Abstract

We present a detailed derivation of the phase-retrieval formula based on the phase-attenuation duality that we recently proposed in previous brief communication. We have incorporated the effects of x-ray source coherence and detector resolution into the phase-retrieval formula as well. Since only a single image is needed for performing the phase retrieval by means of this new approach, we point out the great advantages of this new approach for implementation of phase tomography. We combine our phase-retrieval formula with the Feldkamp-Davis-Kresss (FDK) cone-beam reconstruction algorithm to provide a three-dimensional phase tomography formula for soft tissue objects of relatively small sizes, such as small animals or human breast. For large objects we briefly show how to apply Katsevich’s cone-beam reconstruction formula to the helical phase tomography as well.

© 2005 Optical Society of America

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References

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  1. P A. Snigirev and I. Snigireva and V. Kohn, et al, "On the possibilities of x-ray phase contrast micro-imaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492, (1995).
    [CrossRef]
  2. S. Wilkins, T. Gureyev, D. Gao, A. Pogany and A. Stevenson, �??Phase contrast imaging using polychromatic hard x-ray,�?? Nature 384, 335-338 (1996).
    [CrossRef]
  3. K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, �??Quantitative phase imaging using hard x rays,�?? Phys. Rev. Lett. 77, 2961-2965, (1996).
    [CrossRef] [PubMed]
  4. A. Pogany, D. Gao and S. Wilkins, �??Contrast and resolution in imaging with a microfocus x-ray source,�?? Rev. Sci. Instrum. , 2774-2782 (1997).
    [CrossRef]
  5. D. Paganin and K. Nugent, �??Noninterferometric phase imaging with partialcoherent light,�?? Phys. Rev. Lett. 80, 2586-2589 (1998).
    [CrossRef]
  6. F . Arfelli, V. Bonvicini, et al, �??Mammography with synchrotron radiation: phase-detected Techniques,�?? Radiology 215, 286-293 (2000).
  7. X. Wu and H. Liu, "A general formalism for x-ray phase contrast imaging," J. X-Ray Sci. Technol. 11, 33-42 (2003).
  8. X. Wu and H. Liu, "Clinical implementation of phase contrast x-ray imaging: theoretical foundation and design considerations," Med. Phys. 30, 2169-2179 (2003).
    [CrossRef] [PubMed]
  9. X. Wu and H. Liu, �??A dual detector approach for X-ray attenuation and phase imaging,�?? J. X-Ray Sci. Technol. 12, 35-42, 2004.
  10. X. Wu and H. Liu, �??An experimental method of determining relative phase-contrast factor for x-ray imaging systems," Med. Phys. 31, 997-1002 (2004).
    [CrossRef]
  11. X. Wu and H. Liu, �??A new theory of phase-contrast x-ray imaging based on Wigner distributions," Med. Phys. 31, 2380-2384 (2004).
    [CrossRef]
  12. E. Donnelly, R. Price and D. Pickens, �??Experimental validation of the Wigner distributions theory of phase-contrast imaging,�?? Med. Phys. 32, 928-931 (2005).
    [CrossRef] [PubMed]
  13. X. Wu, A. Dean and H Liu, "X-ray diagnostic techniques," in Biomedical photonics handbook, T. VoDinh ed., Chapter 26, p.26-1 to p.26-34, (CRC Press, Tampa, Florida, 2003).
  14. A. Bronnikov, �??Reconstruction formulas in phase-contrast tomography,�?? Optics Commun. 171, 29-244 (1999)
    [CrossRef]
  15. A. Bronnikov, �??Theory of quantitative phase-contrast computed tomography,�?? J. Opt. Soc. Am. A 19, 472-480 (2002).
    [CrossRef]
  16. D. Paganin, S. Mayo, T. Gureyev, P. Miller and S. Wilkins, �??Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,�?? J. Microsc. 206, 33-40 (2002).
    [CrossRef] [PubMed]
  17. X. Wu, H. Liu, and A. Yan, �??X-ray phase-attenuation duality and phase retrieval,�?? Optics Lett. 30, 379-381 (2005).
    [CrossRef]
  18. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (IEEE Press, New York, 1987).
  19. F. Natterer, The Mathematics of Computerized Tomography, (SIAM, Philadelphia, 2001).
    [CrossRef]
  20. A. Katsevich, �??Analysis of an exact inversion algorithm for spiral cone-beam CT,�?? Phys. Med. Biol. 47, 2583-2597 (2003).
    [CrossRef]
  21. A. Katsevich, �??An improved exact filtered backprojection algorithm for spiral computed tomography,�?? Advances in Applied Mathematics, 35, 681-697 (2004).
    [CrossRef]
  22. F. Noo, J. Pack and D. Heuscher, �??Exact helical reconstruction using native cone-beam geometry,�?? Phys. Med. Biol. 48, 3787-3818 (2003).
    [CrossRef]
  23. M. Born and E. Wolf, Principle of Optics, 6th ed. (Pergamon, Oxford 1980).
  24. H. Wiedemann, Synchrotron Radiation, (Springer-verlag, Berlin Heidelberg 2003).
    [CrossRef]
  25. N. A. Dyson, X-rays in atomic and Nuclear physics, (Longman, 1973).
  26. ICRU, Tissue Substitutes in Radiation Dosimetry and Measurement, Report 44 of the International Commission on Radiation Units and Measurements (Bethesda, MD, 1989).
  27. X. Wu and H. Liu, �??A reconstruction formula for soft tissue X-ray phase tomography,�?? J. X-Ray Sci. Technol. 12, 273-279 (2004).

Advances in Applied Mathematics (1)

A. Katsevich, �??An improved exact filtered backprojection algorithm for spiral computed tomography,�?? Advances in Applied Mathematics, 35, 681-697 (2004).
[CrossRef]

Biomedical photonics handbook, Ch. 26 (1)

X. Wu, A. Dean and H Liu, "X-ray diagnostic techniques," in Biomedical photonics handbook, T. VoDinh ed., Chapter 26, p.26-1 to p.26-34, (CRC Press, Tampa, Florida, 2003).

J. Microsc. (1)

D. Paganin, S. Mayo, T. Gureyev, P. Miller and S. Wilkins, �??Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,�?? J. Microsc. 206, 33-40 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

J. X-Ray Sci. Technol. (3)

X. Wu and H. Liu, �??A reconstruction formula for soft tissue X-ray phase tomography,�?? J. X-Ray Sci. Technol. 12, 273-279 (2004).

X. Wu and H. Liu, "A general formalism for x-ray phase contrast imaging," J. X-Ray Sci. Technol. 11, 33-42 (2003).

X. Wu and H. Liu, �??A dual detector approach for X-ray attenuation and phase imaging,�?? J. X-Ray Sci. Technol. 12, 35-42, 2004.

Med. Phys. (4)

X. Wu and H. Liu, �??An experimental method of determining relative phase-contrast factor for x-ray imaging systems," Med. Phys. 31, 997-1002 (2004).
[CrossRef]

X. Wu and H. Liu, �??A new theory of phase-contrast x-ray imaging based on Wigner distributions," Med. Phys. 31, 2380-2384 (2004).
[CrossRef]

E. Donnelly, R. Price and D. Pickens, �??Experimental validation of the Wigner distributions theory of phase-contrast imaging,�?? Med. Phys. 32, 928-931 (2005).
[CrossRef] [PubMed]

X. Wu and H. Liu, "Clinical implementation of phase contrast x-ray imaging: theoretical foundation and design considerations," Med. Phys. 30, 2169-2179 (2003).
[CrossRef] [PubMed]

Nature (1)

S. Wilkins, T. Gureyev, D. Gao, A. Pogany and A. Stevenson, �??Phase contrast imaging using polychromatic hard x-ray,�?? Nature 384, 335-338 (1996).
[CrossRef]

Optics Commun. (1)

A. Bronnikov, �??Reconstruction formulas in phase-contrast tomography,�?? Optics Commun. 171, 29-244 (1999)
[CrossRef]

Optics Lett. (1)

X. Wu, H. Liu, and A. Yan, �??X-ray phase-attenuation duality and phase retrieval,�?? Optics Lett. 30, 379-381 (2005).
[CrossRef]

Phys. Med. Biol. (2)

A. Katsevich, �??Analysis of an exact inversion algorithm for spiral cone-beam CT,�?? Phys. Med. Biol. 47, 2583-2597 (2003).
[CrossRef]

F. Noo, J. Pack and D. Heuscher, �??Exact helical reconstruction using native cone-beam geometry,�?? Phys. Med. Biol. 48, 3787-3818 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, �??Quantitative phase imaging using hard x rays,�?? Phys. Rev. Lett. 77, 2961-2965, (1996).
[CrossRef] [PubMed]

D. Paganin and K. Nugent, �??Noninterferometric phase imaging with partialcoherent light,�?? Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

Radiology (1)

F . Arfelli, V. Bonvicini, et al, �??Mammography with synchrotron radiation: phase-detected Techniques,�?? Radiology 215, 286-293 (2000).

Rev. Sci. Instrum. (2)

A. Pogany, D. Gao and S. Wilkins, �??Contrast and resolution in imaging with a microfocus x-ray source,�?? Rev. Sci. Instrum. , 2774-2782 (1997).
[CrossRef]

P A. Snigirev and I. Snigireva and V. Kohn, et al, "On the possibilities of x-ray phase contrast micro-imaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492, (1995).
[CrossRef]

Other (6)

M. Born and E. Wolf, Principle of Optics, 6th ed. (Pergamon, Oxford 1980).

H. Wiedemann, Synchrotron Radiation, (Springer-verlag, Berlin Heidelberg 2003).
[CrossRef]

N. A. Dyson, X-rays in atomic and Nuclear physics, (Longman, 1973).

ICRU, Tissue Substitutes in Radiation Dosimetry and Measurement, Report 44 of the International Commission on Radiation Units and Measurements (Bethesda, MD, 1989).

C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (IEEE Press, New York, 1987).

F. Natterer, The Mathematics of Computerized Tomography, (SIAM, Philadelphia, 2001).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Cone-beam tomography with a source orbiting on a circle in (X 1, X 2) -plane. The fixed frame is attached with object, while the detector-plane denotes the 2-D imaging detector a distance of R2 downstream. The imaging detector is always kept perpendicular to the line joining the origin and the source. For definitions of the angles, see the text for details.

Fig. 2.
Fig. 2.

The moving frame attached to the virtual detector plane.

Fig. 3.
Fig. 3.

Schematic of an undulator source.

Fig. 4.
Fig. 4.

Sampled Radon space for the FDK reconstruction algorithm.

Fig. 5.
Fig. 5.

Helical scanning and the PI-segment.

Fig. 6.
Fig. 6.

Parallel beam tomography with a source orbiting along a circle.

Equations (39)

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ϕ = 2 π λ Δ ( s ) ds
g ( θ , y ) = 0 Δ ( R 1 θ + t ( y R 1 θ ) y R 1 θ ) dt
θ = ( cos β , sin β , 0 ) , θ = ( sin β , cos β , 0 ) , e 3 = 0,0,1
y = y 2 θ + y 3 e 3
ϕ ( y ; θ ) = 2 π λ g ( θ , y )
I ˜ ( u M ; θ ) = I in μ ˜ in ( λ R 2 u M ) OTF det ( u M ) ×
× { cos ( π λ R 2 u 2 M ) ( F ̂ ( A o 2 ) i λ R 2 M u F ̂ ( ϕ A o 2 ) ) + 2 sin ( π λ R 2 u 2 M ) ( F ̂ ( A o 2 ϕ ) i λ R 2 M u F ̂ ( A o 2 ) ) }
A o 2 ( y ; θ ) = Exp ( 4 π λ 0 B ( R 1 θ + t ( y R 1 θ ) y R 1 θ ) dt )
= Exp ( 0 μ ( R 1 θ + t ( y R 1 θ ) y R 1 θ ) dt )
μ ˜ in ( λ R 2 u M ) = OTF G . U . ( u M )
μ ˜ in ( λ R 2 u M ) = MTF G . U . ( u M ) = 2 J 1 ( πf ( M 1 ) u M ) πf ( M 1 ) u M
μ ˜ in ( λ R 2 u M ) = exp [ ( ( ( M 1 ) u 1 σ 1 M ) 2 + ( ( M 1 ) u 2 σ 2 M ) 2 ) ]
I ˜ ( u M ; θ ) = I in μ ˜ in ( λ R 2 u M ) OFT det ( u M ) { ( F ̂ ( A o 2 ) i λ R 2 M u . F ̂ ( ϕ A o 2 ) ) + 2 πλ R 2 u 2 M F ̂ ( A o 2 ϕ ) }
ρ e , p ( y ; θ ) = 0 ρ e ( R 1 θ + t ( y R 1 θ ) y R 1 θ ) dt )
ϕ ( y ; θ ) = λ r e ρ e , p ( y ; θ ) A o 2 ( y ; θ ) = exp ( σ KN ρ e , p ( y ; θ ) )
σ KN = 2 π r e 2 { 1 + η η 2 [ 2 ( 1 + η ) 1 + 2 η 1 η log ( 1 + 2 η ) ] + 1 2 η log ( 1 + η ) ( 1 + 3 η ) ( 1 + 2 η ) 2 }
2 πλ R 2 u 2 M F ̂ ( A o 2 ϕ ) i λ R 2 M u . F ̂ ( ϕ A o 2 ) = 2 π r e λ 2 R 2 u 2 M σ KN F ̂ ( A o 2 )
I ˜ ( u M ; θ ) = I in μ ˜ in ( λ R 2 u M ) OTF det ( u M ) { 1 + 2 π r e λ 2 R 2 u 2 M σ KN } F ̂ ( A o 2 )
ρ e , p ( y ; θ ) = 1 σ KN log e ( F ̂ 1 { F ̂ [ M 2 I ( M y ; θ ) ] I in μ ˜ in ( λ R 2 u M ) OTF det ( u M ) ( 1 + 2 π ( r e λ 2 R 2 M σ KN ) u 2 ) } )
ϕ ( y ; θ ) = λ r e σ KN log e ( F ̂ 1 { F ̂ [ M 2 I ( M y ; θ ) ] I in μ ˜ in ( λ R 2 u M ) OTF det ( u M ) ( 1 + 2 π ( r e λ 2 R 2 M σ KN ) u 2 ) } )
Δ ( r o ) = λ 4 π 0 2 π R 1 2 ( R 1 2 r o θ ) 2 ρ ρ v ( y 2 y ' 2 ) ϕ ( y ' 2 θ + y 3 e 3 ; θ ) R 1 d y ' 2 R 1 2 + ( y ' 2 ) 2 + y 3 2 ,
y 2 = R 1 R 1 r o θ r o θ y 3 = R 1 R 1 r o θ x 3
Δ ̂ ( s , n ) = Δ ( r o ) δ ( r o n s ) d r o
a ( s ) = [ R 1 cos ( s ) , R 1 sin ( s ) , x 30 + P s 2 π ]
f ( r o ) = ( 1 2 π 2 ) I PI ( r o ) 1 r o a ( s ) 0 2 π q g ( Θ ( s , r o , γ ) , a ( q ) ) q = s sin γ ds
Δ ( r o ) = ( λ 8 π 2 ) s b s 1 ( R 1 r o θ ( s ) ) ϕ F ( y * ( s , r o ) , s ) ds
y * s r o = y 2 * ( s , r o ) θ + y 3 * ( s , r o ) e 3
y 2 * ( s , r o ) = R 1 ( r o a ( s ) ) θ ( s ) ( r o a ( s ) ) θ ( s ) = R 1 r o θ ( s ) ( R 1 r o θ ( s ) )
y 3 * ( s , r o ) = R 1 ( r o a ( s ) ) e 3 ( s ) ( r o a ( s ) ) θ ( s ) = R 1 ( x 30 ( z 0 + Ps 2 π ) ) ( R 1 r o θ ( s ) )
ϕ F ( y * ( s , r o ) , s ) = ( λ 8 π 2 ) + 1 ( π ) ( y 2 * ( s , r o ) y 2 ' ) [ R 1 y 2 ' 2 + y 3 κ 2 + R 1 2 ϕ ' ( y 2 ' , y 3 κ y 2 ' ψ ̂ , s ) ] dy 2 '
ϕ ' ( y 2 , y 3 , s ) = ( ϕ s + y 2 2 + R 1 2 R 1 ϕ y 2 + y 2 y 3 R 1 ϕ y 3 )
ϕ ' ( y 2 , y 3 , s ) = λ r e σ KN A o 2 ( A o 2 s + y 2 2 + R 1 2 R 1 A o 2 y 2 + y 2 y 3 R 1 A o 2 y 3 )
A o 2 ( y ; θ ) = F ̂ 1 { F ̂ [ M 2 I ( M y ; θ ) ] I in μ ˜ in ( λ R 2 u M ) OTF det ( u M ) ( 1 + 2 π ( r e λ 2 R 2 M σ KN ) u 2 ) }
y 3 κ y 2 ψ = P 2 π ( ψ + ψ tan ( ψ ) R 1 y 2 R 1 )
y 3 * s r o = P 2 π ( ψ ̂ + ψ ̂ tan ( ψ ̂ ) y 2 * s r o R 1 )
Δ ( r o ) = λ 8 π 3 0 π sin ω [ 0 π 2 ϕ ( y ; θ ) δ ( y n D s ) d y ]
Δ ( r o ) = λ 8 π 3 0 π q ( y ; r o ) 2 ϕ ( y ; θ ) d y
q ( y , r o ) = x 3 y 3 ( r o θ y 2 ) 2 + ( x 3 y 3 ) 2
Δ ( r o ) = λ 8 π 3 0 π q 2 ϕdβ

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