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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 16 — Aug. 8, 2005
  • pp: 6000–6014
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X-Ray cone-beam phase tomography formulas based on phase-attenuation duality

Xizeng Wu and Hong Liu  »View Author Affiliations


Optics Express, Vol. 13, Issue 16, pp. 6000-6014 (2005)
http://dx.doi.org/10.1364/OPEX.13.006000


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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

1. Introduction

A goal of many research efforts in the current x-ray diagnostic imaging is to greatly enhance the lesion-detection sensitivity. While the digital imaging has currently been a major effort toward to this goal in x-ray diagnostic imaging, but its potential for improving lesion-detection sensitivity is limited by the tissue’s radiological contrast per se. In the over100-years history of x-ray diagnostic imaging, the biological tissue contrast has always been based on the biological tissue’s differences in x-ray attenuation until quite recently. Recently a new class of tissue contrast, that is, the tissue phase-contrast brings new momentum into x-ray imaging research [1–13

1. P A. Snigirev, 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]

]. In fact when x-ray traverses body parts the tissues cause x-ray phase changes as well. The amount of the phase change is determined by biological tissue dielectric susceptibility, or equivalently, by the refractive index of the tissue. The refractive index n(r o ) of a tissue voxel at r o is complex and equal to n(r o ) = 1-Δ(r o ) + iB(r o ), where Δ(r o ) is the refractive index decrement at k o , and it is responsible for x-ray phase shift. In contrast, the imaginary part B(r o )of n(r o ) is responsible for x-ray attenuation. When x-ray traverses tissues, the x-ray phase change by tissue is given by [1–3

1. P A. Snigirev, 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]

]

ϕ=2πλΔ(s)ds
(1)

where λ is the x-ray wavelength, and the integral is over the ray path. In other words, the phase change is proportional to the projection of refractive index decrement Δ. It is interesting to note that tissue’s Δ is much larger than B. We have estimated Δ and B for biological tissues, and found that the issue’s Δ (10-6-10-8) is about 1000 times greater thanB (10-9-10-11) for x-rays of 10 keV to 150keV[13

13. X. Wu, A. Dean, and H Liu, “X-ray diagnostic techniques,” in Biomedical photonics handbook, T. VoDinh ed., Chapter 26, p.2626–34, (CRC Press, Tampa, Florida, 2003).

]. Therefore, the differences in x-ray phase shifts between different tissues are about 1000 times greater than the differences in the projected linear attenuation coefficients. Hence phase-contrast imaging may greatly increase the lesion-detection sensitivity for x-ray imaging.

The x-ray phase tomography is a technique for three-dimensional imaging of tissue refractive index decrement Δ(r o ). With the high phase-sensitivity x-ray phase tomography has a great potential for three-dimensional imaging of soft tissues. In order to develop x-ray phase tomography, it is essential to develop the basic image reconstruction formula, and then reconstruction algorithms can be further developed from the basic reconstruction formula. Bronnikov first developed a basic reconstruction formula for phase tomography of objects with weak attenuations [14–15

14. A. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Optics Commun. 171, 29–244 (1999) [CrossRef]

]. His reconstruction formula can be applied to phase tomography of very thin biological samples imaged by almost parallel x-ray beams from synchrotron-based sources. Assuming a parallel x-ray beam and pure phase objects, his reconstruction formulas require only a single phase-contrast image per tomographic view (projection). For objects with substantial attenuation, he assumed that the attenuation for a given tomographic view is a constant across the field of view, but the attenuation may vary for different views. To apply his reconstruction formula to these cases, one needs to acquire two images per view: one is the “contact print” of the object and another is the phase-contrast image for the same view [15

15. A. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19, 472–480 (2002). [CrossRef]

].

In general for phase retrieval one need acquire at least two images that are separated by a distance in x-ray propagation direction [3

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]

, 5

5. D. Paganin and K. Nugent, “Noninterferometric phase imaging with partialcoherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998). [CrossRef]

, 7–9

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

, 11

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]

]. According to Eq. (1) the phase-image for a given tomographic view provides the ray-sum map of Δ(ro ) needed for 3-D image reconstruction. While a phase-contrast image for a tomographic view contains contributions from both the ray sums of Δ(ro ) and B(ro ), but acquisition of the single phase-contrast image is not enough to disentangle the ray-sums of Δ(ro ) and B(ro ) for phase retrieval. For this reason in general at least two images separated by a distance in x-ray propagation direction are needed for the phase retrieval for a given view. This requirement of at least two acquired images for each tomographic view makes the implementation of phase tomography much more difficult than for the projectional phase-contrast imaging, because of the limitations imposed by body motion and radiation dose associated with the large number of tomographic views [9

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.

]. In addition this multiple-image requirement impedes the dynamical imaging such as cardiac imaging.

It should be pointed out there is some exception of the multiple-image requirement for phase retrieval. For a pure phase objects with negligible attenuation a single phase-contrast image is enough for the phase-retrieval. However in many practical cases, and especially in clinical application, object attenuation is significant and cannot be ignored at all for phase-retrieval. The average ray-sum of the linear attenuation coefficients μ(r o ) of an abdomen along an anterior-posterior projection is about 4 to 6 for x-rays of 60 keV, and even for small animal the average ray-sum of μ(ro ) can be as high as 1 at this energy. Another special case is the case of homogeneous objects of single-material. In this case the ratio of Δ and B is a constant over the object. In this case hence the phase map and attenuation-map of the object are determined by the thickness-map, hence one single phase-contrast image is enough to retrieve the thickness-map [16

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]

]. However interesting objects for tomography are all inhomogeneous and this approach cannot be applied.

2. Phase-attenuation duality and cone beam tomography for soft tissues

Consider a soft-tissue object irradiated by a monochromatic x-ray of wavelength λ emitted from a point source S that is moving on a circle, as is shown in Fig. 1. Note a fixed coordinate frame (x 1,x 2,x 3) is attached with the object. As is the convention in the cone-beam CT literature [18–19

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

], we denote a virtual detector plane passing the origin as the (y 2,y 3)-plane shown in Fig. 1. This virtual detector plane represents the real 2-D imaging detector plane a distance of R2 downstream from the origin. In the cone beam scan geometry, the virtual detector plane is kept perpendicular to the line joining the origin and the source moving along the circle.

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.

A cone-beam transform g(θ,y) of Δ(ro ) along an x-ray is given by:

g(θ,y)=0Δ(R1θ+t(yR1θ)yR1θ)dt
(2)

θ=(cosβ,sinβ,0),θ=(sinβ,cosβ,0),e3=0,0,1
(3)

In addition, the position vector in virtual detector plane D is denoted by:

y=y2θ+y3e3
(4)

In fact the cone-beam transform g(θ,y) is equal to a measurement of the line integral of refractive index decrement Δ(ro )along the ray. Obviously all the line integrals acquired for a view θ represent the object’s phase-map for this view. According to Eq. (1) the phase-map ϕ(y;θ) is given by:

ϕ(y;θ)=2πλg(θ,y)
(5)

Therefore the cone-beam transforms g(θ,y) can be found by the phase retrieval of ϕ(θ,y ) and a three-dimensional map of Δ(ro ) can be reconstructed if ϕ(θ,y ) is retrieved for all the

I˜(uM;θ)=Iinμ˜in(λR2uM)OTFdet(uM)×
×{cos(πλR2u2M)(F̂(Ao2)iλR2MuF̂(ϕAo2))+2sin(πλR2u2M)(F̂(Ao2ϕ)iλR2MuF̂(Ao2))}
(6)
Fig. 2. The moving frame attached to the virtual detector plane.

where the symbol denotes the 2-D Fourier transform and Iin is the intensity of the incident x-ray upon the object, and u is the spatial frequency vector. Here we denote the distance from a x-ray source to the plane of the imaged object as R1, and the distance from object plane to the detector plane as R2, hence the geometric magnification factor M is equal to (R1 + R2)/ R1. Interesting readers are referred to ref. [11

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]

] for details of derivation of Eq. (6). In Eq. (6) Ao2 represents tissue’s attenuation:

Ao2(y;θ)=Exp(4πλ0B(R1θ+t(yR1θ)yR1θ)dt)
=Exp(0μ(R1θ+t(yR1θ)yR1θ)dt)
(7)

where μ(R 1 θ + t(y - R 1 θ)/|y - R 1 θ|), is the linear attenuation coefficient along the cone beam R 1 θ + t(y - R 1 θ)/|y - R 1 θ|) is the imaginary part of the refractive index. In Eq. (6) OTF det(u ) is the detector’s spatial frequency response, which is determined by detector pixel 11

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]

].

Fig. 3. Schematic of an undulator source.

It should be noted that the effects on image intensity of spatial coherence of the incident x-ray is accounted for by a multiplication factor μ˜in(λR2uM) in Eq. (6). For the perfectly coherent incident x-ray one has μ˜in(λR2uM)=1 for all |λR2uM|. For partially coherent incident x-ray μ˜in(λR2uM)1 and decreases with increasing |λR2uM|. X-ray tubes are incoherent sources, hence we can apply the van Cittert-Zernike theorem to them [23

23. M. Born and E. Wolf, Principle of Optics, 6th ed. (Pergamon, Oxford1980).

] and we found [11

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]

]:

μ˜in(λR2uM)=OTFG.U.(uM)
(8)

where OTFG.U.(uM) the optical transfer function (OTF) for the geometrical unsharpness associated with a finite focal spot [23

23. M. Born and E. Wolf, Principle of Optics, 6th ed. (Pergamon, Oxford1980).

]. For example, for an anode source with a round focus spot we found [11

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]

]

μ˜in(λR2uM)=MTFG.U.(uM)=2J1(πf(M1)uM)πf(M1)uM
(9)

where f is the diameter of the focus spot, and J1(x) is a Bessel function of the 1st kind. Currently synchrotron radiation such as the undulator radiation is the most common x-ray source for phase contrast imaging experiments. An undulator source is an insertion device installed along the electron beam path in synchrotron, in which a spatially-periodic magnetic field makes electrons to deflect periodically along the beam path as is shown schematically in Fig. 3 [24

24. H. Wiedemann, Synchrotron Radiation, (Springer-verlag, Berlin Heidelberg2003). [CrossRef]

]. In undulators the electron’s deflection is in order of microns. Due to the interferences of x-ray waves emitted by wiggling electrons under each pole of the undulator, a bright and narrow forward-cone of x-ray beam are generated with photon-energies in resonance-harmonics [24

24. H. Wiedemann, Synchrotron Radiation, (Springer-verlag, Berlin Heidelberg2003). [CrossRef]

]. Different from the incoherent anode sources, an undulator source is partially coherent at the source level, that is why undulator radiation is narrowly collimated. Hence the van Cittert-Zernike theorem cannot be applied to undulator radiation in general.

For a synchrotron a fully damped and stored electron beam can be well approximated by a Gaussian distribution in transverse position and velocity-divergence [24

24. H. Wiedemann, Synchrotron Radiation, (Springer-verlag, Berlin Heidelberg2003). [CrossRef]

]. Under the Gaussian-beam assumption we found that

μ˜in(λR2uM)=exp[(((M1)u1σ1M)2+((M1)u2σ2M)2)]
(10)

Here σ 1 denotes total effective horizontal-axis electron beam size, which accounts for the electron-beam’s divergence as well [24

24. H. Wiedemann, Synchrotron Radiation, (Springer-verlag, Berlin Heidelberg2003). [CrossRef]

]. The total effective vertical-axis electron beam size is denoted by σ 2.

In many applications including clinical imaging one has πλR2u2M1. For these cases Eq. (6) is reduced to

I˜(uM;θ)=Iinμ˜in(λR2uM)OFTdet(uM){(F̂(Ao2)iλR2Mu.F̂(ϕAo2))+2πλR2u2MF̂(Ao2ϕ)}
(11)

Since both the phase-map ϕ(y ;θ) and attenuation-map Ao2(y ;θ) are unknown, one cannot solve for ϕ(y ;θ) by means of computing I˜(uM;θ) from the image intensity measurement for the view θ. That is why one needs in general at least two acquired images for retrieving a phase-map ϕ(y ;θ). As we pointed out earlier this requirement of multiple-image for each

As the tissue phase arises from the x-ray coherent scattering from tissue, the tissue attenuation arises from three x-ray-tissue interactions: the photoelectric absorption, coherent scattering and incoherent scattering [13

13. X. Wu, A. Dean, and H Liu, “X-ray diagnostic techniques,” in Biomedical photonics handbook, T. VoDinh ed., Chapter 26, p.2626–34, (CRC Press, Tampa, Florida, 2003).

, 25

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

] in the x-ray energy range employed for medical imaging. Soft tissues are oftentimes imaged in clinical exams and experiments at synchrotron centers. Soft tissues are composed by atoms of predominantly the light elements with Z<10, except trace heavier elements in soft tissue [26

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

]. Quite recently in a brief communication we proposed a new concept of phase-attenuation duality and a new phase-retrieval approach for soft tissues [17

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

], by means of which only a single phase-contrast image is acquired for phase retrieval for objects with any strong attenuation such as thick body parts. As is shown in [17

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

], ϕ(y ;θ) and attenuation-map Ao2(y ;θ) of soft tissues are really determined by the same map of the projected electron density ρe,p (y ;θ).

Considering the different origins in x-ray-tissue interactions for soft-tissue phase (from coherent x-ray scattering) and attenuation (from incoherent x-ray scattering), we called this complementary relationship between phase and attenuation as the phase-attenuation duality [17

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

]. The phase-attenuation duality roots deeply in the complementary quantum-mechanical relationship between the atomic form factor for x-ray coherent scattering and the atomic incoherent scattering function. More specifically, the projected electron density ρe,p (y ;θ) is defined as

ρe,p(y;θ)=0ρe(R1θ+t(yR1θ)yR1θ)dt)
(12)

where ρe (ro ) is tissue electron density at position ro . The phase-attenuation duality manifests as [17

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

]:

ϕ(y;θ)=λreρe,p(y;θ)Ao2(y;θ)=exp(σKNρe,p(y;θ))
(13)

In Eq. (13

13. X. Wu, A. Dean, and H Liu, “X-ray diagnostic techniques,” in Biomedical photonics handbook, T. VoDinh ed., Chapter 26, p.2626–34, (CRC Press, Tampa, Florida, 2003).

)re is the classical electron’s radius, and σKN is the total cross section for x-ray Compton scattering from a single free electron:

σKN=2πre2{1+ηη2[2(1+η)1+2η1ηlog(1+2η)]+12ηlog(1+η)(1+3η)(1+2η)2}
(14)

where η=Ephotonmec2,, here we denote the photon energy of the primary x-ray beam by Ephoton , and mec 2 is the rest electron energy and equal to 511 keV [25

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

].

Taking advantage of phase-attenuation duality, we substitute the duality condition Eq. (13) back into Eq. (11) for simplification. Note that the duality condition Eq. (13) means that

2πλR2u2MF̂(Ao2ϕ)iλR2Mu.F̂(ϕAo2)=2πreλ2R2u2MσKNF̂(Ao2)
(15)

Combining Eq. (11), Eq. (13) and Eq. (15), after a straight-forward calculation we found:

I˜(uM;θ)=Iinμ˜in(λR2uM)OTFdet(uM){1+2πreλ2R2u2MσKN}F̂(Ao2)
(16)

Rewriting Eq. (16) explicitly we found eventually the phase retrieval formula when phase-attenuation duality holds:

ρe,p(y;θ)=1σKNloge(F̂1{F̂[M2I(My;θ)]Iinμ˜in(λR2uM)OTFdet(uM)(1+2π(reλ2R2MσKN)u2)})
(17)

where I(My;θ) is the measured phase-contrast image intensity. Note that Eq. (17) generalize the phase retrieval formula in [17

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

] by including the effects of partial coherence and detector resolution. Equivalently, we can rewrite Eq. (17) in terms of the retrieved phase:

ϕ(y;θ)=λreσKNloge(F̂1{F̂[M2I(My;θ)]Iinμ˜in(λR2uM)OTFdet(uM)(1+2π(reλ2R2MσKN)u2)})
(18)

Eq. (18) is a key result of this work, and it forms the basis of phase imaging based on the phase-attenuation duality.

3. Cone beam reconstruction formula of Δ(ro )

Currently phase tomography experiments mostly involve small size objects. For small objects such as small animals or small human body parts such as breast, one can apply the Feldcamp-Davis-Kress (FDK) reconstruction algorithm for 3-D tomography. In this section we combine the new phase-retrieval formula of Eq. (18) with the FDK reconstruction algorithm to derive a reconstruction formula for 3-D cone-beam phase tomography.

Δ(ro)=λ4π02πR12(R12roθ)2ρρv(y2y'2)ϕ(y'2θ+y3e3;θ)R1dy'2R12+(y'2)2+y32,
(19)

y2=R1R1roθroθy3=R1R1roθx3
(20)

Let us briefly explain the meaning of terms in Eq. (19

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

) as follows. In phase tomography the ray-sums for each given tomographic view are given by the retrieved phase ϕ(y ;θ) of Eq. (18

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

) for that view. The retrieved phase-maps are first weighted by the factor R1R12+(y'2)2+y32 to account for the fan-beam coordinates conversion from the parallel beam coordinates. The weighted data are then filtered row by row along horizontal lines in by means of one-dimensional convolution in y´2 from -ρ to ρ(the data are assumed be zero for |y´2|> ρ). The convolution kernel v(y 2) in Eq. (19) is the conventional ramp filter in 2D tomography [18–19

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

]. Note that the filtering is applied view by view for all the views, though different views are really not co-planar. The front factor R12(R12roθ)2 in Eq. (19) is the x 3 -axis (Fig. 2) away from the mid-plane the reconstruction is approximate, though the average along x 3 - direction is exact 18–19

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

]. The reason of this inexactness roots in the incomplete 3-D Radon-transform data limited by the co-planar source orbit associated with

Fig. 4. Sampled Radon space for the FDK reconstruction algorithm.

FDK algorithm. In order to understand this note that a three-dimensional Radon transform of Δ(ro ) is a plane-integral of Δ(ro ):

Δ̂(s,n)=Δ(ro)δ(rons)dro
(21)

where n is the plane’s normal vector, and δ(ron - s) is a 1-D Dirac-function. As is shown in [18–19

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

], Eq. (21) represents a 3-D Radon transform of Δ(ro ) along the plane ron = s. Hence a 3-D Radon transform Δ^ (s,n ) is represented by a point (s,n )in the space of Radon transforms, or the Radon space. Fig. 4 plots the all such data-points in Radon space that aare sampled by a source in a circular trajectory and the figure shows clearly where the Radon transform data are missing [18–19

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

]. In spite of this the FDK algorithm is the most widely used algorithm for cone beam attenuation-based tomography of small-size objects, such as small animal and human breast. Therefore by means of our duality-based phase retrieval approach Eqs. (18–19) provide good reconstruction formulas of cone beam phase tomography for small animal and human breast.

Although current phase tomography experiments mostly involve small-size objects only, but for sake of completeness let us briefly comment on cone-beam phase-tomography for large objects. As is well known, currently clinical CT imaging with multislice scanners undergoes explosive development with advent of commercially available 64-slice CT scanners. Obviously the helical scanning approach for large objects can be applied to the phase tomography as well. Moreover, recently an exact filtered backprojection reconstruction formula for a helical cone beam CT has been discovered by Katsevich [20–22

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

], and its implementation and generalization is currently an rapidly evolving research area of the attenuation-based tomography. In this work we combine our duality-based phase retrieval approach with Katsevich’s formulas [20–22

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

] to derive a phase tomography formula valid for helical scanning of large objects. Consider now that the source traces a helix of radius R 1 around the x 3 -axis of Fig. 1, and the source’s position is specified by a (s)as is shown in Fig. 5:

a(s)=[R1cos(s),R1sin(s),x30+Ps2π]
(22)

where P is the pitch of this helical scanning. Obviously s denotes the source-rotation angle. According to Katsevich’s original formula, a three-dimensional object function f(ro )can reconstructed from its cone beam projections acquired as the point-source traces a helix:

f(ro)=(12π2)IPI(ro)1roa(s)02πqg(Θ(s,ro,γ),a(q))q=ssinγds
(23)
Fig. 5. Helical scanning and the PI-segment.

In Eq. (23) we assume that the point-source traces a helical trajectory {a (s)}as is shown in Fig. 5. Let’s first briefly explain what Eq. (23) does for reconstruction. We start with g(Θ ,a(s)) which denotes a cone beam projection of f(ro ) with a ray emitted from the source at a (s) and the ray-direction unit-vector Θ→. Eq. (23) says that one takes the derivative of g(Θ ,a (s)) with respect to source angle s for the same Θ; and then convolve the derivatives 1sinγ[20–22

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

]. Therefore the expression inside the square bracket in Eq. (23) accounts for the differentiating and filtering of the cone-beam projection data. The filtered projection data are then multiplied by a factor of 1roa(s) for distance weighting to account for transverse magnification associated with the cone beam projection. Finally the integral over source angle s is to sum the backprojection of the filtered projections over the different views. Note that the integral interval is IPI (ro ), which is an interval [sb (ro ),st (ro )]of source angle s such that sb (ro ) and st (ro ) are the endpoints of the PI–line segment that passing ro [20–22

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

]. By definition a PI-line segment means the ‘bottom” endpoint sb (ro ) and the “top” endpoint.st (ro ) are separated by less than a helix turn. Putting together, the Katsevich formula Eq. (23) provides an exact filtered backprojection reconstruction formula for cone beam reconstruction.

We can apply Eq. (23) for cone beam phase tomography as well. The source’s position in the rest frame is specified by a (s), and its orthogonal projection onto the(x 1,x 2)-plane is denoted by R 1 θ(s) in according to similar notation adopted in Fig. 1. In this way the coordinates frame on the virtual detector is set up such that the orthogonal projection of the source point is the origin and {θ (s),e 3(s)} is the basis in the same way as shown in Fig. 2. By means of Katsevich formula Eq. (23) we found the reconstruction formula for cone beam phase tomography valid for helical scanning:

Δ(ro)=(λ8π2)sbs1(R1roθ(s))ϕF(y*(s,ro),s)ds
(24)

where y * (s,ro ) is the projection of an object point ro onto the virtual detector plane when the source is at a (s) :

y*sro=y2*(s,ro)θ+y3*(s,ro)e3
(25)

Generalizing Eq. (20) for the circular source-orbit to the case for the helical orbit we found that: for helical scanning

y2*(s,ro)=R1(roa(s))θ(s)(roa(s))θ(s)=R1roθ(s)(R1roθ(s))
y3*(s,ro)=R1(roa(s))e3(s)(roa(s))θ(s)=R1(x30(z0+Ps2π))(R1roθ(s))
(26)

Note that in Eq.(24) ϕF (y *(s,ro ), s) is the filtered phase defined as

ϕF(y*(s,ro),s)=(λ8π2)+1(π)(y2*(s,ro)y2')[R1y2'2+y3κ2+R12ϕ'(y2',y3κy2'ψ̂,s)]dy2'
(27)

where ϕ´ (y 2,y 3, s)is the derivative of the retrieved phase with respective to s at constant ray direction, which is given by [22

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

]

ϕ'(y2,y3,s)=(ϕs+y22+R12R1ϕy2+y2y3R1ϕy3)
(28)

Hence ϕF (y *(s,ro ),s) of Eq. (27) is essentially the phase derivative convolved with an one-dimensional Hilbert filter in y 2. Taking advantage of phase-attenuation duality, we calculate ϕ´(y 2,y 3,s) in terms of Ao2 as well:

ϕ'(y2,y3,s)=λreσKNAo2(Ao2s+y22+R12R1Ao2y2+y2y3R1Ao2y3)
(29)

In Eq. (29) Ao2 as can be determined directly from Eq. (17) we have:

Ao2(y;θ)=F̂1{F̂[M2I(My;θ)]Iinμ˜in(λR2uM)OTFdet(uM)(1+2π(reλ2R2MσKN)u2)}
(30)

It should be noted that Eq. (29) is more computationally efficient that Eq. (28), since unlike Eq. (18) there is no need to calculate logarithms in Eq. (30). Finally we should explain how to define y 3κ(y 2, ψ^) in Eq. (27). As is explained by Katsevich, the one-dimensional convolution is performed along a so-called κ–line in virtual detector plane and (y 2,y 3κ)is a point on a Κ–line [20–22

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

]:

y3κy2ψ=P2π(ψ+ψtan(ψ)R1y2R1)
(31)

and angle ψ is an offset to source angle s. Therefore κ–lines are labeled by offset angle ψ. On the other hand, the angle ψ^ in y 3κ(y 2,ψ^) is of the smallest |ψ| that labels the κ-line passing through the object-point’s cone beam projection onto virtual detector plane. In other words, ψ^ is of the smallest |ψ| satisfies the following equation [22

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

]:

y3*sro=P2π(ψ̂+ψ̂tan(ψ̂)y2*sroR1)
(32)

4. Discussion and conclusions

The conventional computed tomography a provides three-dimensional map of tissue’s linear attenuation coefficients, and it finds widely spread clinical applications [18–20

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

]. In contrast to conventional computed tomography, x-ray phase tomography provides a three-dimensional map of the tissue refractive index decrement Δ(ro ). With the notion of high sensitivity associated with tissue phase changes, as discussed at the beginning of this paper, x-ray phase tomography is expected to be much more sensitive than the conventional CT, hence it has a great potential for clinical applications.

To develop x-ray phase tomography it is important to establish the phase-tomography reconstruction formulas, since such formulas form the basis for various reconstruction algorithms that can be developed. As is shown earlier, for phase tomography at least two acquired images for each tomographic view are needed. This requirement increases the radiation dose to body parts, and makes the scanner design much more difficult as explained earlier. That is why in Bronnikov’s approach of phase tomography one has to acquire two images per view, since one need acquire a “contact print” image in addition to the phase-contrast image [15

15. A. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19, 472–480 (2002). [CrossRef]

]. To alleviate this difficulty for phase tomography implementation, we applied the idea of phase-attenuation duality to phase tomography, and in this new approach one needs only one image per view for three-dimensional tomography reconstruction. In this work we presented for the first time the detailed derivation of the phase-retrieval formula based the duality. Furthermore in this derivation we have explicitly included the effects of partial coherence of the incident x-rays. We hope this inclusion of the partial coherence effects can be more useful for phase retrieval works involving the synchrotron and anode sources. Combining the derived phase-retrieval formula with the FDK cone-beam reconstruction algorithm we derived the reconstruction formula Eqs. (18–19) for the three-dimensional phase tomography of small objects. For phase tomography of large objects we combined our duality-based phase retrieval approach with Katsevich’s reconstruction formulas [20–22

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

] to derive the phase tomography formulas, that is, Eq. (24), Eq. (27) and Eqs. (29–30) for helical scanning of large objects.

Fig. 6. Parallel beam tomography with a source orbiting along a circle.

Δ(ro)=λ8π30πsinω[0π2ϕ(y;θ)δ(ynDs)dy]
(33)

Note that Eq. (33) is essentially the 3-D inverse Radon transform formula [19

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

]. The phase map ϕ(y ;θ) in Eq. (33) is retrieved according to Eq. (18) in the same way as in cone-beam phase tomography described earlier. In Eq. (33) n = (cosθ∙sinω,sinθ∙sinω,cosω), s = ron , the unit vectors n D = (sin ω,cos ω), as is depicted in Fig. 6. Note also that while the FDK reconstruction is a filtered back projection algorithm, Eq. (33) is not. But following the approach of Bronnikov [14–15

14. A. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Optics Commun. 171, 29–244 (1999) [CrossRef]

] one can rewrite the 3-D inverse Radon transform in a filtered backprojection form by means of the integration over angle ω,

Δ(ro)=λ8π30πq(y;ro)2ϕ(y;θ)dy
(34)

where the 2-D convolution kernel (filter) is defined as

q(y,ro)=x3y3(roθy2)2+(x3y3)2
(35)

Obviously Eq. (35) can be rewritten as

Δ(ro)=λ8π30πq2ϕdβ
(36)

However note that in Eq. (36) the convolution symbol ⊗ denotes the two-dimensional convolution, rather the convolutions in Eq. (19) and Eq. (27) for cone-beam tomography are one-dimensional convolutions.

In summary in this work we first presented a detailed derivation of the phase-retrieval formula based on phase-attenuation duality. Moreover we have included the effects of x-ray source coherence and detector resolution into the phase-retrieval formula as well. As only a single image is needed for performing the phase retrieval in this approach, this new approach has great advantages for implementation of phase tomography. We then combine our phase-retrieval formula with the FDK cone beam reconstruction algorithm to provide a three-dimensional phase tomography formula for soft tissue objects of relatively small size, such as small animals or human breast. For large objects we briefly discussed how to apply Katsevich’s cone beam reconstruction formula to the helical phase tomography for large objects as well.

Acknowledgments

The research is supported in part by grants from the National Institute of Health (CA104773, EB002604). The authors would like to acknowledge Mr. Hong-gang Liu for the assistance in preparation of the figures. The authors would like to acknowledge the support of Charles and Jean Smith Chair endowment fund as well.

References and links

1.

P A. Snigirev, 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. 68, 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 and 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.2626–34, (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, Oxford1980).

24.

H. Wiedemann, Synchrotron Radiation, (Springer-verlag, Berlin Heidelberg2003). [CrossRef]

25.

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

26.

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).

OCIS Codes
(030.1670) Coherence and statistical optics : Coherent optical effects
(340.7440) X-ray optics : X-ray imaging

ToC Category:
Research Papers

History
Original Manuscript: June 28, 2005
Revised Manuscript: July 20, 2005
Published: August 8, 2005

Citation
Xizeng Wu and Hong Liu, "X-Ray cone-beam phase tomography formulas based on phase-attenuation duality," Opt. Express 13, 6000-6014 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-16-6000


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References

  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).

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