x-ray cone-beam phase tomography formulas

7/31/2019 X-Ray Cone-beam Phase Tomography Formulas

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X-Ray cone-beam phase tomography formulasbased on phase-attenuation duality

Xizeng Wu

Department of Radiology, University of Alabama at Birmingham, Birmingham, AL 35233, USA

[email protected]

Hong Liu

Center for Bioengineering and School of Electrical and Computer Engineering, University of Oklahoma, Norman,OK 73019, USA

[email protected]

Abstract: We present a detailed derivation of the phase-retrieval formulabased on the phase-attenuation duality that we recently proposed in previousbrief communication. We have incorporated the effects of x-ray sourcecoherence and detector resolution into the phase-retrieval formula as well.Since only a single image is needed for performing the phase retrieval bymeans of this new approach, we point out the great advantages of this newapproach for implementation of phase tomography. We combine our phase-

retrieval formula with the Feldkamp-Davis-Kresss (FDK) cone-beamreconstruction algorithm to provide a three-dimensional phase tomographyformula for soft tissue objects of relatively small sizes, such as smallanimals or human breast. For large objects we briefly show how to applyKatsevichs cone-beam reconstruction formula to the helical phasetomography as well.

2005 Optical Society of America

OCIS codes: (340.7440) X-ray imaging); (030.1670) Coherent optical effects

References and links

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

2. S. Wilkins, T. Gureyev, D. Gao, A. Pogany and A. Stevenson, Phase contrast imaging using polychromatichard x-ray, Nature 384, 335-338 (1996).

3. K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, Quantitative phase imaging using hard xrays, Phys. Rev. Lett. 77,2961-2965, (1996).

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).

5. D. Paganin and K. Nugent, Noninterferometric phase imaging with partialcoherent light, Phys. Rev. Lett. 80,2586-2589 (1998).

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 designconsiderations," Med. Phys. 30, 2169-2179 (2003).

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 imagingsystems, " Med. Phys. 31, 997-1002 (2004).

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).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).13. X. Wu, A. Dean and H Liu, "X-ray diagnostic techniques," inBiomedical 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)

(C) 2005 OSA 8 August 2005 / Vol. 13, No. 16 / OPTICS EXPRESS 6000

#7953 - $15.00 US Received 28 June 2005; revised 20 July 2005; accepted 24 July 2005
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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15. A. Bronnikov, Theory of quantitative phase-contrast computed tomography, J. Opt. Soc. Am. A 19, 472-480(2002).

16. D. Paganin, S. Mayo, T. Gureyev, P. Miller and S. Wilkins, Simultaneous phase and amplitude extraction froma single defocused image of a homogeneous object, J. Microsc. 206, 33-40 (2002).

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

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).20. A. Katsevich, Analysis of an exact inversion algorithm for spiral cone-beam CT, Phys. Med. Biol. 47, 2583-

2597 (2003).21. A. Katsevich, An improved exact filtered backprojection algorithm for spiral computed tomography,

Advances in Applied Mathematics, 35, 681-697 (2004).22. F. Noo, J. Pack and D. Heuscher, Exact helical reconstruction using native cone-beam geometry, Phys. Med.

Biol. 48, 3787-3818 (2003).23. M. Born and E. Wolf, Principle of Optics, 6th ed. (Pergamon, Oxford 1980).24. H. Wiedemann, Synchrotron Radiation, (Springer-verlag, Berlin Heidelberg 2003).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).

1. Introduction

A goal of many research efforts in the current x-ray diagnostic imaging is to greatly enhancethe lesion-detection sensitivity. While the digital imaging has currently been a major efforttoward to this goal in x-ray diagnostic imaging, but its potential for improving lesion-detection sensitivity is limited by the tissues radiological contrast per se. In the over100-yearshistory of x-ray diagnostic imaging, the biological tissue contrast has always been based onthe biological tissue's differences in x-ray attenuation until quite recently. Recently a newclass of tissue contrast, that is, the tissue phase-contrast brings new momentum into x-rayimaging research [1-13]. In fact when x-ray traverses body parts the tissues cause x-ray phasechanges as well. The amount of the phase change is determined by biological tissue dielectricsusceptibility, or equivalently, by the refractive index of the tissue. The refractive index

)( orn of a tissue voxel at or is complex and equal to n(ro ) =1(ro )+ i(ro ) , where (ro ) is

the refractive index decrement at or , and it is responsible for x-ray phase shift. In contrast, the

imaginary part(ro )ofn(ro ) is responsible for x-ray attenuation. When x-ray traversestissues, the x-ray phase change by tissue is given by [1-3]

= dss)(2

(1)

whereis the x-ray wavelength, and the integral is over the ray path. In other words, thephase change is proportional to the projection of refractive index decrement . It isinteresting to note that tissues is much larger than. We have estimated and forbiological tissues, and found that the issue's (10-6-10-8) is about 1000 times greater than(10

-9-10

-11) for x-rays of 10 keV to 150keV[13]. Therefore, the differences in x-ray phase

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

The x-ray phase tomography is a technique for three-dimensional imaging of tissue

refractive index decrement (ro ). With the high phase-sensitivity x-ray phase tomographyhas a great potential for three-dimensional imaging of soft tissues. In order to develop x-rayphase tomography, it is essential to develop the basic image reconstruction formula, and thenreconstruction algorithms can be further developed from the basic reconstruction formula.Bronnikov first developed a basic reconstruction formula for phase tomography of objects




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with weak attenuations [14-15]. His reconstruction formula can be applied to phasetomography of very thin biological samples imaged by almost parallel x-ray beams fromsynchrotron-based sources. Assuming a parallel x-ray beam and pure phase objects, hisreconstruction formulas require only a single phase-contrast image per tomographic view(projection). For objects with substantial attenuation, he assumed that the attenuation for agiven tomographic view is a constant across the field of view, but the attenuation may vary fordifferent 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-contrastimage for the same view [15].

In general for phase retrieval one need acquire at least two images that are separated by adistance in x-ray propagation direction [3, 5, 7-9, 11]. According to Eq. (1) the phase-image

for a given tomographic view provides the ray-sum map of (ro ) needed for 3-D image

reconstruction. While a phase-contrast image for a tomographic view contains contributions

from both the ray sums of (ro )and ro( ), but acquisition of the single phase-contrast imageis not enough to disentangle the ray-sums of(ro )and ro( ) for phase retrieval. For thisreason in general at least two images separated by a distance in x-ray propagation directionare needed for the phase retrieval for a given view. This requirement of at least two acquiredimages for each tomographic view makes the implementation of phase tomography muchmore difficult than for the projectional phase-contrast imaging, because of the limitationsimposed by body motion and radiation dose associated with the large number of tomographic

views [9]. In addition this multiple-image requirement impedes the dynamical imaging suchas cardiac imaging.

It should be pointed out there is some exception of the multiple-image requirement forphase retrieval. For a pure phase objects with negligible attenuation a single phase-contrastimage is enough for the phase-retrieval. However in many practical cases, and especially inclinical application, object attenuation is significant and cannot be ignored at all for phase-

retrieval. The average ray-sum of the linear attenuation coefficients ro( ) of an abdomenalong 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 ro( ) can be as high as 1 at this energy. Another special caseis the case of homogeneous objects of single-material. In this case the ratio of and is aconstant over the object. In this case hence the phase map and attenuation-map of the objectare determined by the thickness-map, hence one single phase-contrast image is enough toretrieve the thickness-map [16]. However interesting objects for tomography are all

inhomogeneous and this approach cannot be applied.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], for which only a single phase-contrast image is needed for phase retrieval for objects with any strong attenuation such asthick body parts. The duality-based phase retrieval formula was presented without derivationin that brief communication due to space limitation in [17]. Besides, only the projectionalimaging of soft-tissue was discussed in that work and it is natural to extend this new phase-

retrieval approach to three-dimensional imaging of refraction index-decrement (ro ). In this

work, we first present detailed derivation of the phase-retrieval formula based on the phase-attenuation duality. In addition we have included the effects of x-ray source coherence anddetector resolution into the phase-retrieval formula. We then combine this formula with theFDK cone-beam reconstruction algorithm [18-19] to derive a three-dimensional phasetomography formula for soft tissue objects of relatively small sizes, such as small animals or

small human body parts such as breast. For large objects the FDK reconstruction formula isinaccurate and one may have to use helical scanning for phase tomography. We combined ourduality-based phase retrieval approach with the exact cone-beam reconstruction algorithmdeveloped recently by Katsevich [20-22] to derive a phase tomography formula valid forhelical scanning of large objects. In the final section we compare these new reconstruction




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formulas of cone-beam phase tomography with that for the parallel beams in a previous work[27].

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

Consider a soft-tissue object irradiated by a monochromatic x-ray of wavelength emittedfrom a point source S that is moving on a circle, as is shown in Fig. 1. Note a fixed

coordinate frame (x1,x

2,x

3) is attached with the object. As is the convention in the cone-

beam CT literature [18-19], we denote a virtual detector plane passing the origin as the(y2 ,y3 )-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 sourcemoving along the circle.

Fig. 1. Cone-beam tomography with a source orbiting on a circle in(x1,x2 )-plane. Thefixed frame is attached with object, while the detector-plane denotes the 2-D imaging detector adistance 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(,y) of(ro ) along an x-ray is given by:

g(,y) = R1+ tyR1( ) yR1( )0

dt (2)

where , ,e3{ are the unit basis vectors of the moving frame attached to the virtualdetector plane D, which passes through the origin. The basis vectors are related to the source-

angle by (Figs. 1-2)

= cos,sin,0( ), = sin,cos,0( ), e3 = 0,0,1( ) (3)

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

y =y2 +y3e3 (4)

In fact the cone-beam transform g(,y) is equal to a measurement of the line integral of

refractive index decrement (ro )along the ray. Obviously all the line integrals acquired for a

view represent the objects phase-map for this view. According to Eq. (1) the phase-map

y;( )is given by:




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y;( )= 2

g(,y) (5)

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

In order to understand how to retrieve the phase of a soft-tissue object, lets recall howtissues phase distribution determines the phase-contrast image of an object. As is pointed out

Fig. 2. The moving frame attached to the virtual detector plane.

in our previous work, in addition to tissue phase distribution, the limited spatial coherence ofthe incident x-ray, tissue attenuation and the detector resolutions are all the important factorsdetermining the image intensity [8, 11]. In order to quantitatively account for the effects ofspatial coherence of the incident x-ray, we developed a theory based on the Wigner

distribution formalism [11]. We found that I u;( ), the Fourier transform of the phase-contrastimage intensity measured by the detector for a tomographic view , satisfies the followingequation [11]:

I

u

M;

=Iinin

R2u

M

OTFdet

u

M

cosR2u

2

M

F(Ao

2 ) iR2M

u F(Ao2 )

+ 2sin

R2u2

M

F(Ao

2)+ iR24M

u F(Ao2 )

(6)

where the symbol F(.) 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 thedistance from a x-ray source to the plane of the imaged object as R1, and the distance fromobject plane to the detector plane as R2, hence the geometric magnification factor M is equalto (R1+ R2)/ R1. Interesting readers are referred to ref. [11] for details of derivation of Eq. (6).

In Eq. (6) Ao2 represents tissues attenuation:

Ao2 (y;)=Exp

4

R1+ tyR1( ) yR1( )

0

dt

=Exp R1+ tyR1( ) yR1( )0

dt

(7)




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where R1+ tyR1( ) yR1( is the linear attenuation coefficient along the cone beamray, and R1+ tyR1( ) yR1( is the imaginary part of the refractive index. In Eq. (6)OTFdet u( ) is the detectors spatial frequency response, which is determined by detector pixelpitch and shape [11].

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 inR2u

M

in Eq. (6). For the perfectly

coherent incident x-ray one has inR2u

M

=1 for all

R2u

M. For partially coherent incident

x-ray inR2u

M

1 and decreases with increasing

R2u

M. X-ray tubes are incoherent

sources, hence we can apply the van Cittert-Zernike theorem to them [23] and we found [11]:

inR2u

M

= OTFG.U.

u

M

(8)

where OTFG.U.u

M

the optical transfer function (OTF) for the geometrical unsharpness

associated with a finite focal spot [23]. For example, for an anode source with a round focusspot we found [11]

inR2u

M

=MTFG.U.

u

M

=

2J1(f(M1)u/M)

f(M1)u/M(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-raysource for phase contrast imaging experiments. An undulator source is an insertion deviceinstalled along the electron beam path in synchrotron, in which a spatially-periodic magneticfield makes electrons to deflect periodically along the beam path as is shown schematically in

Fig. 3 [24]. In undulators the electrons deflection is in order of microns. Due to theinterferences of x-ray waves emitted by wiggling electrons under each pole of the undulator, abright and narrow forward-cone of x-ray beam are generated with photon-energies inresonance-harmonics [24]. Different from the incoherent anode sources, an undulator sourceis 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.




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Feldcamp-Davis-Kress (FDK) reconstruction algorithm for 3-D tomography. In this sectionwe combine the new phase-retrieval formula of Eq. (18) with the FDK reconstructionalgorithm to derive a reconstruction formula for 3-D cone-beam phase tomography.

Different from the parallel beam assumption made in previous works [14-15, 27] theobject is illuminated by a cone beam of x-ray from source orbiting in a circle enclosing theobject. In principle a planar circular orbit is an incomplete orbit for cone beam reconstruction.But for small cone angles the approximate FDK cone-beam reconstruction formula give

satisfactory tomographic reconstruction [18-19]. The idea of applying FDK reconstructionalgorithm to phase tomography is to first calculate the contribution to (ro ) from a given

viewby means of the conventional fan-beam filtered backprojection. Apparently thecontributions to(ro )from different viewscome from different planes, and these planes forms

a sheaf with its vertex at ro . Disregarding this fact the FDK algorithm sums all these

contributions over all different views to reconstruct a good approximation of(ro ). Consider a

x-ray source orbits a circle enclosing the object, as is shown in Fig. 1. The radius of the circle

is R1 , and the coordinates-frame and source angle are defined in the same way as shown in

Fig. 2. Note that the moving frame basis vectorsand are defined in Eq. (3). Applying theFDK cone-beam reconstruction formula to our case we found by means of Eq. (6) [18-19]:

(ro

) =

4

R12

R12 ro ( )

20

2

v y

2

y2

'

( )

y

2

'

+y3

e3

;

( )

R1 dy2'

R12 + y2

'( )2+y3

2

d, (19)

where y2 ,y3( ) are related to object point ro as

y2 =R1

R1 ro ro y3 =

R1

R1 ro x3 (20)

Let us briefly explain the meaning of terms in Eq. (19) as follows. In phase tomographythe

ray-sums for each given tomographic view are given by the retrieved phase y;( ) of Eq.(18) for that view. The retrieved phase-maps are first weighted by the factor

R1

R1

2 + y2

'

( )

2+y

3

2

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 iny2' from to(the data are assumed be zero for

y2' >). The convolution kernel v y2( ) in Eq. (19) is the conventional ramp filter in 2D

tomography [18-19]. Note that the filtering is applied view by view for all the views, though

different views are really not co-planar. The front factorR1

2

R12 ro ( )

2in Eq. (19) is the

distance-weighting factor to account for transverse magnification from the cone beamprojection. Although the FDK reconstruction is approximate, the reconstructed image in the

mid-plane is exact and identical to the 2-D fan-beam reconstruction. Along the 3x -axis (Fig.

2) away from the mid-plane the reconstruction is approximate, though the average along3

x -

direction is exact [18-19]. The reason of this inexactness roots in the incomplete 3-D Radon-transform data limited by the co-planar source orbit associated with




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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(ro )is a plane-integral of(ro ):

(s,n) = (ro )(ro n s)dro (21)

where n is the planes normal vector, and (ro n s) is a 1-D Dirac-function. As is shown in

[18-19], Eq. (21) represents a 3-D Radon transform of(ro ) along the plane ro n = s . Hence

a 3-D Radon transform (s,n) is represented by a point (s,n) in the space of Radontransforms, or the Radon space. Fig. 4 plots the all such data-points in Radon space that aaresampled by a source in a circular trajectory and the figure shows clearly where the Radontransform data are missing [18-19]. In spite of this the FDK algorithm is the most widelyused algorithm for cone beam attenuation-based tomography of small-size objects, such assmall animal and human breast. Therefore by means of our duality-based phase retrievalapproach Eqs. (18-19) provide good reconstruction formulas of cone beam phase tomographyfor 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 forlarge objects. As is well known, currently clinical CT imaging with multislice scannersundergoes explosive development with advent of commercially available 64-slice CT

scanners. Obviously the helical scanning approach for large objects can be applied to thephase tomography as well. Moreover, recently an exact filtered backprojection reconstructionformula for a helical cone beam CT has been discovered by Katsevich [20-22], and itsimplementation and generalization is currently an rapidly evolving research area of theattenuation-based tomography. In this work we combine our duality-based phase retrievalapproach with Katsevichs formulas [20-22] to derive a phase tomography formula valid forhelical scanning of large objects. Consider now that the source traces a helix of

radiusR1 around thex3 -axis of Fig. 1, and the sources position is specified by a s( )as is shownin Fig. 5:

a s( )= R1 cos s( ),R1 sin s( ),x30 + Ps

2

(22)

where Pis the pitch of this helical scanning. Obviously s denotes the source-rotation angle.

According to Katsevichs original formula, a three-dimensional object function f(ro )canreconstructed from its cone beam projections acquired as the point-source traces a helix:

f(ro )=1

22

1

ro a s( )

q0

2

g s,ro ,( ),a q( )( )q=sd

sin

IPI(ro )ds (23)




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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 inFig. 5. Lets first briefly explain what Eq. (23) does for reconstruction. We start

with g ,a s( )( ), which denotes a cone beam projection off(ro ) with a ray emitted from the

source at a s( ) and the ray-direction unit-vector . Eq. (23) says that one takes the derivativeof g ,a s( )( )with respect to source angle s for the same , and then convolve the derivatives

with a one-dimensional filter1

sin[20-22]. 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 of1

ro a 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 isIPI 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 PIlinesegment that passing ro [20-22]. By definition a PIline segment means the bottomendpoint sb ro( ) and the top endpoint st ro( ) are separated by less than a helix turn. Puttingtogether, the Katsevich formula Eq. (23) provides an exact filtered backprojectionreconstruction formula for cone beam reconstruction.

We can apply Eq. (23) for cone beam phase tomography as well. The sources position in

the rest frame is specified by a s( ), and its orthogonal projection onto the x1,x2( )-plane is

denoted byR1 s( ) in according to similar notation adopted in Fig. 1. In this way thecoordinates frame on the virtual detector is set up such that the orthogonal projection of the

source point is the origin and s( ),e3 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 beamphase tomography valid for helical scanning:

(ro ) =

82

1

R1 ro s( )( )F y* s,ro( ), s( )ds

sb

s t

(24)




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y3 y2 ,( )=P

2+

tan ( )y2

R1

(31)

and angle is an offset to source angle s . Therefore lines are labeled by offset angle.

On the other hand, the angle in y3 y2 ,( ) is of the smallest that labels the linepassing through the object-points cone beam projection onto virtual detector plane. In other

words, is of the smallest satisfies the following equation [22]:

y3*

s,ro( )=P

2+

tan ( )y2

*s,ro( )

R1

(32)

4. Discussion and conclusions

The conventional computed tomography a provides three-dimensional map of tissues linearattenuation coefficients, and it finds widely spread clinical applications [18-20]. In contrast toconventional computed tomography, x-ray phase tomography provides a three-dimensional

map of the tissue refractive index decrement (ro ). With the notion of high sensitivity

associated with tissue phase changes, as discussed at the beginning of this paper, x-ray phasetomography is expected to be much more sensitive than the conventional CT, hence it has agreat potential for clinical applications.

To develop x-ray phase tomography it is important to establish the phase-tomographyreconstruction formulas, since such formulas form the basis for various reconstructionalgorithms that can be developed. As is shown earlier, for phase tomography at least twoacquired images for each tomographic view are needed. This requirement increases theradiation dose to body parts, and makes the scanner design much more difficult as explainedearlier. That is why in Bronnikovs approach of phase tomography one has to acquire twoimages per view, since one need acquire a contact print image in addition to the phase-contrast image [15]. To alleviate this difficulty for phase tomography implementation, weapplied the idea of phase-attenuation duality to phase tomography, and in this new approachone needs only one image per view for three-dimensional tomography reconstruction. In thiswork we presented for the first time the detailed derivation of the phase-retrieval formulabased 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 coherenceeffects can be more useful for phase retrieval works involving the synchrotron and anodesources. Combining the derived phase-retrieval formula with the FDK cone-beamreconstruction algorithm we derived the reconstruction formula Eqs. (18-19) for the three-dimensional phase tomography of small objects. For phase tomography of large objects wecombined our duality-based phase retrieval approach with Katsevichs reconstructionformulas [20-22] to derive the phase tomography formulas, that is, Eq. (24), Eq. (27) and Eqs.(29-30) for helical scanning of large objects.

Another difference of this work from the previous ones is that we deal with the cone beamreconstruction rather that the parallel beams in [14-15, 27]. It should be noted that eventhough synchrotron radiation, though narrowly forward collimated, is not really parallel, andsynchrotron radiation beams have smaller divergence in the vertical direction than that inhorizontal direction [24], the cone beam reconstruction will be more accurate than thereconstruction based on parallel beam assumption. For tomography with x-ray tube sources,

obviously the cone beam reconstruction is necessary and the parallel beam reconstructioncannot be used for tomography reconstruction. When an x-ray source is far away from theobject, the beam can be treated parallel. For example, the Spring-8 synchrotron in Japan has abeam line of length of 1.3km and phase contrast imaging experiments has been performed inthis beam line [24]. For parallel beams the reconstruction formulas Eq. (18-19) still holds butthe magnification factor M becomes 1,




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

and the reduced complex degree of coherence inR2u

M

=1 for any values of R2u .

Furthermore, with a parallel beam source one can use an exact reconstruction formula basedon the 3-D radon transform inversion [14-15, 27]. This is because with parallel beams one

can find all the plane integrals of(ro )by the phase retrieval for all tomographic views. Withthis complete data set one can reconstruct the object by means of the 3-D inverse Radontransform [27]. In contrast with a cone-beam source orbiting in a circle, some plane integrals

of (ro )are not measured and the data set is incomplete for the exact reconstruction, as is

shown in Fig. 4.That is why the FDK reconstruction is approximate and good only for casessmall cone angles. For sake of comparison we rewrite the phase tomography reconstructionformula in following, which was derived in previous work for cases with a parallel-beamsource orbiting in a circle [27]:

(ro ) =

83sin

0

2(y;) (y nD s)dy d0

d (33)

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

map y;( )in Eq. (33) is retrieved according to Eq. (18) in the same way as in cone-beamphase tomography described earlier. In Eq. (33) n = (cossin,sin sin,cos) ,

nrs o = , the unit vectors nD = (sin,cos), as is depicted in Fig. 6. Note also that while

the FDK reconstruction is a filtered back projection algorithm, Eq. (33) is not. But followingthe approach of Bronnikov [14-15] one can rewrite the 3-D inverse Radon transform in afiltered backprojection form by means of the integration over angle ,

(ro ) =

83q y,ro( )

2(y;)dy0

d (34)

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

qy,ro( )=x3 y3

(ro y2 )2 + (x3 y3)

2(35)

Obviously Eq. (35) can be rewritten as

(ro ) =

83q2

0

d (36)




15/15

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

In summary in this work we first presented a detailed derivation of the phase-retrievalformula based on phase-attenuation duality. Moreover we have included the effects of x-raysource coherence and detector resolution into the phase-retrieval formula as well. As only asingle image is needed for performing the phase retrieval in this approach, this new approachhas 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 assmall animals or human breast. For large objects we briefly discussed how to applyKatsevichs cone beam reconstruction formula to the helical phase tomography for largeobjects 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 inpreparation of the figures. The authors would like to acknowledge the support of Charles andJean Smith Chair endowment fund as well.



x-ray cone-beam phase tomography formulas

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