Quantitative phase-contrast tomography using polychromatic radiation

Glenn R. Myers*a, Timur E. Gureyevb, David M. Paganina,b, Karen K. W. Siu a,c, Sheridan C. Mayob,

Stephen W. Wilkinsb

aSchool of Physics, Monash University, VIC 3800, Australia

bCSIRO Materials Science and Engineering, PB33, Clayton VIC 3169, Australia cMonash Centre for Synchrotron Science, Monash University, VIC 3800, Australia

ABSTRACT

We discuss theoretical, experimental and numerical aspects of several new techniques for quantitative phase-contrast tomography using, for example, unfiltered radiation from a polychromatic X-ray microfocus source. The proposed algorithms allow one to reconstruct the three-dimensional distribution of complex refractive index in a sample consisting of one or more constituent materials, given one or more projection images per view angle. If the sample is weakly absorbing or consists predominantly of a single material, these reconstruction algorithms can be simplified and fewer projections may be required for an unambiguous quantitative reconstruction of the spatial distribution of the refractive index. In the case of weakly absorbing samples, the reconstruction algorithm is shown to be achromatic and stable with respect to high-spatial-frequency noise, in contrast to conventional tomography. A variation of the algorithm exploits the natural combination of binary tomography with a phase-retrieval method that makes explicit use of the single-material nature of the sample. Such consistent use of a priori knowledge dramatically reduces the number of required projections, implying significantly reduced dose and scanning time when compared to most alternative phase-contrast tomography methods. Experimental demonstrations are also given, using data from a point-projection X-ray microscope. The refractive index distribution, in test samples of both a polymer fibre scaffold and an adult mouse, is accurately reconstructed from polychromatic phase-contrast data. Applications of the new techniques to rapid non-destructive testing in materials science and biomedical imaging are considered. Keywords: computed tomography, binary tomography, diffraction, phase contrast

1. INTRODUCTION It is well known [1, Section III.2] that in order to achieve a quantitatively accurate reconstruction of a generic sample at a desired spatial resolution in the case of conventional parallel-beam computed tomography (CT) using monochromatic X-rays, one needs to acquire approximately )1)(2/( −nπ evenly spaced projections over the full 180o angular range, where n is the number of pixels in each row of the detector. This means, of course, that the linear and angular resolution in a CT scan are coupled. With the constant quest for ever finer spatial resolution in CT and the development of new X-ray detectors with more pixels, the number of projections has to be increased accordingly, which may lead to increased scanning time and larger X-ray doses delivered to the sample. Both of these consequences are highly undesirable, particularly in the context of biomedical imaging and, more generally, in any application area where the samples tend to be sensitive to the X-ray dose or may exhibit structural changes on a time scale comparable to that of the scan. In view of these facts, it is important to investigate options for reducing the number of angular projections without sacrificing the accuracy of the reconstruction. It appears that this can be achieved only by utilizing a priori information about the sample. For example, if a sample is known to be rotationally symmetric, then a single projection is sufficient for the CT reconstruction [1, Section II.2]. Reconstruction of convex homogeneous objects can be achieved from 4 projections [1, Section VI.6]. There is also a large body of literature demonstrating that binary objects, i.e. objects with a refractive index that can only take one of two known values in each voxel, can be reconstructed from a few projections (of the order of 10), regardless of the spatial resolution [2-10].

Developments in X-Ray Tomography VI, edited by Stuart R. Stock, Proc. of SPIE Vol. 7078, 707819, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.795999

Proc. of SPIE Vol. 7078 707819-12008 SPIE Digital Library -- Subscriber Archive Copy

Let us take a closer look at the general nature of a priori information about the sample that can be useful in CT reconstruction. It is possible to classify any such information as "geometric", i.e. containing a priori knowledge about the internal structure of the object, or "compositional", i.e. knowledge about the refractive indices of the constituent materials. According to such a classification, the information that the object is binary is of a compositional nature, while the restriction of a class of samples to those with rotational symmetry or convex shape is an example of geometric a priori information. A more specialized example with strong geometric a priori information was studied in [11], where it was shown that certain wood samples may be considered "quasi-one-dimensional", i.e. the density in axial slices of such samples vary only very slowly along one of the Cartesian coordinates in the plane. This was later generalized [12] to a class of objects with the density distribution in axial slices representable in the form )],([),( yxSGyx =ρ , where ),( yxS is an a priori known real-valued function and )(tG is an unknown bounded function which can take non-negative values. A known "structure function" ),( yxS maps units of area into units of length. It defines lines (regions) with constant density, as ),( yxρ is constant whenever ),( yxS is constant. An unknown "1D density function" )(tG maps units of length into units of density; it determines the actual density values for the regions of constant density defined by

),( yxS . The CT reconstruction in this situation is equivalent to the determination of the function )(tG . It was shown in [12] that the density distribution in such objects can be reconstructed from just three projections, regardless of the required spatial resolution, provided that the structure function varies sufficiently slowly along one of the coordinates. In the present paper, we consider several classes of samples with strong compositional a priori information. We start from a rather generic "compositional" model, according to which the sample consists of N different materials with a priori known chemical composition and density. The location (distribution) of these materials inside the sample is not known a priori, but each voxel of the sample is allowed to contain only one of the N constituent materials. We go beyond the conventional CT problem in that we aim at reconstruction of the 3D distribution of the complex refractive index inside the sample, rather than just its imaginary part, as in conventional CT. The sensitivity to the real part of the refractive index is achieved via a phase-contrast imaging setup [13-20]. As the allowed values of the complex refractive index in each voxel are constrained to be equal to one of the N known values (corresponding to the constituent materials and a given X-ray wavelength), it is feasible that the number of projections required for quantitative reconstruction of the refractive index distribution in such a sample can be smaller than in the general case. Indeed, this conjecture has been already proven for the case N = 1, i.e. for binary objects where any voxel is either empty, or is fully filled with a known material [2-10]. We investigate below the cases of samples consisting of one, two or three different materials. Using the technique described in this paper, a sample with an arbitrary number N of constituent materials can be effectively "dissected" into N non-overlapping binary objects by acquiring images at N' ( NN ′≤ ) different suitably selected X-ray energies at each projection angle. This can be performed using a tunable monochromator or an energy-sensitive detector. The total number of required projection images will be equal to 1pNp ′=′ , where p1 is the number of projections required for the reconstruction of a binary object. If the number of different materials in the sample is small, and the required spatial resolution, nh /1~ , is reasonably high, one can expect p' to be much smaller than )1)(2/( −= np π (the required number of projection images in the conventional approach). Generally, there exists a trade-off between the number of images collected under different settings at each projection angle and the number of different projection angles required for the CT reconstruction. We develop our phase-contrast CT (PCT) algorithms in the polychromatic cone-beam X-ray setting following [20]. Accounting for the effects of beam divergence and polychromaticity is essential for quantitative CT of samples using laboratory microfocus X-ray sources [21]. Polychromaticity often presents a significant difficulty for quantitative CT reconstruction, as it leads to beam hardening [1]. However, in the case of phase-contrast imaging of weakly absorbing objects in the near-field propagation regime, beam hardening no longer presents a problem, as the image contrast mechanism is based primarily on refraction, rather than absorption, of X-rays. As a consequence, PCT can be essentially achromatic [20, 22], which allows us to come up with simple algorithms for tomographic reconstruction with polychromatic X-rays.

2. POLYCHROMATIC CONE-BEAM PHASE-CONTRAST TOMOGRAPHY

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P3,

r3 +X2

xl

ii6bject planeT12

R2Mx1

Detector plane

Reconstruction of a complex refractive index in an object in PCT is typically a two-stage process. The first stage involves the recovery of the object-plane phase and intensity at various projection angles θ ∈ [0,π ) . In the second stage, the 3D distribution of the spectrally-averaged complex refractive index )()(1)( rrr βin +∆−= in the sample is reconstructed from the object-plane phase and intensity. Assuming the projection approximation is valid, when the object is illuminated by X-rays with incident spectral density )(λinS , the spectral density ),,(0 λθxS and phase ),,(0 λθφ x in the contact plane are related to the projected values of the refractive index according to [21]:

)],,)((2exp[)(),,(0 λθβλλθ xx PkSS in −= , (1) ),,)((),,(0 λθλθφ xx ∆−= Pk ; (2)

here the projection operator P is defined by

( ) ∫∞

∞−⎟⎟⎠

⎞⎜⎜⎝

⎛+= dttnPn λθλθ ,

||)(,,)(

ppsx , (3)

where ),(),( 23121121 ppRppRxx −==x is a Cartesian coordinate system in the object plane (see Fig. 1), and

),sincos,sincos(),,( 32112321 rrrrrppp θθρθθ −−−==p is a rotated coordinate system centered at the X-ray source position (see Fig. 1). The object-plane phase and spectral density can be obtained from monochromatic projection images using, for example, the Transport of Intensity equation (TIE) (cf. eq.(5)). From these quantities, the projected values of the real and imaginary parts of refractive index can be trivially found using eqs.(1)-(2). Let the object lie entirely within a sphere of radius d centred at the origin of the Cartesian coordinates ),,( 321 rrr=r . Consider first a point source located at position )0,sin,cos()( θρθρθ =s , emitting radiation with wavenumber

λπ /2=k . This source traces a circle of radius ρ in the 21 rr − plane centred at r = 0, where )2,0[ πθ ∈ is the angle from the r1 axis to the source (see Fig. 1). The distance from the source to the object plane is dR += ρ1 .

Fig.1. Schematic of experimental layout for polychromatic cone-beam phase-contrast tomography.

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Given knowledge of the object-plane phase for projections )2,0[ πθ ∈ , we can reconstruct the real refractive-index decrement ),( λr∆ using, for example, the well-known Feldkamp-Davis-Kress (FDK) reconstruction algorithm [23, 24]:

∫ −−=∆π

θλθφξρ

λ2

0011

112

2

1 )]),,([|(|12

),( dpk

R xFFr , (4)

where 1F is the one-dimensional Fourier transform with 1ξ dual to 1x . A similar equation allows the recovery of the imaginary part of the refractive index from the logarithm of the object-plane spectral density. For certain types of object, the two-stage PCT process described above can be merged into a single-step process with potential gains in the speed and numerical stability of the reconstruction. Recall that an object is called “monomorphous” with respect to the incident radiation if the distribution of its complex refractive index ),(),(1),( λβλλ rrr in +∆−= is such that the ratio ),(/),()( λλβλε rr ∆= is independent of position r , at all wavelengths present in the detected spectrum. The assumption of monomorphicity is valid for X-ray-illuminated or neutron-illuminated objects composed of a single material [25, 26], and for objects composed of light elements (atomic number Z<10) illuminated by high energy (60–500 keV) X-rays [27]. Note that non-absorbing objects constitute a subset of monomorphous objects, with 0)( ≡λε for all incident wavelengths. For weakly-absorbing monomorphous objects, )(λε is assumed small. To render the phase of the object-plane wave-field visible as intensity variations in a propagation-based phase-contrast image, a propagation distance 02 >R is introduced between the object plane and the detector (see Fig. 1). It is assumed that the transmitted wave is paraxial, and that the distance R2 is sufficiently small. Then the free-space propagation of the transmitted wave can be described by the monomorphous version of the Transport of Intensity equation (TIE) [25, 20]:

])],,([)]([)([21)(),,( 02221

22

2λθφλεξπλλλθ xFxFFx MPRSMSM MinR +′+= − , (5)

where SR2

(Mx,θ ,λ) is the spectral density in the detector plane, Sin (λ) is the spectral density incident on the object, ξ 2 = ξ1

2 +ξ22, 2F is the two-dimensional Fourier transform with ),( 21 ξξ dual to ),( 21 xx=x and

)(* ))1/(()( detectorsource xxx PMPPM −= , the PSF of the imaging system which accounts for both the finite size of the fully incoherent source and the spatial resolution of the detector (note that an asterisk denotes two-dimensional convolution), MRR /2=′ and )/( 2121 RRRRM += . The paraxial approximation made in deriving eq.(5) is equivalent to assuming a small cone-beam angle and consequently neglecting terms of second or higher order in 1/ Rd . Applying this approximation to eq.(4), subsequently making use of both eq.(5) and the identity 0]1[|| 1 ≡2Fξ , and then integrating over X-ray wavelengths, yields:

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛

+′−=∆ −

π

θλλεξπ

θξπρ 2

0 2002

21122

2

21

)]([]/)([)],([||1

8)( 2 d

MPRMI

pIMR

M

R

in xFxF

Fr . (6)

In obtaining this equation, we assumed that: (i) )(/ λεπλRd ′< ; (ii) the wavelength spread is sufficiently narrow that

00 /)(/)( λλελλε ≅ , where 0λ is some suitable central or characteristic wavelength of the incident beam. Also, in this

equation we have introduced the spectrally averaged refractive index decrement λλλλ dDSI inin ),()()()(0

1 xx ∆=∆ ∫∞− , the

polychromatic (time-averaged) incident detected intensity λλλ dDSI inin )()(0∫∞

= , and the registered intensity

λλλθθ dDSI RR )(),,(),(0 22 ∫∞

= xx in the detector plane, where )(λD is the conversion efficiency of the detector.

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Equation (6) allows one to reconstruct the spectrally-averaged complex refractive index ]1)()[(1)( 0 −∆+= λεin rr of an object from a single polychromatic phase-contrast image taken at each projection angle )2,0[ πθ ∈ . The phase-and-amplitude computed tomography (PACT) algorithm for plane incident waves [28] is a special case of eq.(4), corresponding to the limit ∞→ρ . Equation (6) defines the polychromatic cone-beam PACT algorithm for weakly-absorbing monomorphous objects, where the phase-retrieval step is merged with the deconvolution of the PSF and with the filter used in the standard FDK algorithm. This results in a partial cancellation of singularities of the CT filter function, phase-retrieval kernel and the PSF of the imaging system, in a manner similar to that demonstrated in [29] in the case of 2D imaging. Furthermore, the weak absorption term acts as a regularizer in the denominator of eq.(4), increasing the low-frequency stability (cf. [25]). Note that unlike an equivalent reconstruction formula in conventional CT [1], eq.(6) is stable with respect to high-frequency noise due to the suppression of the growth of the ramp filter, || 1ξ , by the "phase retrieval" kernel, 2ξ , in the denominator [20, 30]. Note also that when a scaling law of the type ),(),(),( 00 λλλλ xx ∆=∆ f is applicable (e.g.

),()/(),( 02

0 λλλλ xx ∆=∆ , as typically holds outside X-ray absorption edges of constituent materials), the true 3D distribution of refractive index can be recovered from the spectrally-averaged refractive index according to

)()(),( 00 xx ∆=∆ λλ C , provided the position-independent spectral factor λλλλλλ dfDSIC inin ),()()(/)( 000 ∫∞

≡ is

known. When any scaling law of the above type holds, but )( 0λC is not known, eq.(6) still provides a distribution equal to the refractive index ),( 0λx∆ up to a multiplicative factor.

3. ALGORITHMS FOR POLYCHROMATIC PHASE-CONTRAST COMPUTED TOMOGRAPHY OF BINARY, TERNARY AND QUATERNARY OBJECTS

3.1. Binary objects A binary object is composed of only a single material and void (see Fig. 2(a)) such that the complex refractive index

),(),(1),( λβλλ rrr in +∆−= of the object can assume only the values )()(1 11 λβλ i+∆− and 1.

Fig.2. Sample (a) binary (one material and void), (b) ternary (two materials and void), and (c) quaternary (three materials and void) objects. Different grey levels denote different materials.

Under the weak absorption assumption, 1),,)((2 <<λθβ xPk , assumed for all X-ray wavelengths present in the spectrum, the polychromatic TIE can be written as [31]:

),()(),( 2

12

θεθ xx j

J

jjR PRK ⊥

=

∇′−= ∑ , (7)

(a) (b) (c)

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where the sum is taken over all constituent materials Jj ,...,2,1= , ),)((),( θθ xx jj PP ∆≡ , jjj ∆= /βε ,

∫∞− ∆=∆

0

1 )()()( λλλλ dDSI jininj , ∫∞−=

0

1 )()()( λλβλλβ dDSI jininj and inRR IIK /),(1),(22

θθ xx −= is a "contrast"

function corresponding to the polychromatic image collected at a distance R2 from the sample at the projection angle θ. Equation 7 indicates that, under the weak-object approximation, the propagated contrast transfer function

2RK is a linear sum of the propagated signals due to each of the component materials j. In the case of a binary object, we have J=1, and eq.(7) can be easily solved for ),( θxjP :

),()(),(2

1211 θεθ xx RKRP −

⊥∇′−= . (8) Having solved for the projected refractive index at a variety of projection angles we must then use this information to reconstruct the 3D distribution of refractive index within the object. Conventional methods for tomographic reconstruction such as filtered backprojection (see eq.(4)) typically require information at hundreds of projection angles in order to produce a reconstruction at a useful resolution [1]. In contrast, methods developed specifically for “binary tomography” can significantly reduce the required number of projection angles by making use of the a priori knowledge that the complex refractive index (averaged over the X-ray wavelength) is either 1− ∆1 + iβ1 or 1 in any given voxel [2-10]. We make an initial estimate )(0 r∆

( of the real decrement to the refractive index using filtered backprojection from the

limited number of collected projection images:

))((|)]),([|(|)( 10

),sincos(),(1111

10 31221rxFFr PdP rrrxx B≡−=∆ ∫ −→

−π

θθ θθξ(

, (9)

where F1 is the one-dimensional Fourier transform and B represents the filtered backprojection operator [1]. Note that the subscript on

( ∆ 0 (r) denotes that this is the estimated real decrement to the refractive index at the zeroeth iteration of

the reconstruction process. We then apply the known constraints to the real refractive index via a thresholding procedure:

⎩⎨⎧

Ω≥∆∆Ω<∆

≡∆=∆,)(if ,

,)(if ,0))(()(

01

000, r

rrr (

(((

TT (10)

where the numeric subscript on

( ∆ T , 0 (r) again denotes the zeroeth iteration of the reconstruction procedure, and Ω is a

constant chosen such that the correct amount of material is present in the object, i.e. such that:

xxrrRR

dPdT ∫∫ =∆23

),()( 10, θ(

(11)

for all θ . This zeroeth order estimate of the refractive index is then iteratively refined:

⎪⎩

⎪⎨⎧

∆=∆

∆−+∆=∆

++

+

).)(()(

)],,)((),([)()(

11,

,11

rr

xxrr

iiT

iTiii PP((

(((

T

B θθγ (12)

At each iteration the parameter 0 < γ i ≤ 1 is optionally varied to ensure a convergent reconstruction. If necessary, the step is performed for several values of γ i , beginning at 1 and halving until a value of γ i is found for which:

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∫∫∈∈

+ ∆−<∆−22

|),)((),(||),)((),(| ,11,1RxRx

xxxxxx dPPdPP iTiT θθθθ((

, (13)

ensuring that at each step the projected refractive index ),)(( 1, θx+∆ iTP

( is closer to the retrieved projected refractive

index ),(1 θxP than at the previous iteration. The process terminates if no successful step is possible without reducing γ i

below some pre-determined lower limit, or if ),)(( 1, θx+∆ iTP(

is deemed to be sufficiently close to ),(1 θxP (i.e. within one standard deviation of the known noise level). It has been demonstrated (using both simulated and experimental data) that this method typically requires measurements at 20 or even fewer projection angles in order to produce an accurate reconstruction [10].

3.2. Ternary objects In this section, following the results presented in the case of plane monochromatic X-rays in [32], we extend the method presented in Section 3.1 for 3D imaging of binary objects to objects composed of two materials and void, which we term “ternary” objects (see Fig. 2(b)). The spectrally averaged refractive index decrement of a ternary object can assume the values 1− ∆1 + iβ1 , 1− ∆2 + iβ2 and 1. Note that we have assigned the subscripts “1” and “2” in order to differentiate between the two materials, and all equations are equally valid if all such subscripts are exchanged. One possible course of action would be to take measurements at additional projection angles and “relax” the second (tomographic) stage of the reconstruction process to accommodate 3 possible values of the refractive index when thresholding (eq.(10)), rather than two. Instead, we consider our ternary object to be composed of two non-overlapping binary objects (that may be superimposed in projection) such that:

⎩⎨⎧ +∆−

=⎩⎨⎧ +∆−

==1

1)( ,

11

)( ),()()( 222

11121

ββ in

innnn rrrrr . (14)

The first step of our reconstruction process is then altered to recover the projected refractive indices of materials one and two ( ),(1 θxP and ),(2 θxP respectively) at each projection angle. Due to the additional degree of freedom of the ternary object problem in comparison to the binary object considered in Section 3.1, in the absence of additional a priori information phase retrieval forms an integral part of this reconstruction method and is not used solely to compensate for poor absorption contrast as in Section 3.1. Assuming a ternary object that satisfies the projection approximation, with two images collected at different defocus distances 2R and −

2R , we obtain the following system of linear partial differential equations from eq.(7):

),()(),()(),( 22

212

12θεθεθ xxx PRPRK R ⊥⊥ ∇′−+∇′−= , (15)

),()(),()(),( 22

212

12

θεθεθ xxx PRPRKR ⊥−⊥− ∇′−+∇′−=− , (16)

where )/( 2121

−−− +=′ RRRRR . Fourier transformation can be used to convert eqs.(15)-(16) into a system of two linear

algebraic equations, which can then be easily solved for ),(1 θxP and ),(2 θxP , provided that 21 εε ≠ and −≠ 22 RR . In some cases the total projected thickness of the object:

2211 /),(/),(),( ∆+∆= θθθ xxx PPA (17) may be known a priori. If the sample is known to contain no internal voids then A(x,θ ) can be found by measuring the surface of the object using, for example, laser profilometry [33]. Note that even if there are no voids within the sample, it still constitutes a ternary (rather than binary) object due to the (in general) non-trivial shape of the interface between the object and the surrounding void. Reconstruction may then proceed using only a single image at each projection angle. If

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images taken in the contact plane display weak, but sufficient, absorption contrast, then ),(1 θxP and ),(2 θxP can be recovered by solving the system of linear eqs.(15) (with R' = 0) and (17), and phase-contrast imaging is no longer necessary. If the total projected thickness is known a priori but images taken in the contact plane display insufficient absorption contrast, then we again introduce some propagation distance R between the contact plane and detector and take a single propagated image at each projection angle. Given knowledge of both ),(1 θxP and ),(2 θxP , the iterative reconstruction process defined in eqs.(9)-(13) can be employed to separately reconstruct n1 (r) and n2 (r) . We have satisfied the increased information requirements for tomography of ternary objects by collecting additional information at each projection angle. Imaging of ternary and quaternary (see Section 3.3) objects may thus be performed using measurements at no more projection angles than are required to reconstruct a binary object. This approach is complementary to conventional CT methods, in which the required additional information is typically collected by increasing the number of projection angles and the resolution of the measurements taken at each projection angle [1].

3.3. Quaternary and N-material objects The extension of our reconstruction methods from ternary to quaternary objects (see Fig. 2(c)) proceeds as follows. In a quaternary object (i.e. an object composed of three materials and void), the refractive index may assume the values 1− ∆1 + iβ1 , 1− ∆2 + iβ2 , 1− ∆ 3 + iβ3 and 1. Following the path we took in Section 3.2, we describe the quaternary object as three non-overlapping binary objects with complex refractive indices n1 (r), n2 (r) and n3 (r) such that:

⎩⎨⎧ +∆−

=⎩⎨⎧ +∆−

=⎩⎨⎧ +∆−

==1

1)( ,

11

)( ,1

1)( ),()()()( 33

322

211

1321βββ i

ni

ni

nnnnn rrrrrrr . (18)

Assuming that two images can be collected at different defocus distances 2R and −

2R at each projection angle, we obtain as above:

),()(),()(),()(),( 32

322

212

12θεθεθεθ xxxx PRPRPRK R ⊥⊥⊥ ∇′−+∇′−+∇′−= , (19)

),()(),()(),()(),( 32

322

212

12

θεθεθεθ xxxx PRPRPRKR ⊥−⊥−⊥− ∇′−+∇′−+∇′−=− . (20)

Unlike the previous case with a ternary object, this linear system cannot be solved with respect to ),(1 θxP , ),(2 θxP and

),(3 θxP , unless additional information is available. Bearing this in mind, let us define the total projected thickness:

332211 /),(/),(/),(),( ∆+∆+∆= θθθθ xxxx PPPA . (21) If this total projected thickness is known, we can solve the linear system of simultaneous equations (19)-(21) for

),(1 θxP , ),(2 θxP and ),(3 θxP . As always, the system of linear equations will be stable provided the condition number is sufficiently close to 1. This requirement is violated if any two materials have almost identical complex refractive indices, or if the ratio nnn ∆= /βε is almost identical for all three materials. An appropriate method for tomographic reconstruction of binary objects (see, for example, eqs.(9)-(13)) from projected refractive indices is then employed to reconstruct the 3D distributions of complex refractive index n1 (r), n2 (r) and n3 (r). Further intensity measurements at different defocus distances at each projection angle are unlikely to contribute additional information (or may contribute only weak additional information with poor signal-to-noise ratio). Note that in the monochromatic case the beam is completely characterized by its phase and intensity, which can be obtained from intensity measurements in two planes orthogonal to the optic axis. Therefore, we cannot generalize this imaging method to quinary (four material and void) objects unless some other parameter, such as the wavelength of the illuminating X-rays, is altered [31]. Using the latter approach, however, the present method can be easily extended to a general case of

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objects consisting of N different materials, provided that multiple ( N≥ ) images can be collected at each projection angle with different X-ray spectra. Ideally, quasi-monochromatic images should be collected near different absorption edges of constituent materials, which could make a matrix of equations (7) (written for different X-ray spectra) almost diagonal, leading to a stable solution. After solving this linear system for the projected refractive indices NPPP ,...,, 21 , the method for binary CT, as described in Section 3.1 above (eqs.(9)-(13)), can be applied separately for each of the N materials. Thus, the main idea of the proposed approach is to "dissect" an object consisting of N materials into N non-overlapping binary objects that can be CT-reconstructed separately using methods of binary CT.

4. EXPERIMENTAL RECONSTRUCTION OF BINARY AND TERNARY OBJECTS We now turn to an experimental implementation of our method. A tomographic dataset was acquired of a phantom consisting of a ~2 mm wide Polyethylene Terephthalate (PET) fibre scaffold on a point-projection X-ray ultra Microscope (XuM) based on an FEI XL-30 scanning electron microscope [26]. The X-rays were generated by focusing a 20 keV electron beam onto a 500 nm thick tantalum foil target, with the resulting characteristic radiation and bremsstrahlung yielding a divergent polychromatic X-ray source of approximately 0.2 µm diameter, and a mean photon energy of approximately E0 = 4 keV. The presence of characteristic Ta L lines in the spectrum had the effect of making the data effectively more monochromatic, but the contribution of the bremsstrahlung was large enough to prohibit the use of a quasi-monochromatic approximation. A total of 360 images of the sample were acquired with equal rotational steps of 1º between the views. Each projection image took 1 minute to acquire, and the total CT scan time was more than 15 hours. The source-to-sample distance 1R was 24.6 mm, with a source-to-detector distance 21 RR + of 259 mm. This geometry gave a geometric magnification of M = 9.4× at the surface of the detector, producing propagation-based phase-contrast images (see Fig. 3(a)). Note that the phase contrast present in these projections was an inevitable consequence of the geometric magnification achieved in point-projection X-ray microscopy. The present experimental approach allowed one to collect high-resolution images without X-ray focusing elements and without a high-resolution detector. Using the collected projection data, a numerical implementation of eq.(6) was used for the tomographic reconstruction with

0016.0)( 0 =λε . A sample of the reconstructed distribution of ),()(),( 000 λλελβ rr ∆≡ is shown in Fig. 3(b). The spatial resolution of this reconstruction was approximately 3.8 µm per cubic voxel.

Fig.3. (a) Sample image from the data set, displaying propagation induced phase-contrast around the fibre edges; (b) Volume rendered, reconstructed distribution of the imaginary part of the refractive index in the phantom.

500 µm 500 µm

(a) (b)

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3 mm

\h ,,/J J

Note that despite the value of ε being quite small, the observed X-ray attenuation reached almost 50 % in some projections. It was difficult to avoid significant absorption, as our XuM microscope becomes less efficient at higher X-ray energies. Nevertheless, one can see from Fig.3 that a reasonable accuracy has been achieved in the reconstruction, using the CT data collected with an unfiltered broad-band cone beam of X-rays. The next sample was an adult C57Bl/6 mouse weighing 18-20 grams, humanely killed with anaesthesia overdose and placed in a plastic mounting assembly (see the tube and rods in Fig. 4(a), [34]). 1200 images at equal angular intervals over the range of [0, 180] degrees were collected with a 17 keV, monochromatic beam at the bending magnet Beamline 20B2, Biomedical Imaging Centre, SPring-8 synchrotron facility, Hyogo, Japan. The source-to-sample distance was approximately 206 m and the sample-to-detector distance was 65 cm. Filtered backprojection of ln(IR (x,θ ) / I in ) at all 1200 projection angles allowed us to approximate the imaginary part of the refractive index of the mouse. Note that, as we possessed only a single propagated image per projection angle, it was impossible to exactly retrieve the contact plane intensity and phase without some a priori information. Consequently, the propagation-based phase contrast in the collected images led to some edge enhancement in the reconstructed slice (see Fig. 4(a)). In order to perform a ternary reconstruction from only one collected image at each projection angle it was necessary to simulate knowledge of the total projected thickness of the mouse, as well as the position of the plastic mounting assembly which consisted of three rods and a tube surrounding the object. The “known” position of the plastic mounting assembly was then used to remove the associated signal from the collected images. Finally, the monochromatic limiting case of eqs.(9)-(13), (15), and (17) was used to reconstruct the distributions of bone and soft tissue within the mouse (see Fig. 4(b)) from data collected at 48 of the 1200 projection angles.

Figure 4: Reconstructions of a slice through a mouse from (a) 1200 projections and (b) 48 projections. The slice on the left is the result of a qualitative reconstruction performed using filtered backprojection, and the slice on the right is reconstructed using eqs.(9)-(13), (15), and (17) for ternary tomography. Voxels are ~12 µm per side. Some bony features are absent from Fig. 4(b): these features are for the most part less dense than the majority of the bone in the image and are only several pixels in width. Furthermore, there is a small void in the interior of the mouse (visible in Fig. 4(a)). In simulating knowledge of the total projected thickness we have assumed ignorance of this void, as it would not be detected by surface profilometry. Although the fine bone features near the void show some inaccuracies, more distant features in Fig. 4(b) are reconstructed accurately, indicating that the algorithm is relatively stable with respect to inaccuracies in the total projected thickness. The phase retrieval performed in order to recover the projected thicknesses eliminates the streaking artifacts that can be observed in Fig. 4(a). It is also worth noting that as Fig. 4(b) only uses 1/25th of the available intensity data the effective signal-to-noise ratio is 5 times lower for the purposes of tomographic reconstruction.

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ACKNOWLEDGEMENTS

Financial support from the Australian Research Council is gratefully acknowledged, for funding from both the Discovery-Project (DMP) and Australian Postgraduate Award (GRM) schemes. TEG, SCM and SWW are grateful to XRT Limited for encouragement of this work. KKWS and her collaborator Dr D. W. Parsons (Women's and Children's Hospital and Department of Paediatrics, University of Adelaide) would like to acknowledge support of the National Health and Medical Research Council and the Access to Major Research Facilities program for enabling the work carried out at the SPring-8 synchrotron in Japan.

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