Abstract

Using computer simulations we examine the ranges of validity of the first Born and first Rytov approximations employed in diffraction tomography. To that end we apply the filtered backpropagation (FBP) algorithm in conjunction with the first Born approximation and the hybrid FBP algorithm in conjunction with the first Rytov approximation. We find that the range of validity of the first Born approximation is approximately 3 times smaller than that of the first Rytov approximation and that the range of validity of each approximation can be expressed in terms of the product of the refractive-index difference between the object and the background and the size of the object. Also, we establish precise criteria for the validity of diffraction tomography within each of these two approximations. For the first Rytov approximation the validity of the hybrid FBP algorithm is found to be limited by phase-unwrapping problems.

© 1998 Optical Society of America

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References

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  1. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).
  2. N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1991), Vol. 18, pp. 401–411.
    [CrossRef]
  3. N. Sponheim, L.-J. Gelius, I. Johansen, J. J. Stamnes, “Quantitative results in ultrasonic tomography of large objects using line sources and curved detector arrays,” IEEE Trans. UFFC 38, 370–379 (1991).
    [CrossRef]
  4. M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
    [CrossRef]
  5. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1987), Chap. 6.
  6. T. C. Wedberg, J. J. Stamnes, “Experimental examination of the quantitative imaging properties of optical diffraction tomography,” J. Opt. Soc. Am. A 12, 493–500 (1995).
    [CrossRef]
  7. M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE 58, 140–141 (1970).
    [CrossRef]
  8. J. B. Keller, “Accuracy and validity of the Born and Rytov approximations,” J. Opt. Soc. Am. 59, 1003–1004 (1969).
  9. W. J. Hadden, D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,” J. Acoust. Soc. Am. 63(5), 1279–1286 (1978).
    [CrossRef]
  10. M. L. Oristaglio, “Accuracy of the Born and Rytov approximations for reflection and refraction at a plane interface,” J. Opt. Soc. Am. A 2, 1987–1993 (1985).
    [CrossRef]
  11. A. J. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. 6, 374–376 (1981).
    [CrossRef]
  12. M. A. Fiddy, “Inversion of optical scattered field data,” J. Phys. D: Appl. Phys. 19, 301–317 (1986).
    [CrossRef]
  13. T. C. Wedberg, J. J. Stamnes, W. Singer, “Comparison of the filtered backpropagation and the filtered backprojection algorithms for quantitative tomography,” Appl. Opt. 34, 6575–6581 (1995).
    [CrossRef]
  14. H. T. Yura, C. C. Sung, S. F. Clifford, R. J. Hill, “Second-order Rytov approximation,” J. Opt. Soc. Am. 73, 500–502 (1983).
    [CrossRef]
  15. W. P. Brown, “Validity of the Rytov approximation in optical propagation calculations,” J. Opt. Soc. Am. 56, 1045–1052 (1966).
    [CrossRef]
  16. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 7.
  17. K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov’s approximation,” Jap. J. Appl. Phys. 14, 1921–1927 (1975).
  18. L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955), Chap. 7.
  19. J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
    [CrossRef]
  20. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 7.
  21. B. Chen, J. J. Stamnes, “Scattering by simple and nonsimple shapes by the combined method of ray tracing and diffraction: application to circular cylinders,” Appl. Opt. 37, 1999–2010 (1998).
    [CrossRef]
  22. D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 194–229.

1998 (1)

1995 (2)

1991 (1)

N. Sponheim, L.-J. Gelius, I. Johansen, J. J. Stamnes, “Quantitative results in ultrasonic tomography of large objects using line sources and curved detector arrays,” IEEE Trans. UFFC 38, 370–379 (1991).
[CrossRef]

1986 (1)

M. A. Fiddy, “Inversion of optical scattered field data,” J. Phys. D: Appl. Phys. 19, 301–317 (1986).
[CrossRef]

1985 (1)

1984 (1)

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

1983 (2)

H. T. Yura, C. C. Sung, S. F. Clifford, R. J. Hill, “Second-order Rytov approximation,” J. Opt. Soc. Am. 73, 500–502 (1983).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

1982 (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).

1981 (1)

1978 (1)

W. J. Hadden, D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,” J. Acoust. Soc. Am. 63(5), 1279–1286 (1978).
[CrossRef]

1975 (1)

K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov’s approximation,” Jap. J. Appl. Phys. 14, 1921–1927 (1975).

1970 (1)

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE 58, 140–141 (1970).
[CrossRef]

1969 (1)

1966 (1)

Brown, W. P.

Chen, B.

Clifford, S. F.

Devaney, A. J.

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).

A. J. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. 6, 374–376 (1981).
[CrossRef]

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1991), Vol. 18, pp. 401–411.
[CrossRef]

Fiddy, M. A.

M. A. Fiddy, “Inversion of optical scattered field data,” J. Phys. D: Appl. Phys. 19, 301–317 (1986).
[CrossRef]

Gelius, L.-J.

N. Sponheim, L.-J. Gelius, I. Johansen, J. J. Stamnes, “Quantitative results in ultrasonic tomography of large objects using line sources and curved detector arrays,” IEEE Trans. UFFC 38, 370–379 (1991).
[CrossRef]

Hadden, W. J.

W. J. Hadden, D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,” J. Acoust. Soc. Am. 63(5), 1279–1286 (1978).
[CrossRef]

Hill, R. J.

Iwata, K.

K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov’s approximation,” Jap. J. Appl. Phys. 14, 1921–1927 (1975).

Johansen, I.

N. Sponheim, L.-J. Gelius, I. Johansen, J. J. Stamnes, “Quantitative results in ultrasonic tomography of large objects using line sources and curved detector arrays,” IEEE Trans. UFFC 38, 370–379 (1991).
[CrossRef]

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1991), Vol. 18, pp. 401–411.
[CrossRef]

Kak, A. C.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1987), Chap. 6.

Keller, J. B.

Larsen, L. E.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Mintzer, D.

W. J. Hadden, D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,” J. Acoust. Soc. Am. 63(5), 1279–1286 (1978).
[CrossRef]

Nagata, R.

K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov’s approximation,” Jap. J. Appl. Phys. 14, 1921–1927 (1975).

Oristaglio, M. L.

Pedersen, H. M.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Robinson, D. W.

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 194–229.

Sancer, M. I.

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE 58, 140–141 (1970).
[CrossRef]

Schiff, L. I.

L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955), Chap. 7.

Singer, W.

Slaney, M.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1987), Chap. 6.

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Sponheim, N.

N. Sponheim, L.-J. Gelius, I. Johansen, J. J. Stamnes, “Quantitative results in ultrasonic tomography of large objects using line sources and curved detector arrays,” IEEE Trans. UFFC 38, 370–379 (1991).
[CrossRef]

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1991), Vol. 18, pp. 401–411.
[CrossRef]

Stamnes, J. J.

B. Chen, J. J. Stamnes, “Scattering by simple and nonsimple shapes by the combined method of ray tracing and diffraction: application to circular cylinders,” Appl. Opt. 37, 1999–2010 (1998).
[CrossRef]

T. C. Wedberg, J. J. Stamnes, “Experimental examination of the quantitative imaging properties of optical diffraction tomography,” J. Opt. Soc. Am. A 12, 493–500 (1995).
[CrossRef]

T. C. Wedberg, J. J. Stamnes, W. Singer, “Comparison of the filtered backpropagation and the filtered backprojection algorithms for quantitative tomography,” Appl. Opt. 34, 6575–6581 (1995).
[CrossRef]

N. Sponheim, L.-J. Gelius, I. Johansen, J. J. Stamnes, “Quantitative results in ultrasonic tomography of large objects using line sources and curved detector arrays,” IEEE Trans. UFFC 38, 370–379 (1991).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 7.

Sung, C. C.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 7.

Varvatsis, A. D.

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE 58, 140–141 (1970).
[CrossRef]

Wedberg, T. C.

Yura, H. T.

Appl. Opt. (2)

IEEE Trans. Microwave Theory Tech. (1)

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

IEEE Trans. UFFC (1)

N. Sponheim, L.-J. Gelius, I. Johansen, J. J. Stamnes, “Quantitative results in ultrasonic tomography of large objects using line sources and curved detector arrays,” IEEE Trans. UFFC 38, 370–379 (1991).
[CrossRef]

J. Acoust. Soc. Am. (1)

W. J. Hadden, D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,” J. Acoust. Soc. Am. 63(5), 1279–1286 (1978).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

J. Phys. D: Appl. Phys. (1)

M. A. Fiddy, “Inversion of optical scattered field data,” J. Phys. D: Appl. Phys. 19, 301–317 (1986).
[CrossRef]

Jap. J. Appl. Phys. (1)

K. Iwata, R. Nagata, “Calculation of refractive index distribution from interferograms using the Born and Rytov’s approximation,” Jap. J. Appl. Phys. 14, 1921–1927 (1975).

Opt. Acta (1)

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE 58, 140–141 (1970).
[CrossRef]

Ultrason. Imag. (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).

Other (6)

N. Sponheim, I. Johansen, A. J. Devaney, “Initial testing of a clinical ultrasound mammograph,” in Acoustical Imaging, H. Lee, G. Wade, eds. (Plenum, New York, 1991), Vol. 18, pp. 401–411.
[CrossRef]

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1987), Chap. 6.

L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955), Chap. 7.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 7.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 7.

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 194–229.

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Figures (8)

Fig. 1
Fig. 1

Comparison between the reconstructed and the original diametrical section of the real part of the refractive index of a circular cylinder of radius a = 24λ. The cylinder and the background had real refractive indices of n = 1.001 and n 0 = 1.000, respectively.

Fig. 2
Fig. 2

Two-dimensional scattering by a circular cylinder. Comparisons of scattered-field intensities obtained in the first Born approximation with corresponding exact results. The refractive index of the background is n 0 = 1.00, and the radius and the refractive index of the cylinder are a and n, respectively. (a) a = 5λ, n = 1.01. (b) a = 5λ, n = 1.02. (c) a = 10λ, n = 1.01.

Fig. 3
Fig. 3

Reconstructions based on the FBP algorithm and the first Born approximation of diametrical sections of circular cylinders with a radius of a = 20λ. The refractive index of the background is n 0 = 1.000. (a) n = 1.0005. (b) n = 1.001. (c) n = 1.004. (d) n = 1.005. (e) n = 1.008. (f) n = 1.012.

Fig. 4
Fig. 4

Reconstructions based on the FBP algorithm and the first Born approximation of circular cylinders with a radius of a = 10λ and refractive index n. The refractive index of the background is n 0 = 1.000. (a) n = 1.008. (b) n = 1.009.

Fig. 5
Fig. 5

Reconstruction results based on the first Born approximation of cylinders with a radius of a = 5λ and refractive index n. The refractive index of the background is n 0 = 1.000. (a) n = 1.016. (b) n = 1.017.

Fig. 6
Fig. 6

Reconstructions based on the hybrid FBP algorithm and the first Rytov approximation of circular cylinders with a radius of a = 20λ and a refractive index n. The refractive index of the background is n 0 = 1.000. (a) n = 1.0005. (b) n = 1.001. (c) n = 1.005. (d) n = 1.012. (e) n = 1.0125.

Fig. 7
Fig. 7

Reconstructions based on the hybrid FBP algorithm and the first Rytov approximation of circular cylinders with a radius of a = 10λ and different refractive indices n. The refractive index of the background is n 0 = 1.000. (a) n = 1.0245. (b) n = 1.025.

Fig. 8
Fig. 8

Reconstructions based on the hybrid FBP algorithm and the first Rytov approximation of circular cylinders with a radius of a = 5λ and different refractive indices n. The refractive index of the background is n 0 = 1.000. (a) n = 1.049. (b) n = 1.050.

Tables (5)

Tables Icon

Table 1 Results of the Error Analyses for Figs. 2(a)2(c)

Tables Icon

Table 2 Results of the Error Analyses for Figs. 3(a)3(f)

Tables Icon

Table 3 Results of the Error Analyses for Figs. 4(a), 4(b), 5(a), and 5(b)

Tables Icon

Table 4 Results of the Error Analyses for Figs. 6(a)6(e)

Tables Icon

Table 5 Results of the Error Analyses of Figs. 7(a), 7(b), 8(a), and 8(b)

Equations (27)

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2 u sc r + k 2 u sc r = k 2 O r u r ,
O r = 1 - n 2 r n 0 2 - 2 δ n r n 0 .
u r = u i r + u sc r .
| u sc r |     | u i r | ,
2 u sc r + k 2 u sc r - 2 k 2 δ n r n 0   u i r .
k   | δ n r | n 0   a     1 .
u B sc r A   k 2 O r u i r G r - r d 2 r ,
ψ = ln   u r ,
ψ i = ln   u i r ,
δ ψ = ψ - ψ i = ln u r u i r .
2 u = 2 u i exp δ ψ = exp δ ψ u i 2 δ ψ + δ ψ 2 + 2 ψ i · δ ψ + 2 u i ,
2 u i δ ψ = u i 2 δ ψ + 2 ψ i · δ ψ + δ ψ 2 u i .
2 + k 2 u i δ ψ = k 2 O - δ ψ 2 u i = - 2 k 2 δ n n 0 + k 2 δ n 2 n 0 2 + δ ψ 2 u i .
2 + k 2 u i r δ ψ R r - 2 k 2 δ n r n 0   u i r ,
| δ n r |     n 0 , | δ ψ R |     | ψ i | = k ,
| δ n r |     n 0 , δ ψ 1 r δ ψ 1 r *     δ ψ 0 r δ ψ 0 r * ,
u B r u i r + u B sc r .
δ ψ = ln 1 + u sc u i u sc u i u B sc u i .
δ ψ R u B sc u i ,
u R r = u i r exp δ ψ R r u i r exp u B sc r u i r u i r 1 + u B r u i r + 1 2 ! u B r u i r 2 + 1 3 ! u B r u i r 3 + .
E % n = 1 N | r | a | n re r - n 0 | 2 n - n 0 2 1 / 2 100 % ,
E % | u B sc | 2 = 1 N j 1 st lobe | u B sc | j 2 - | u E sc | j 2 2 | u E sc | j 4 1 / 2 100 % ,
a δ n p ,
a δ n 0.08 λ ,
a δ n 0.24 λ ,
a δ n 0.08 λ .
a δ n 0.24 λ .

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