Abstract

An early paper by McCutchen [J. Opt. Soc. Am. 54, 240 (1964)] relates the modulation of a convergent spherical wave front to the Fourier transform of the complex amplitude near the geometrical focal point. This implies additional useful Fourier relationships between the wave-front modulations and the cross sections of the diffracted field, which contain the geometrical focal point. We show how these relations can be applied to diffraction tomography. To make use of McCutchen’s relations, particular emphasis is given to the analysis of diffraction tomography with point-source illumination. We derive a sufficient condition under which linear tomographic reconstruction can be applied to arbitrary incident fields and synthetic apertures. This suggests a modified filtered backpropagation algorithm. In addition, we use the results of McCutchen’s paper to obtain information about the symmetry of the object from the scattered field.

© 1999 Optical Society of America

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References

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  1. E. Wolf, “Principles and developments of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), pp. 83–110.
  2. A. C. Kak, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), Chap. 6, pp. 203–274.
  3. F. C. Lin, M. A. Fiddy, “Image estimation from scattered field data,” Int. J. Imaging Syst. Technol. 2, 76–95 (1990).
    [CrossRef]
  4. J. D. Sanchez-de-la-Llave, A. Morales-Porras, M. Testorf, R. V. McGahan, M. Fiddy, “Nonlinear filtering of backpropagated fields,” J. Opt. Soc. Am. A 16, 1799–1805 (1999).
    [CrossRef]
  5. A. Schatzberg, A. J. Devaney, “Super-resolution in diffraction tomography,” Invest. Radiol. 8, 149–164 (1992).
  6. A. Devaney, “The limited-view problem in diffraction tomography,” Inverse Probl. 5, 501–521 (1989).
    [CrossRef]
  7. B. Chen, J. J. Stamnes, “Validity of diffraction tomography based on the first Born and the first Rytov approximations,” Appl. Opt. 37, 2996–3006 (1998).
    [CrossRef]
  8. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yoadh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
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  9. Y. Yao, Y. Wang, Y. Pai, W. Zhu, R. L. Barbour, “Frequency-domain optical imaging and scattering distribution by a Born iterative method,” J. Opt. Soc. Am. A 14, 325–341 (1997).
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  10. H. Jiang, K. D. Paulsen, U. Österberg, B. Pogue, M. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
    [CrossRef]
  11. S. J. Norton, T. Vo-Dinh, “Diffraction tomographic imaging with photon density waves: an explicit solution,” J. Opt. Soc. Am. A 15, 2670–2677 (1998).
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  12. C. L. Matson, “A diffraction tomographic model for the forward problem using diffuse photon density waves,” Opt. Express 1, 6–11 (1997).
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  13. D. L. Lasocki, C. L. Matson, P. J. Collins, “Analysis of diffuse photon-density waves in turbid media: a diffraction tomography approach to an analytic solution,” Opt. Lett. 23, 558–560 (1998).
    [CrossRef]
  14. A. J. Devaney, “Inversion formula for inverse scattering within the Born approximation,” Opt. Lett. 7, 111–112 (1982).
    [CrossRef] [PubMed]
  15. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
    [CrossRef] [PubMed]
  16. R. P. Porter, “Image formation with arbitrary holographic type surfaces,” Phys. Lett. A 29, 193–194 (1996).
    [CrossRef]
  17. R. P. Porter, W. C. Schwab, “Electromagnetic image formation with holograms of arbitrary shape,” J. Opt. Soc. Am. 61, 789–796 (1971).
    [CrossRef]
  18. K. J. Langenberg, M. Fischer, M. Berger, G. Weinfurter, “Imaging performance of generalized holography,” J. Opt. Soc. Am. A 3, 329–339 (1986).
    [CrossRef]
  19. A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
    [CrossRef]
  20. A. J. Devaney, G. Beylkin, “Diffraction tomography using arbitrary transmitter and receiver surfaces,” Ultrason. Imaging 6, 181–193 (1984).
    [CrossRef] [PubMed]
  21. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  22. G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), Chaps. 8, 11, and 12.
  23. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1993), Chap. 8.2, pp. 370–375.
  24. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  25. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 5–29.
  26. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 440.
  27. A. C. Kak, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), Chap. 3, pp. 75–99.

1999

1998

1997

1996

1995

1994

1992

A. Schatzberg, A. J. Devaney, “Super-resolution in diffraction tomography,” Invest. Radiol. 8, 149–164 (1992).

1990

F. C. Lin, M. A. Fiddy, “Image estimation from scattered field data,” Int. J. Imaging Syst. Technol. 2, 76–95 (1990).
[CrossRef]

1989

A. Devaney, “The limited-view problem in diffraction tomography,” Inverse Probl. 5, 501–521 (1989).
[CrossRef]

1986

K. J. Langenberg, M. Fischer, M. Berger, G. Weinfurter, “Imaging performance of generalized holography,” J. Opt. Soc. Am. A 3, 329–339 (1986).
[CrossRef]

A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
[CrossRef]

1984

A. J. Devaney, G. Beylkin, “Diffraction tomography using arbitrary transmitter and receiver surfaces,” Ultrason. Imaging 6, 181–193 (1984).
[CrossRef] [PubMed]

1982

A. J. Devaney, “Inversion formula for inverse scattering within the Born approximation,” Opt. Lett. 7, 111–112 (1982).
[CrossRef] [PubMed]

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

1971

1964

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), Chaps. 8, 11, and 12.

Barbour, R. L.

Berger, M.

Beylkin, G.

A. J. Devaney, G. Beylkin, “Diffraction tomography using arbitrary transmitter and receiver surfaces,” Ultrason. Imaging 6, 181–193 (1984).
[CrossRef] [PubMed]

Boas, D. A.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1993), Chap. 8.2, pp. 370–375.

Chance, B.

Chen, B.

Collins, P. J.

Devaney, A.

A. Devaney, “The limited-view problem in diffraction tomography,” Inverse Probl. 5, 501–521 (1989).
[CrossRef]

Devaney, A. J.

A. Schatzberg, A. J. Devaney, “Super-resolution in diffraction tomography,” Invest. Radiol. 8, 149–164 (1992).

A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
[CrossRef]

A. J. Devaney, G. Beylkin, “Diffraction tomography using arbitrary transmitter and receiver surfaces,” Ultrason. Imaging 6, 181–193 (1984).
[CrossRef] [PubMed]

A. J. Devaney, “Inversion formula for inverse scattering within the Born approximation,” Opt. Lett. 7, 111–112 (1982).
[CrossRef] [PubMed]

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

Feng, T.-C.

Fiddy, M.

Fiddy, M. A.

F. C. Lin, M. A. Fiddy, “Image estimation from scattered field data,” Int. J. Imaging Syst. Technol. 2, 76–95 (1990).
[CrossRef]

Fischer, M.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 5–29.

Haskell, R. C.

Jiang, H.

Kak, A. C.

A. C. Kak, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), Chap. 3, pp. 75–99.

A. C. Kak, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), Chap. 6, pp. 203–274.

Langenberg, K. J.

Lasocki, D. L.

Lin, F. C.

F. C. Lin, M. A. Fiddy, “Image estimation from scattered field data,” Int. J. Imaging Syst. Technol. 2, 76–95 (1990).
[CrossRef]

Matson, C. L.

McAdams, M. S.

McCutchen, C. W.

McGahan, R. V.

Morales-Porras, A.

Norton, S. J.

O’Leary, M. A.

Österberg, U.

Pai, Y.

Patterson, M.

Paulsen, K. D.

Pogue, B.

Porter, R. P.

R. P. Porter, “Image formation with arbitrary holographic type surfaces,” Phys. Lett. A 29, 193–194 (1996).
[CrossRef]

R. P. Porter, W. C. Schwab, “Electromagnetic image formation with holograms of arbitrary shape,” J. Opt. Soc. Am. 61, 789–796 (1971).
[CrossRef]

Sanchez-de-la-Llave, J. D.

Schatzberg, A.

A. Schatzberg, A. J. Devaney, “Super-resolution in diffraction tomography,” Invest. Radiol. 8, 149–164 (1992).

Schwab, W. C.

Stamnes, J. J.

Svaasand, L. O.

Testorf, M.

Tromberg, B. J.

Tsay, T.-T.

Vo-Dinh, T.

Wang, Y.

Weber, H. J.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), Chaps. 8, 11, and 12.

Weinfurter, G.

Wolf, E.

E. Wolf, “Principles and developments of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), pp. 83–110.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1993), Chap. 8.2, pp. 370–375.

Yao, Y.

Yoadh, A. G.

Zhu, W.

Appl. Opt.

Int. J. Imaging Syst. Technol.

F. C. Lin, M. A. Fiddy, “Image estimation from scattered field data,” Int. J. Imaging Syst. Technol. 2, 76–95 (1990).
[CrossRef]

Inverse Probl.

A. J. Devaney, “Reconstructive tomography with diffracting wavefields,” Inverse Probl. 2, 161–183 (1986).
[CrossRef]

A. Devaney, “The limited-view problem in diffraction tomography,” Inverse Probl. 5, 501–521 (1989).
[CrossRef]

Invest. Radiol.

A. Schatzberg, A. J. Devaney, “Super-resolution in diffraction tomography,” Invest. Radiol. 8, 149–164 (1992).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Phys. Lett. A

R. P. Porter, “Image formation with arbitrary holographic type surfaces,” Phys. Lett. A 29, 193–194 (1996).
[CrossRef]

Ultrason. Imaging

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

A. J. Devaney, G. Beylkin, “Diffraction tomography using arbitrary transmitter and receiver surfaces,” Ultrason. Imaging 6, 181–193 (1984).
[CrossRef] [PubMed]

Other

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), Chaps. 8, 11, and 12.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1993), Chap. 8.2, pp. 370–375.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 5–29.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), p. 440.

A. C. Kak, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), Chap. 3, pp. 75–99.

E. Wolf, “Principles and developments of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), pp. 83–110.

A. C. Kak, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), Chap. 6, pp. 203–274.

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Figures (6)

Fig. 1
Fig. 1

Diffraction of a spherical incident wave field. The scattered field is detected at a spherical surface of radius rD centered at the source S.

Fig. 2
Fig. 2

Scattered field, as detected in Fig. 1, corresponding to the convolution of the plane-wave spectra of the object and the incident field at a centered spherical surface in reciprocal space.

Fig. 3
Fig. 3

Diffraction of a modulated convergent spherical wave front. The image can be calculated as the Fourier transform of the generalized aperture A.

Fig. 4
Fig. 4

Diffraction tomography with spherical incident illumination. The source is moved on a sphere around the object, and the scattered field is detected at a sphere centered at the source position.

Fig. 5
Fig. 5

Generalized aperture A(ϕ, ϑ), averaged over the azimuth angle ϕ, corresponding to the Fourier transform of the diffracted field along the line z through the focal point.

Fig. 6
Fig. 6

Test of the cylindrical symmetry of the object. The symmetry axis z corresponds to a line through the source location.

Equations (29)

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u(r)=ui(r)+V(r)ui(r)G(|r-r|)d3r.
ui(r)=exp(ik0|r|)|r|,
G(|r-r|)=exp(ik0|r-r|)4π|r-r|.
|r-r|r-r|r| r.
u(r)=ui(r)+exp(ik0|r|)4π|r| V(r)ui(r)×exp-ik0 r|r| rd3r.
u(r)=ui(r)1+4πV(r)ui(r)×exp-ik0 r|r| rd3r.
ud(rp)=-iλ W exp(-ik0rD)rD A(r)G(|rp-r|)d3r,
ud(rp)=-i4πλ ΩAr|r|exp-ik0 r|r| rpdΩ,
Ar|r|=4πV(r)ui*(r)expik0 r|r| rd3r.
ud(rp)=-iλ V(r)ui*(r)j0(k0|rp-r|)d3r.
ud(rr, ri)=-iλ V(rr)ui*(rr-ri)j0(k0|rr-rr|)d3rr.
Vest(rr)=ud(rr, ri)uc(rr, ri)p(ri)d3ri.
Vest(rr)=-iλ V(rr)j0(k0|rr-rr|)g(rr, rr)d3rr,
g(rr, rr)=ui*(rr-ri)uc(rr, ri)p(ri)d3ri.
h(rr, rr)=j0(k0|rr-rr|) exp(-ik0|rr-ri|)|rr-ri|.
ui*(rr-ri)=exp(-ik0ri)ri expik0 ri|ri| rr.
h(|rr-rr|)=j0(k0|rr-rr|)× expik0 ri|ri| (rr-rr)d3ri=4πj02(k0|rr-rr|).
h˜(|k|)=4πj02(k0|r|)exp(-ikr)r2drdΩ=4π2k02 1k rectk-k02k0,
ui*(rr, ri)=i4πk0l=0m=-lljl(k0rr)hl*(k0ri)×Ylm*(ϕ, ϑ)Ylm(ϕi, ϑi),
uc(rr, ri)=4πiri2ki l=0m=-ll jl(k0rr)hl*(k0ri)
×Ylm(ϕ, ϑ)Ylm*(ϕi, ϑi).
g(rr, rr)=(4π)2l=0m=-lljl(k0rr)×jl(k0rr)Ylm(ϕ, ϑ)Ylm*(ϕ, ϑ)=4πj0(k0|rr-rr|);
ud(z)=-iλ -1102πA(ϑ, ϕ)dϕ×exp(-ik0z cos ϑ)sin ϑdϑ.
V(r, ϕ, ϑ)=l=0Vl(c)rlYl0(ϕ, ϑ)+l=0m=-llVlm(n)rlYlm(ϕ, ϑ),
A(ϕ, ϑ)
=(4π)2l=0m=-ll0il02π-π/2π/2V(r, ϕ, ϑ)ui*(r)×jl(k0r)Ylm*(ϕ, ϑ)Yml(ϕ, ϑ)sin ϑdϑdϕdr.
02πAmod(ϕ, ϑ)A(ϕ, ϑ)dϕ
=(4π)2l=00ilrlui*(r)Vl,1(n)
×jl(k0r)Yl1(0, ϑ)Ym1*(0, ϑ)dr,

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