Abstract

A recently developed inverse scattering algorithm [ A. J. Devaney and M. Dennison, Inverse Probl. , 19, 855 (2003) and M. Dennison and A. J. Devaney, Inverse Probl. , 20, 1307 (2004) ] is described and applied in a computer simulation study of optical diffraction tomography (ODT). The new algorithm is superior to standard ODT reconstruction algorithms, such as the filtered backpropagation algorithm, in applications employing a limited number of scattering experiments (the so-called limited-view case) and also in cases where multiple scattering occurs between the object being interrogated and the (known) background in which the object is embedded. The new algorithm is compared and contrasted with the filtered backpropagation algorithm in a computer simulation of ODT of weakly inhomogeneous cylindrical objects being interrogated in a limited number of scattering experiments employing incident plane waves. Our study has potential applications in biomedical imaging and tomographic microscopy.

© 2005 Optical Society of America

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References

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  1. A. J. Devaney, M. Maleki, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
    [CrossRef]
  2. 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]
  3. T. C. Wedberg, W. C. Wedberg, “Tomographic reconstruction of the cross-sectional complex refractive index of semitransparent, birefringent fibers,” J. Microsc. 177, 53–67 (1995).
    [CrossRef]
  4. M. Maleki, A. J. Devaney, “Phase retrieval and intensity-only reconstruction algorithm for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086–1092 (1993).
    [CrossRef]
  5. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
    [CrossRef] [PubMed]
  6. A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. GE- 22, 3–12 (1984).
    [CrossRef]
  7. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).
  8. A. J. Devaney, M. Dennison, “Inverse scattering in inhomogeneous background media,” Inverse Probl. 19, 855–870 (2003).
    [CrossRef]
  9. M. Dennison, A. J. Devaney, “Inverse scattering in inhomogeneous background media: II. Multi-frequency case and SVD formulation,” Inverse Probl. 20, 1307–1324 (2004).
    [CrossRef]
  10. J. H. Taylor, Scattering Theory (Wiley, 1972).
  11. T. J. Hall, A. M. Darling, M. A. Fiddy, “Image compression and restoration incorporating prior knowledge,” Opt. Lett. 7, 467–468 (1982).
    [CrossRef] [PubMed]
  12. C. L. Byme, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, A. M. Darling, “Image restoration and resolution enhancement,” J. Opt. Soc. Am. 73, 1481–1487 (1983).
    [CrossRef]
  13. P. Guo, A. J. Devaney, “Digital microscopy using phase-shifting digital holography with two reference waves,” Opt. Lett. 29, 857–859 (2004).
    [CrossRef] [PubMed]
  14. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction phase pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  15. R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. 66, 961–964 (1976).
    [CrossRef]
  16. A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. 30, 377–386 (1983).
    [CrossRef] [PubMed]
  17. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
    [CrossRef]

2004 (2)

M. Dennison, A. J. Devaney, “Inverse scattering in inhomogeneous background media: II. Multi-frequency case and SVD formulation,” Inverse Probl. 20, 1307–1324 (2004).
[CrossRef]

P. Guo, A. J. Devaney, “Digital microscopy using phase-shifting digital holography with two reference waves,” Opt. Lett. 29, 857–859 (2004).
[CrossRef] [PubMed]

2003 (1)

A. J. Devaney, M. Dennison, “Inverse scattering in inhomogeneous background media,” Inverse Probl. 19, 855–870 (2003).
[CrossRef]

1995 (2)

T. C. Wedberg, W. C. Wedberg, “Tomographic reconstruction of the cross-sectional complex refractive index of semitransparent, birefringent fibers,” J. Microsc. 177, 53–67 (1995).
[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]

1993 (1)

1992 (1)

1984 (1)

A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. GE- 22, 3–12 (1984).
[CrossRef]

1983 (2)

1982 (2)

T. J. Hall, A. M. Darling, M. A. Fiddy, “Image compression and restoration incorporating prior knowledge,” Opt. Lett. 7, 467–468 (1982).
[CrossRef] [PubMed]

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[CrossRef] [PubMed]

1976 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction phase pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

Byme, C. L.

Darling, A. M.

Dennison, M.

M. Dennison, A. J. Devaney, “Inverse scattering in inhomogeneous background media: II. Multi-frequency case and SVD formulation,” Inverse Probl. 20, 1307–1324 (2004).
[CrossRef]

A. J. Devaney, M. Dennison, “Inverse scattering in inhomogeneous background media,” Inverse Probl. 19, 855–870 (2003).
[CrossRef]

Devaney, A. J.

M. Dennison, A. J. Devaney, “Inverse scattering in inhomogeneous background media: II. Multi-frequency case and SVD formulation,” Inverse Probl. 20, 1307–1324 (2004).
[CrossRef]

P. Guo, A. J. Devaney, “Digital microscopy using phase-shifting digital holography with two reference waves,” Opt. Lett. 29, 857–859 (2004).
[CrossRef] [PubMed]

A. J. Devaney, M. Dennison, “Inverse scattering in inhomogeneous background media,” Inverse Probl. 19, 855–870 (2003).
[CrossRef]

M. Maleki, A. J. Devaney, “Phase retrieval and intensity-only reconstruction algorithm for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086–1092 (1993).
[CrossRef]

A. J. Devaney, M. Maleki, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
[CrossRef]

A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. GE- 22, 3–12 (1984).
[CrossRef]

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. 30, 377–386 (1983).
[CrossRef] [PubMed]

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[CrossRef] [PubMed]

Fiddy, M. A.

Fitzgerald, R. M.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction phase pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Gonsalves, R. A.

Guo, P.

Hall, T. J.

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Maleki, M.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction phase pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Schatzberg, A.

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Stamnes, J. J.

Taylor, J. H.

J. H. Taylor, Scattering Theory (Wiley, 1972).

Wedberg, T. C.

T. C. Wedberg, W. C. Wedberg, “Tomographic reconstruction of the cross-sectional complex refractive index of semitransparent, birefringent fibers,” J. Microsc. 177, 53–67 (1995).
[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]

Wedberg, W. C.

T. C. Wedberg, W. C. Wedberg, “Tomographic reconstruction of the cross-sectional complex refractive index of semitransparent, birefringent fibers,” J. Microsc. 177, 53–67 (1995).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. 30, 377–386 (1983).
[CrossRef] [PubMed]

IEEE Trans. Geosci. Remote Sens. (1)

A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. GE- 22, 3–12 (1984).
[CrossRef]

Inverse Probl. (2)

A. J. Devaney, M. Dennison, “Inverse scattering in inhomogeneous background media,” Inverse Probl. 19, 855–870 (2003).
[CrossRef]

M. Dennison, A. J. Devaney, “Inverse scattering in inhomogeneous background media: II. Multi-frequency case and SVD formulation,” Inverse Probl. 20, 1307–1324 (2004).
[CrossRef]

J. Microsc. (1)

T. C. Wedberg, W. C. Wedberg, “Tomographic reconstruction of the cross-sectional complex refractive index of semitransparent, birefringent fibers,” J. Microsc. 177, 53–67 (1995).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Lett. (2)

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction phase pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Ultrason. Imaging (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[CrossRef] [PubMed]

Other (3)

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

J. H. Taylor, Scattering Theory (Wiley, 1972).

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

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Figures (10)

Fig. 1
Fig. 1

Optically semitransparent object is mounted in such a way that it can achieve varying orientations relative to the direction of incident light.

Fig. 2
Fig. 2

Mathematic model of DT.

Fig. 3
Fig. 3

Cross-sectional images of an optical fiber’s refractive-index distribution. (a) Actual cross-sectional refractive index of an optical fiber. (b) Reconstruction of the refractive index of the optical fiber using the new algorithm. (c) Reconstruction of the refractive index of the optical fiber using the FBP algorithm. Pixel size in (a)–(c) is δ x = λ = 633 nm . The following parameters have been used for reconstruction: measurement distance l 0 40 mm , N s = 16 views , N ξ = 41 CCD pixels per view, and distance between adjacent CCD pixels d = 26.8 μ m .

Fig. 4
Fig. 4

(a) Comparison of the scattered data (solid curve) generated by the actual object (optical fiber) and the scattered data (dot curve) generated by the reconstructed object that were obtained by the new algorithm. (b) Comparison of the scattered data (solid curve) generated by the actual object (optical fiber) and the scattered data (dot curve) generated by the reconstructed object that were obtained by the FBP algorithm.

Fig. 5
Fig. 5

Reconstruction of a phantom with sparse (limited) data. (a) Actual index of refraction of the phantom. (b) Reconstruction of the refractive index of the phantom using the new algorithm. (c) Reconstruction of the refractive index of the phantom using the FBP algorithm. Pixel size in (a)–(c) is δ x = λ = 633 nm . The following parameters have been used for reconstruction: measurement distance l 0 45 mm , N s = 40 views , N ξ = 61 CCD pixels per view, and distance between adjacent CCD pixels d = 67 μ m .

Fig. 6
Fig. 6

Comparison of the scattered data (solid curve) generated by the actual object (phantom) and the scattered data generated by the reconstructed object obtained using the FBP algorithm (dotted curve) and the new algorithm (represented by the dashed curve, which agrees with the solid curve). (a)–(d) show the scattered data for four different view angles, respectively.

Fig. 7
Fig. 7

Reconstruction of a phantom with dense data. (a) Actual index of refraction of a phantom. (b) Reconstruction of the refractive index of the phantom using the new algorithm. (c) Reconstruction of the refractive index of the phantom using the FBP algorithm. Pixel size in (a)–(c) is δ x = λ = 633 nm . The following parameters have been used for reconstruction: measurement distance l 0 190 μ m , N s = 40 views , N ξ = 61 CCD pixels per view, and distance between adjacent CCD pixels d = 1.9 μ m .

Fig. 8
Fig. 8

Comparison of PSFs between the new and the FBP algorithms. (a) Real part of the reconstructed point scatter by the new algorithm. (b) Spectrum of the new algorithm PSF. (c) Real part of the reconstructed point scatter by the FBP algorithm. (d) Spectrum of the FBP algorithm PSF.

Fig. 9
Fig. 9

Reconstructions by the new algorithm. (a) Object. (b) Reconstruction by the new algorithm from noise-free scattered data. (c) Reconstruction by the new algorithm from noisy data [Signal-to-noise ratio ( SNR ) = 20 dB ]. (d) Reconstruction by the new algorithm using a truncated eigenvalue spectrum with noisy data ( SNR = 20 dB ) .

Fig. 10
Fig. 10

Reconstructions with noisy data. (a) Reconstruction by the new algorithm from noise-free scattered data. (b) Reconstruction by the FBP algorithm from noise-free scattered data. (c) Reconstruction by the new algorithm from noisy data ( SNR = 20 dB ) without using a truncated eigenvalue spectrum. (d) Reconstruction by the new algorithm using a truncated eigenvalue spectrum with noisy data ( SNR = 20 dB ) . (e) Reconstruction by the FBP algorithm with noisy data ( SNR = 20 dB ) . Pixel size in (a)–(e) is δ x = λ = 633 nm .

Equations (18)

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ψ ̃ ( K ; l 0 ) = i k 0 2 n b γ exp ( i γ l 0 ) δ ̃ n ( K , γ K ) ,
γ = { ( k b 2 K 2 ) 1 2 , K k b i ( K 2 k b 2 ) 1 2 , K > k b } ,
δ n ( x , y ) = n ( x , y ) n b
δ ̂ n ( x , y ) = i exp ( i k b l 0 ) 4 π k b 0 2 π d ϕ k k d K K ψ ̃ ( K ; ϕ ) exp [ i ( γ k b ) ( x cos ϕ + y sin ϕ l 0 ) ] exp [ i K ( y cos ϕ x sin ϕ ) ] ,
ψ ( s ) ( ρ j ; s i ) = V d 2 r G b ( ρ j , r ) O ( r ) ψ i n ( r ; s i ) = V d 2 r G b ( ρ j , r ) O ( r ) exp ( i k b s i r ) ,
O ( r ) = k 0 2 [ n b 2 n 2 ( r ) ] 2 k 0 2 n b δ n ( r ) ,
π n ( r ) = { G b * ( ρ j , r ) exp ( i k b s i r ) , if r V 0 , otherwise }
L n = d 2 r π n * ( r ) ,
ψ n s = ψ s ( ρ j ; s i ) = L n O ( r ) = π n ( r ) , O ( r ) ,
f 1 , f 2 = d 2 r f 1 * ( r ) f 2 ( r )
O ̂ ( r ) = n = 1 N C n π n ( r ) ,
ψ n s = m = 1 N C m π n ( r ) , π m ( r ) V for n = 1 , 2 , , N .
A ͇ c ̱ = d ̱ ,
c ̱ = ( C 1 , C 2 , , C N ) T ,
d ̱ = ( ψ 1 s , ψ 2 s , , ψ N s ) T
A ͇ ( i j ) = π i ( r ) , π j ( r ) = d 2 r π i * ( r ) π j ( r ) ,
A ͇ = U ͇ Λ ͇ U ͇ ,
c ̱ ̂ = U ͇ Λ ͇ 1 U ͇ d ͇ ,

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