Abstract

The relationship between computed tomography (CAT) and diffraction tomography (DT) is investigated. A simple condition with a clear physical meaning is derived for the applicability of CAT. Corrections due to scattering are incorporated into CAT, and it is shown that the effect of scattering may be characterized by a two-dimensional fractional Fourier transform. The implications of these results for the three-dimensional imaging of weakly scattering objects are also discussed.

© 2001 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 7th, expanded ed. (Cambridge U. Press, Cambridge, UK, 1999).
  2. G. T. Herman, Image Reconstruction from Projections (Academic, Orlando, Fla., 1980).
  3. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
  4. E. Wolf, “Principles and development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), pp. 83–110.
  5. Very recently a method somewhat analogous to DT was proposed that circumvents the need for measurement of the phase. Instead, one measures the power scattered by the object when it is illuminated simultaneously by pairs of plane waves [P. S. Carney, E. Wolf, G. S. Agarwal, “Diffraction tomography using power extinction measurements,” J. Opt. Soc. Am. A 16, 2643–2648 (1999)]. However, as yet the method has not been tested experimentally.
    [CrossRef]
  6. This question was briefly addressed by A. J. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. 6, 374–376 (1981).
    [CrossRef] [PubMed]
  7. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  8. Equation (17) is equivalent to Eq. (8) of Sec. 4.11 of Ref. 1.
  9. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1960), Chap. 2, Sec. 7.
  10. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  11. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  12. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]
  13. D. Mendlovic, Z. Zalevsky, R. G. Dorsch, Y. Bitran, A. W. Lohmann, H. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424–2431 (1995).
    [CrossRef]
  14. D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
    [CrossRef]
  15. For a thorough discussion of the Rayleigh range see, for instance, J. F. Ramsey, “Tubular beams from radiating apertures,” in Advances in Microwaves, L. F. Young, ed. (Academic, New York, 1968), Vol. 3, pp. 127–221.
  16. We have neglected the contribution of the evanescent waves [p2+q2>1in Eq. (6)], because with our choices of the values of kσ0and kd,the contribution of the evanescent waves will be negligible.
  17. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), pp. 667–669.

1999 (1)

1996 (1)

D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

1995 (1)

1994 (1)

1987 (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

1981 (1)

1980 (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Agarwal, G. S.

Bitran, Y.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th, expanded ed. (Cambridge U. Press, Cambridge, UK, 1999).

Carney, P. S.

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1960), Chap. 2, Sec. 7.

Devaney, A. J.

Dorsch, R. G.

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections (Academic, Orlando, Fla., 1980).

James, D. F. V.

D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

Kak, A. C.

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

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Lohmann, A. W.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Mendlovic, D.

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Ozaktas, H.

Pellat-Finet, P.

Ramsey, J. F.

For a thorough discussion of the Rayleigh range see, for instance, J. F. Ramsey, “Tubular beams from radiating apertures,” in Advances in Microwaves, L. F. Young, ed. (Academic, New York, 1968), Vol. 3, pp. 127–221.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), pp. 667–669.

Slaney, M.

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

Wolf, E.

Very recently a method somewhat analogous to DT was proposed that circumvents the need for measurement of the phase. Instead, one measures the power scattered by the object when it is illuminated simultaneously by pairs of plane waves [P. S. Carney, E. Wolf, G. S. Agarwal, “Diffraction tomography using power extinction measurements,” J. Opt. Soc. Am. A 16, 2643–2648 (1999)]. However, as yet the method has not been tested experimentally.
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th, expanded ed. (Cambridge U. Press, Cambridge, UK, 1999).

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

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Zalevsky, Z.

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

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

Opt. Commun. (1)

D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

Opt. Lett. (2)

Other (10)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Equation (17) is equivalent to Eq. (8) of Sec. 4.11 of Ref. 1.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1960), Chap. 2, Sec. 7.

M. Born, E. Wolf, Principles of Optics, 7th, expanded ed. (Cambridge U. Press, Cambridge, UK, 1999).

G. T. Herman, Image Reconstruction from Projections (Academic, Orlando, Fla., 1980).

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

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

For a thorough discussion of the Rayleigh range see, for instance, J. F. Ramsey, “Tubular beams from radiating apertures,” in Advances in Microwaves, L. F. Young, ed. (Academic, New York, 1968), Vol. 3, pp. 127–221.

We have neglected the contribution of the evanescent waves [p2+q2>1in Eq. (6)], because with our choices of the values of kσ0and kd,the contribution of the evanescent waves will be negligible.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), pp. 667–669.

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Figures (5)

Fig. 1
Fig. 1

Depiction of the arrangement and notation.

Fig. 2
Fig. 2

Definition of the distance L relevant to inequality (40).

Fig. 3
Fig. 3

First Rytov data function, CAT data function, and fractional Fourier data function for kσ0=40, kd=40. In the figure, the variable ρ=(x2+y2)1/2/(2σ0) and the normalization constant N=A(2π)3/2σ0/k. For this particular choice of kσ0 and kd, all three plots practically coincide.

Fig. 4
Fig. 4

First Rytov data function, CAT data function, and fractional Fourier data function for kσ0=40, kd=500.

Fig. 5
Fig. 5

First Rytov data function, CAT data function, and the fractional Fourier data function for kσ0=2, kd=8.

Equations (47)

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[2+k2]U(r)=-4πF(r)U(r).
U(r)Ui(r)exp[ψ(r)],
ψ(r)=VF(r)exp(ik|r-r|)|r-r|×exp[-iks0(r-r)]d3r.
exp(ik|r-r|)|r-r|=ik2π1mexp[ik[ps1+qs2+ms0](r-r)]dpdq,
m=1-p2-q2whenp2+q21ip2+q2-1 whenp2+q2>1,,
ψ(x, y; d)=ik2πVd3r1mF(r)×exp[ik(m-1)(d-z)]×exp{ik[p(x-x)+q(y-y)]}dpdq.
ψ(x, y; d)
=(2π)2ik1m F˜(k[ps1+qs2+(m-1)s0])×exp[ik(m-1)d]exp[ik(px+qy)]dpdq,
F˜[K]=1(2π)3VF(r)exp(-iKr)d3r
D(x, y; d)ln|U(r)|2|Ui(r)|2z=d=2 Re{ψ(x, y; d)}
F˜[K]0,|K|2π/σ.
kσ2πσλ1.
p2+q21.
1mexp[ik(m-1)(d-z)]
1+[1-ik(d-z)]p2+q22+O[(p2+q2)2],
D(x, y; d)=D0(x,y;d)+D1(x, y; d)+D2(x, y; d)+.
D0(x, y; d)2 Reik2πVd3rF(r)×exp{ik[p(x-x)+q(y-y)]}dpdq.
δ(x-x)=12πexp[iu(x-x)]du.
D0(x, y; d)-4πkIm{F(r)}δ(x-s1r)×δ(y-s2r)d3r.
α(r)=4πkIm{F(r)},
p2exp[ikp(x-x)]=-1k22x2exp[ikp(x-x)],
D1(x, y; d)=2πk3[Im{T2F(r)}-k Re{T2F(r)}×(d-s0r)]δ(x-s1r)×δ(y-s2r)d3r,
(m-1)[-12(p2+q2)].
D(x, y; d)(2π)3kπReim F˜(k[ps1+qs2+(m-1)s0])exp{ik[px+qy-d(p2+q2)/2]}dpdq.
Fθ(2)f(u, v)=i exp(-iθ)2π sin θexp-i2cot θ(u2+v2)×expisin θ(uu+vv)×exp-i2cot θ(u2+v2)×f(u, v)dudv,
F0(2)f(u, v)=f(u, v)(identity),
Fθ1(2)Fθ2(2)f(u, v)=Fθ1+θ2(2)f(u, v)(linearity),
Fπ/2(2)f(u, v)=fˆ(u, v),
fˆ(u, v)12πf(u, v)exp[i(uu+vv)]dudv.
cot θ=kd,
β=k sin θ,
Γ(p, q)=1m F˜(k[ps1+qs2+(m-1)s0]).
D(x, y; d)=2(2π)3β Reexp(iθ)expi2cot θ[(βx)2+(βy)2]Fθ(2)Γ(βx, βy).
Γ(p, q)F˜[k(ps1+qs2)]Γ0(p, q).
limλ0D(x, y; d)=2(2π)3k Re{iFπ/2(2)Γ0(kx, ky)}.
limλ0 D(x, y; d)=-4πkIm{F[r]}δ(x-x)×δ(y-y)d3r,
D[kx, ky; d]D[x, y; d]
Dˆ[u, v; d]=12πexp[-ik(ux+vy)]×D[kx, ky; d]d(kx)d(ky).
Dˆ[u, v; d]=(2π)3ikexp-i2cot θ(u2+v2)Γ(u,v)-expi2cot θ(u2+v2)[Γ(-u, -v)]*,
D^0[u, v; d]=(2π)3ik{F˜(kus1+kvs2)-[F˜(-kus1-kvs2)]*}.
Dˆ[u, v; d]=ikmVF(r)exp[-ik(ux+vy)]×expi2k(u2+v2)(d-z)d3r-VF*(r)exp[-ik(ux+vy)]×exp-i2k(u2+v2)(d-z)d3r.
12k(d-z)[u2+v2]π2
forallu2+v2(2π)2(kσ)2=λσ2,
andforallzV.
Lσ22λ.
F(r)=A exp(-r2/2σ02),
F˜(K)=Aσ03(2π)3/2exp(-K2σ02/2),

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