.8 ( y + 1.5 ) 2 ] + 0.8 exp [ 1.5 ( x 1 ) 2 1.5 y 2 ] } × mask ( x , y , z ) ,
mask ( x , y , z ) = { 1 , if     [ ( x 2 + y 2 ) / 0.3 ] 1 / 2 + z 0 and ( x 2 + y 2 ) 1 / 2 3 nan , else ,
f a ( x , y , z ) = λ f ( x , y , z ) / d ,

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