A computed tomography implementation of multiple-image ...

A computed tomography implementation of multiple-image radiographyJovan G. BrankovDepartment of Electrical and Computer Engineering, Illinois Institute of Technology, Chicago, Illinois60616

Miles N. Wernicka� and Yongyi YangDepartment of Electrical and Computer Engineering, Illinois Institute of Technology, Chicago, Illinois60616 and Department of Biomedical Engineering, Illinois Institute of Technology, Chicago, Illinois 60616

Jun Li and Carol MuehlemanRush University Medical Center, Chicago, Illinois 60616

Zhong ZhongNational Synchrotron Light Source, Brookhaven National Laboratory, Upton, New York 11973

Mark A. AnastasioDepartment of Biomedical Engineering, Illinois Institute of Technology, Chicago, Illinois 60616

�Received 12 April 2005; revised 18 November 2005; accepted for publication 21 November 2005;published 12 January 2006�

Conventional x-ray computed tomography �CT� produces a single volumetric image that representsthe spatially variant linear x-ray attenuation coefficient of an object. However, in many situations,differences in the x-ray attenuation properties of soft tissues are very small and difficult to measurein conventional x-ray imaging. In this work, we investigate an analyzer-based imaging method,called computed tomography multiple-image radiography �CT-MIR�, which is a tomographicimplementation of the recently proposed multiple-image radiography method. The CT-MIR methodreconstructs concurrently three physical properties of the object. In addition to x-ray attenuation,CT-MIR produces volumetric images that represent the refraction and ultrasmall-angle scatteringproperties of the object. These three images can provide a rich description of the object’s physicalproperties that are revealed by the probing x-ray beam. An imaging model for CT-MIR that is basedon the x-ray transform of the object properties is established. The CT-MIR method is demonstratedby use of experimental data acquired at a synchroton radiation imaging beamline, and is comparedto the pre-existing diffraction-enhanced imaging CT method. We also investigate the merit of aniterative reconstruction method for use with future clinical implementations of CT-MIR, which weanticipate would be photon limited. © 2006 American Association of Physicists in Medicine.�DOI: 10.1118/1.2150788�

Key words: diffraction-enhanced imaging, x-ray phase-contrast imaging, image reconstruction,synchrotron radiation

I. INTRODUCTION

X-ray imaging techniques that exploit contrast mechanismsother than x-ray absorption hold great promise for biomedi-cal imaging applications. Recently, we have proposed andinvestigated a new planar imaging method called multiple-image radiography �MIR�.1,2 Multiple-image radiography isa generalization of the diffraction-enhanced imaging �DEI�technique that has been investigated extensively in recentyears.3–9 The imaging system and experimental conditionsthat are needed to acquire MIR images are essentially thesame as those needed in DEI; these include a monochromaticx-ray beam that is used to irradiate the object and an analyzercrystal, placed between the object and a detector system, thatcan reveal information regarding components of the trans-mitted beam that are traveling in certain directions. This im-aging setup is also referred to as a Bonse-Hart camera.10

Both DEI and MIR produce images that represent a pro-jected x-ray refractive-index gradient distribution and an im-
age that is an x-ray radiograph. The MIR method, however,
278 Med. Phys. 33 „2…, February 2006 0094-2405/2006/33„

also produces a third image that represents the ultrasmall-angle scattering properties of the object. The ultrasmall-anglescattering characteristics reflect the textural features of theobject and provide diagnostic information that is comple-mentary to that conveyed by the refraction- and absorption-based images. Moreover, the DEI method does not accountfor contributions to the measured data that arise fromultrasmall-angle scattering and therefore the DEI absorptionand refraction images will generally contain artifacts whenimaging objects that produce such scattering. An example ofsuch artifacts is presented later in this article. The MIRmethod is also superior to the DEI method in that it does notrely on an approximation of the intrinsic rocking curve thatfails for large refraction angles.

It is worthwhile to note that other analyzer-based imagingmethods for circumventing the limitations of DEI have sincebeen investigated. In Ref. 11, Rigon et al. investigated amodified DEI method that statistically accounts for
ultrasmall-angle-scattering effects produced by subpixel-
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279 Brankov et al.: A computed tomography implementation of multiple-image radiography 279

sized structures in the object. Similar to DEI, their methodrequired two measurements; however, unlike DEI these mea-surements corresponded to the peak and toe of the rockingcurve. The refraction angle image produced in DEI was re-placed by a refraction-induced scattering image that revealedinformation about the object on a subpixel scale. However,the effects of measurement noise and beam broadening dueto highly scattering media on this method have not beenassessed systematically. In a different work, Pagot et al. in-dependently proposed3 a method that is essentially identicalto MIR. To our knowledge, neither of these modified DEImethods has been extended for use with three-dimensional�3D� tomographic imaging.

There is an important need to extend MIR into a com-puted tomography �CT� imaging method that can produce3D volumetric images. In addition to revealing the 3D x-rayattenuation coefficient distribution, MIR operating in CTmode, which we will refer to as computed tomography MIR�CT-MIR�, would produce volumetric images of the refrac-tion and ultrasmall-angle scattering properties of the object.These three images would provide a rich description of theobject’s physical properties that are revealed by the probingx-ray beam. The correlation of complementary informationcontained in these images can potentially provide CT-MIRwith dramatically improved diagnostic capabilities as com-pared to conventional x-ray imaging methods.

In this work, the CT-MIR imaging method is implementedand investigated. This article expands on our work first pre-sented in Refs. 12 and 13. An explicit x-ray transform-basedimaging model for CT-MIR is described. This imagingmodel indicates that the volumetric images produced in CT-MIR can be reconstructed on a slice-by-slice basis by use ofreconstruction algorithms from conventional parallel beamx-ray tomography. The CT-MIR method is demonstrated byuse of experimental data. Specifically, the CT-MIR method isemployed for reconstruction of absorption, refractive-indexgradient, and ultrasmall-angle scattering volumetric imagesof a physical phantom and a biological object by use ofmeasurement data produced by a synchrotron light source.An iterative reconstruction method is proposed and investi-

FIG. 1. Definition of the coordinate systems. Note that the figure is drawn oorder of microradians�. Indicated in the figure are: angles � and �, which dethe analyzer crystal �B; the angular setting of the analyzer �A; and the spati

gated for CT-MIR applications that involve photon-limited

Medical Physics, Vol. 33, No. 2, February 2006

measurement data, which we expect will be a feature of fu-ture clinical implementations. We also demonstrate that theabsorption and refractive-index gradient images produced bythe CT-DEI method described in Refs. 14 and 15 �the prede-cessor of CT-MIR� can contain significant artifacts due toultrasmall-angle scattering effects, while the correspondingCT-MIR images do not.

The article is organized as follows. In Sec. II, the MIRmethod is reviewed and the generation of two-dimensional�2D� parametric images that represent the projected objectproperties is described. The CT-MIR imaging model and re-construction procedure are described in Sec. III. Our experi-mental studies and images reconstructed from experimentaldata are presented in Secs. IV and V, respectively. A sum-mary of the CT-MIR method is included in Sec. VI.

II. MIR

A. MIR imaging model

In this section, we review the MIR method upon whichCT-MIR is based. The 2D parametric images produced inMIR represent the projected 3D object properties and, asdescribed in Sec. III, will serve as the tomographic projec-tion data for the CT-MIR method. For a more comprehensivedescription of MIR and specific implementation details, werefer the reader to Refs. 1 and 2. It should be noted that animaging method that is essentially identical to MIR was pro-posed in Ref. 3.

In the MIR method, the object is illuminated with a col-limated and monochromatic x-ray beam, and the angularcontent of the transmitted radiation is analyzed by use of asystem of diffractive optical elements. The spatial coordi-nates �x ,y ,z� and angular coordinates �� ,�� that are definedin Fig. 1 will be employed to describe the MIR imagingsystem that is shown in Fig. 2. The first two crystals in theimaging system form a double crystal monochromator thatserves principally to collimate and monochromate the inci-dent beam, which travels along the z axis. After interactingwith the object, the transmitted beam is incident on a thirdcrystal, called the analyzer. The analyzer diffracts only those

proportion for illustration purposes �in reality, all angles shown are on thehe direction of a ray within the transmitted x-ray beam; the Bragg angle ofordinates �x ,y ,z�.

ut offine tal co

components of the beam traveling at or near the analyzer’s


Bragg angle, thereby rejecting all components outside a nar-row angular range. Note that the analyzer crystal is sensitiveonly to intensity variations in the x-z diffraction plane �i.e.,intensity variations with respect to the angle ��. In MIR, thetransmitted beam components travel at angles described byextremely small values of � �on the order of microradians�with respect to the optical axis of the imaging system. For agiven analyzer orientation, the intensity of the beam that isreflected by the analyzer is measured by an x-ray detectorthat is described by the coordinates �x ,y�. Therefore, themeasurement data in MIR are indexed by three coordinates:the transverse measurement-plane spatial coordinates �x ,y�and the analyzer orientation �.

In MIR, the effect of the object on the beam is regarded asa linear system in terms of the angle �. The angular spectrumof the beam after passing through the object �but before dif-fracting from the analyzer crystal� can be expressed as

I��;x,y� = ��

I0��g��,��;x,y�d��, �1�

where I0�� denotes the angular intensity spectrum of theincident beam and g�� ,�� ;x ,y� is the angular impulse re-sponse function. The angular impulse response functioncharacterizes the object in the sense that it represents theangular intensity spectrum of the transmitted beam thatwould result from illuminating the object with a perfectlycollimated beam, i.e., a beam of the form I0��= I0��,where ��·� denotes the Dirac delta function. The set �=��x ,y� describes the collection of deflected beam anglesthat are intercepted by the detector element at location �x ,y�.The MIR imaging model assumes that, over the ultrasmall-angle window �, the angular intensity pattern caused bysubpixel object structures is approximately invariant to �.This implies that the shape of the angular pattern of thetransmitted beam remains constant when the illuminatingbeam is reoriented by an angle on the order of microradians.This assumption of angle-invariance implies thatg�� ,�� ;x ,y�= f��−�� ;x ,y�. Because its integrand is approxi-mately zero for ��, Eq. �1� can be expressed as

I��;x,y� = ��

I0��f�� − ��;x,y�d� = I0�� * f��;x,y� ,

�2�

where * denotes the one-dimensional convolution operation.
The angular intensity spectrum of the beam that is diffracted

by the analyzer and measured subsequently by the detector isgiven by

y��;x,y� = R�� * f��;x,y� , �3�

where R�� denotes the intrinsic rocking curve2 that repre-sents the intensity that would be measured with no objectpresent when the analyzer crystal is “rocked” �i.e., rotated inangle ��.

The impulse response f�� ;x ,y� that describes the effect ofthe object on the incident beam can be recovered from themeasured intensity y�� ;x ,y� by inverting Eq. �3�, which is aone-dimensional �1D� deconvolution problem. From knowl-edge of f�� ;x ,y�, parametric images can be computed thatprovide convenient characterizations of the absorptive, re-fractive, and ultrasmall-angle scattering properties of the ob-ject. Alternatively, as described later, these parametric im-ages can be computed directly from the measured y�� ;x ,y� ifthe effects of the convolution in Eq. �3� are compensated forappropriately.

B. Computation of 2D parametric images

Later we describe the computation of the absorptive, re-fractive, and ultrasmall-angle scattering parametric imagesthat are produced in MIR. The procedure we employ for thecomputation of these images is different than that employedin Ref. 1 in that they are computed directly from the mea-sured y�� ;x ,y� rather than from the deconvolved quantityf�� ;x ,y�.

In practice, the measured intensity data are recorded at adiscrete set of detector locations at a finite number of angularanalyzer settings. The discrete form of Eq. �3� will be ex-pressed as

ym,n�k� = R�k� * fm,n�k� , �4�

where the detector pixels are referenced by the spatial indi-ces m=1,2 , . . . ,M and n=1,2 , . . . ,N, and k=1,2 , . . . ,K de-scribe discrete �angular� analyzer crystal settings. Therefore,at each pixel location �m ,n� on the detector, we measure theangular intensity spectrum ym,n�k� at the K angular settingsof the analyzer crystal. The measured data set is accordinglycomposed of K images, each of dimension M �N.

It will be useful to define the following intermediatequantities. Let the total intensity at a given pixel �m ,n� be

FIG. 2. Schematic diagram of the MIR imaging system.

denoted by


Tm,n = �k=1

K

ym,n�k� , �5�

and define a normalized angular intensity spectrum as

Ym,n�k� =ym,n�k�

Tm,n. �6�

The intensity that would be measured at each pixel in theabsence of the object will be denoted as

I0 = �k=1

K

R�k� . �7�

1. Attenuation image

In the ultrasmall-angle regime, attenuation of the x-raybeam is caused both by absorption and by scattering intoangles outside the measured angular range. These sources ofbeam attenuation can be summarized collectively by the pa-rameter

pm,n = − lnTm,n

I0, �8�

which corresponds to an inversion of a discrete exponentialloss law.

2. Refraction image

Refraction induces an overall deflection of the beam thatproduces an angular shift of the beam centroid �as comparedto its position when no object is present�. This angular shiftcan be computed as

��m,n = �k=1

K �k −K + 1

2�Ym,n�k�� − �R�, �9�

where � is the angular spacing of the measured samples, and

�R� =1

I0�k=1

K �k −K + 1

2�R�k�� . �10�

Note that the second term in Eq. �9� compensates for the factthat the angular shift is computed from knowledge of Ym,n�k�rather than from direct knowledge of fm,n�k�.

3. Ultrasmall-angle scatter image

Ultrasmall-angle scatter by subpixel object structurescauses an angular broadening of the transmitted beam. Thisangular broadening can be described by the beam’s angulardivergence about the angle ��m,n that can be described bythe second central moment of the normalized angular inten-
sity spectrum as

Mm,n = �k=1

K �k −K + 1

2�� − ��m,n2

Ym,n�k�

−1

I0�k=1

K �k −K + 1

2�� − �R�2

R�k� . �11�

Equation �11� can be understood readily from Eq. �3� bynoting that the second central moment of the convolution oftwo functions is equal to the sum of the second central mo-ments of the individual functions.

III. CT-MIR

As discussed in the introduction, it is highly desirable toperform MIR in CT mode. Such a procedure, which we referto as computed tomography MIR �CT-MIR�, can producethree volumetric images that contain detailed 3D informationabout the object’s absorption, refractive, and ultrasmall-anglescattering properties. These three images would provide arich description of the object’s physical properties that arerevealed by the probing x-ray beam. Later we establish anddiscuss a tomographic imaging model for CT-MIR. At eachtomographic view angle, the 2D parametric images producedby the MIR method will represent the raw projection datafrom which the 3D object property images will be recon-structed tomographically. In effect, the process of computingthe MIR parametric images at each view angle serves todecouple the tomographic inverse problems for determina-tion of the three different object properties. We demonstratethat each object property is related to its associated MIRparametric image by an x-ray transform. Consequently, theimages of the 3D object properties can be reconstructedreadily on a slice-by-slice basis by use of 2D parallel-beamCT reconstruction algorithms.

A. Scanning geometry

In the tomographic imaging model, we will consider theobject to be fixed and assume that the x-ray source andanalyzer/detector system are rotated simultaneously aboutthe x axis. This is done simply as a matter of notationalconvenience; the reconstruction formulas derived later areapplicable immediately to the case where the x-ray sourceand analyzer/detector system are fixed and the object is ro-tated. As shown in Fig. 3, the rotated coordinate system�x ,yr ,zr� is related to the fixed reference coordinate system�x ,y ,z� as yr=y cos �+z sin � and zr=z cos �−y sin �. Thetomographic view angle � is measured from the positive yaxis. In the rotated coordinate system, the zr axis representsthe optical axis and �x ,yr� denotes the corresponding detec-tor plane coordinates. Let L��x ,yr� denote the line that theprobing beam travels along before being measured at detec-tor location �x ,yr� at tomographic view angle �. In terms ofthe rotated coordinates the imaging model in Eq. �3� can be
expressed as


y��;x,yr� = R�� * f��;x,yr� , �12�

where y�� ;x ,yr� and f�� ;x ,yr� denote the measured angu-lar intensity spectrum and object angular impulse responsefunction, respectively, at view angle �.

B. Tomographic imaging model for reconstruction ofattenuation image

In MIR it is assumed that the beam deflections due torefraction and ultrasmall-angle scatter are too small to causesignificant cross talk between adjacent detector pixels.1 Un-der this condition it is true that

T�x,yr,�� = I0exp− �L��x,yr�

��x,yr,zr�dzr , �13�

where

T�x,yr,��

y��;x,yr�d� �14�

is the integrated intensity measured at detector location�x ,yr� at view angle �, I0 is the intensity of the incidentbeam, and ��x ,yr ,zr� is the total x-ray attenuation coefficientof the material �expressed in the rotated coordinate system�that includes the effects of coherent and incoherent atomicscattering.

For a given view angle �, the parametric image pm,n de-fined in Eq. �8� is a discrete approximation of the quantity

p�x,yr,�� − lnT�x,yr,��I0

= �L��x,yr�

��x,yr,zr�dzr,

�15�

which corresponds to the x-ray transform of �. Stated other-wise, for a given x=x0, p�x0 ,yr ,�� corresponds to the 2DRadon transform of ��x0 ,yr ,zr�. It is clear therefore that,from knowledge of the parametric images pm,n computed at acollection of tomographic view angles that span the interval�0,��, a 3D image representing ��x ,yr ,zr� can be recon-structed on a slice-by-slice basis by use of a reconstruction

FIG. 3. An illustration of the tomographic scanning geometry. The angle �denotes the tomographic view angle.

algorithm from conventional parallel-beam CT.


C. Tomographic imaging model for reconstruction ofrefraction image

Refraction can induce an overall deflection of the trans-mitted beam that can be measured as an angular shift of thebeam centroid �as compared to its position when no object ispresent�. The angular shift of a transmitted beam measured atdetector location �x ,yr� at view angle � will be denoted by��x ,yr ,��. Equation �9� can be used to calculate �� fromdiscretely sampled measurement data at a given view angle.It should be noted that the quantity �� is also computed inDEI; however, because DEI neglects the effects ofultrasmall-angle scattering, the value of �� computed inMIR is generally more accurate than that computed inDEI.1,2 A well-known relationship exists14 between �� andthe real-valued refractive index distribution of the objectn�x ,yr ,zr� that is given by

��x,yr,�� L��x,yr�

�n�x,yr,zr��x

dzr. �16�

Equation �16�, which assumes a geometrical optics wavepropagation model, indicates that the computed angularshifts of the beams centroids are related to the x derivative ofthe refractive index distribution by an x-ray transform.

D. Tomographic imaging model for reconstruction ofultrasmall-scatter image

The MIR imaging model in Eq. �12� also provides thebasis for the tomographic reconstruction of a volumetric im-age that reflects the ultrasmall-angle scattering properties ofthe object. We can associate with position r� inside the objecta local angular impulse response function f l�� ;x ,yr ,zr�. Thisfunction describes how the angular intensity spectrum of theprobing beam is perturbed by an infinitesimal volume of theobject at location �x ,yr ,zr�. Here, the superscript “l” is em-ployed to distinguish the local impulse response from theobject impulse response employed in Eq. �12�. Note thatf l�� ;x ,yr ,zr� is independent of the view angle � because theultrasmall-angle scattering properties of each infinitesimalvolume element are assumed to be isotropic. We will assumethat f l�� ;x ,yr ,zr� is a bounded and continuous function of �

and will let f̃ l� ;x ,yr ,zr� denote the 1D Fourier transform off l�� ;x ,yr ,zr� with respect to �. Here, is the Fourier vari-able conjugate to �. After penetrating through the object, thebeam is described by a convolution of its incident angularspectrum with the local angular impulse responses of all in-finitesimal volumes that L��x ,yr� intersects. By use of theFourier-convolution theorem, it must be true therefore that

f̃��;x,yr� = ∀zr�L��x,yr�

f̃ l�;x,yr,zr� , �17�

where f̃�� ;x ,yr� is the 1D Fourier transform of f�� ;x ,yr�with respect to �.

Let Ml�x ,yr ,zr� and M�x ,yr ,�� denote the second centralmoments of f l�� ;x ,yr ,zr� and f�� ;x ,yr�, respectively,
which are defined as


Ml�x,yr,zr� = �−

�� − ��x,yr,��2f l��;x,yr,zr�d� �18�

and

M�x,yr,�� = �−

�� − ��x,yr,��2f��;x,yr�d� , �19�

where ��x ,yr ,�� is the angular shift of the beam as de-scribed in Eq. �16�.

In practice, a discrete approximation of M�x ,yr ,�� can becomputed via Eq. �11�. By use of Eq. �17�, it can be shownthat

M�x,yr,�� = �−

Ml�x,yr,zr�dzr, �20�

which indicates that Ml�x ,yr ,zr� can be reconstructed by in-verting an x-ray transform. A heuristic derivation of Eq. �20�is provided in the Appendix. By use of radiative transporttheory, we have confirmed in Ref. 16 the validity of Eq. �20�.

IV. TOMOGRAPHIC RECONSTRUCTION FROMPHOTON-LIMITED DATA

Due to the flux limitations of currently available benchtopx-ray sources, we expect that, in the foreseeable future, clini-cal implementations of CT-MIR will be photon limited. It istherefore important to develop statistically robust reconstruc-tion methods that can mitigate the effects of high Poissonnoise levels in the measurement data. For such applications,the advantages of statistically motivated reconstruction algo-rithms over transform-based algorithms are well-known inthe medical imaging community.17 Later we describe an it-erative regularized least squares18 �RLS� approach for recon-struction of CT-MIR images.

In the following discussion, tomographic images and pro-jected images are represented as vectors, constructed by lexi-cographic ordering of the pixel values. The vector x refersgenerically to any one of the CT-MIR object properties �forabsorption, refraction, or ultrasmall-angle scatter�. The vec-tor y refers to the corresponding set of MIR parametric im-ages that represent the projection data from which x is re-constructed.

Using these definitions, the RLS algorithm seeks to mini-mize the objective function

J�x� = �y − Hx�2 + ��Qx�2

subject to x � 0 �but not for �� reconstruction� ,�21�

where � · � denotes the L2 norm, x is an object property �i.e.,image� to be reconstructed, H is a matrix operator describingthe 2D Radon transform, and Q is a Laplacian operator. Theleast-squares term in Eq. �21� encourages conformance of theCT-MIR image to the projected MIR images. The regulariza-tion term �Qx�2 represents the “roughness” of the imagessolution; therefore, its presence penalizes noisy solutions.The regularization parameter � controls the degree of
smoothness imposed by the penalty term. Note that the posi-

tivity constraint is not enforced when reconstructing the re-fraction image.

The objective function can be minimized by use of asteepest descent gradient algorithm. An iterative procedurefor accomplishing this is described by

x̂�i+1� = x̂�i� + �i�g�i�, �22�

where x̂�i� denotes an approximation of x obtained at iterationi:

g�i� = �J�x̂�i�� = HTy − �HTH + �QTQ�x̂�i�, �23�

describes the gradient of the object function, and

�i� = −g�i�T

g�i�

g�i�THTHg�i�

. �24�

Equations �22�–�24� are applied until the algorithm con-verges or for a fixed number of iterations. In this work, weadopted the latter approach.

V. EXPERIMENTAL STUDIES

A. Phantoms and experimental data

A physical phantom was constructed that exhibited vari-ous combinations of absorption, refraction, and ultrasmall-angle scattering effects. The phantom, which was utilized inRef. 1 and is shown in Fig. 4, consisted of a Lucite jarcontaining a Lucite rod and a sheet of paper rolled into acylinder. A second sample corresponded to a human talusbone within a left-ankle joint. The soft tissues covering thebones were present. Experimental CT-MIR studies of thesesamples were conducted at the National Synchrotron LightSource X15A beamline. This dedicated imaging beamlinehas been employed previously in studies of DEI and MIR.1,2

The phantom object was imaged using a beam energy of40 keV at each of 11 analyzer positions, ranging from −4 to+4 �rad with 0.8 �rad increments. Additional details regard-ing the data acquisition parameters for the phantom imagingstudy can be found in Ref. 2. The surface exposure dose wasnumerically calculated to be 73.2 mGy �water equivalent�.

FIG. 4. The physical phantom used in CT-MIR study.

This corresponded to a maximum count rate of 1000 photons


per 50 �m�50 �m detector pixel. The talus bone was im-aged using a beam energy of 30 keV at each of the 11 ana-lyzer positions. The surface dose was approximately49.2 mGy. For both objects, the measurement data were ac-quired at 360 evenly spaced tomographic view angles overthe interval �0,2��.

To simulate photon-limited data, we considered the mea-sured synchrotron data y�� ;x ,yr� to be essentially noisefree. The effect of photon noise was simulated by generatingPoisson-distributed data samples in software based on the“noise-free” measurement data. Because the variance of aPoisson random variable is equal to its mean, the noise levelin the measurement data was quantified by the highest meanphoton count per pixel that it contained. In this work, weconsidered two sets of noisy data. The first data set had anoise level of 10 photons per 50 �m�50 �m detector pixel�mean photon count�. This noise level is approximately 100times greater than that of the original synchrotron data andcorresponds to a surface dose of 0.733 mGy �water equiva-lent�. A second, less noisy, data set was generated that cor-responded to a mean photon count of 50 photons per50 �m�50 �m detector pixel and a surface dose of3.66 mGy �water equivalent�.

B. Tomographic reconstruction

At each view angle � the parametric images pm,n, Mm,n,and ��m,n were computed by use of Eqs. �8�, �11�, and �9�,respectively. As described previously, these parametric im-ages represent the x-ray transforms of the object propertiesthat CT-MIR aims to reconstruct. From these data sets, trans-verse slices �i.e., planes of constant x� of the object proper-ties were reconstructed by use of the filtered backprojection�FBP� algorithm and the proposed RLS method. In our
implementation of the RLS method, 100 iterations were em-

ployed, and the smoothing parameter was set to �=0.1. Thiscombination of settings was found to effectively suppress thenoise while preserving important image features. The dimen-sion of all the reconstructed images was 246�246 pixel2

and the pixel size was 380 �m with slice thickness of50 �m.

For purposes of comparison, the CT-DEI method pro-posed in Ref. 14 was also implemented. The DEI measure-ment data corresponded to analyzer settings of one half ofthe full width at half maximum of the rocking curve R��. Ateach tomographic view angle, estimates of pm,n and ��m,n

were calculated by use of the DEI method as described inRef. 4. From these data sets, transverse slices of the objectproperties ��x ,yr ,zr� and �n /�x�x ,yr ,zr� were reconstructedby use of the FBP algorithm. The reconstructed matrix andpixel sizes were the same as for the CT-MIR images de-scribed earlier.

VI. RECONSTRUCTED IMAGES

The CT-MIR images of the phantom object that were re-constructed from the original synchrotron radiation data�without artificially enhanced noise levels� by use of the RLSand FBP algorithms are shown in Fig. 5, top row and middlerow, respectively. In this figure, images of the object’s at-tenuation, refractive index gradient, and ultrasmall-anglescattering properties are displayed from left to right. Becausethe paper roll is composed of fibrous structures that haveradii of the order of 10 �m �which is on subdetector-pixeldimension�, it represents a scattering object that is expectedto broaden the angular spectrum of the probing x-ray beam.This behavior is reflected in the CT-MIR scattering images,shown in the right panels of the top and middle rows of Fig.5, where the paper roll is represented by large values in the

FIG. 5. CT-MIR images of the physical phantom thatwere reconstructed by use of the RLS �upper row� andFBP �middle row� algorithm. The attenuation, refrac-tion, and ultrasmall-angle scattering images are shownin the left, center, and right panels, respectively. Themaximum count was approximately 1000 photons/pixel. The corresponding CT-DEI images are shown inthe bottom row.

scatter image. Alternatively, the paper roll is represented by


relatively lower pixel values in the CT-MIR refraction gradi-ent images that are shown in the center panels of the top andmiddle rows of Fig. 5. The expected strong refraction at the

FIG. 6. Image profiles corresponding to images shown in Fig. 5. The gray

edges of the Lucite rod, and the constant attenuation within


all the Lucite regions �jar and rod�, are conveyed by theimages in the middle and left panels, respectively, of Fig. 5.Note that the Lucite jar does not appear prominently in the

profiles at the bottom represent expected profiles for each object property.
level
refraction images because its walls are approximately paral-


lel to the x axis and therefore �n�x ,yr ,zr� /�x�0 away fromthe ends of the jar. It is also useful to note that images re-constructed by use of the FBP algorithm contain more arti-facts than those reconstructed by use of the RLS method. Forexample, the absorption values inside the paper roll shouldbe the same as those of the background, which is not the casefor the CT-MIR absorption image reconstructed by use of theFBP algorithm shown in the left panel of the middle row ofFig. 5.

The bottom row of Fig. 5 contains the attenuation andrefraction images of the phantom object that were recon-structed by use of the CT-DEI method. The attenuation andrefraction images reconstructed using the CT-DEI methodare seen to be of poorer visual quality than the correspondingCT-MIR images, especially in the vicinity of the paper roll.This can be attributed to the fact that the paper roll producedultrasmall-angle scattering that was not compensated for inthe CT-DEI imaging model. These observations are consis-tent with the results of our previous comparison of planarDEI and MIR.2

Profiles through the central horizontal rows of the imagesin Fig. 5 are contained in Fig. 6. The bottom row contains theexpected profiles. Note that because the true profiles are notknown exactly, the “expected” profiles convey what wewould expect based on our qualitative understanding of the


attenuation, refractive, and ultrasmall-angle scattering prop-erties of the phantom materials. These figures confirm thatthe CT-MIR profiles are more consistent with the expectedprofiles than are the corresponding CT-DEI profiles. A quan-titative comparison of the CT-DEI and CT-MIR methodswithin the context of a well-defined diagnostic task remainsan important topic for future work.

The CT-MIR images of the phantom object that were re-constructed from the artificially enhanced noisy data sets cor-responding to maximum count rates of 50 photons per pixeland 10 photons per pixel are contained in Figs. 7 and 8,respectively. The images in the top and bottom rows werereconstructed by use of the RLS method and FBP algorithm,respectively. Despite the high noise levels in the data, mostof the features that are present in the noise-free images �Fig.5� are still discernable in the reconstructed noisy images. Theiterative reconstruction method was also found to produceCT-MIR images that possessed weaker artifacts and betterapparent separation of certain object structures from thebackground than those reconstructed by the FBP algorithm.These results suggest that photon-limited implementations ofCT-MIR may be viable for biomedical imaging applications.

Figure 9 contains images of the head of a human talusbone reconstructed by use of the CT-MIR method �top row�and the CT-DEI method �lower row�. In both implementa-

FIG. 7. CT-MIR images of the physical phantom thatwere reconstructed from the photon-limited data �50photons/pixel at maximum� by use of the RLS �upperrow� and FBP �lower row� algorithm.

FIG. 8. CT-MIR images of the physical phantom thatwere reconstructed from the photon-limited data �10photons/pixel at maximum� by use of the RLS �upperrow� and FBP �lower row� algorithm.


tions, the FBP algorithm was employed. Our preliminaryanalysis of the images suggests that while the attenuationimage reflects the same information as produced in conven-tional x-ray radiography, the refraction and ultrasmall-anglescatter images reveal additional valuable information. Al-though the cartilage covering the articular surface of the headof the talus is faintly visible in the attenuation image, it isclearly visible in the refraction and ultrasmall-angle scatterimages �between arrows�. This is of clinical significance be-cause cartilage is invisible in conventional radiography. Inaddition, a ligament attached to the bone is visible �lower leftarrow�. Although the border of the cartilage is visible in theCT-DEI refraction image, the image is contaminated heavilyby artifacts. The image artifacts can be attributed to signifi-cant ultrasmall-angle scattering produced by the bone, whichis not acknowledged in the CT-DEI imaging model. The in-ability of the CT-DEI images to clearly reveal referencestructures surrounding the cartilage can hinder the interpre-tation of the images. Figure 10 contains the CT-MIR imagescorresponding to a different transverse slice of the humantalus bone that is displaced by 200 �m.


VII. SUMMARY

The development of novel x-ray imaging techniques thatexploit contrast mechanisms other than x-ray absorption canpotentially revolutionize the field of biomedical x-ray imag-ing. Such techniques hold great promise for imaging lowcontrast soft-tissue structures14 using radiation doses that areless than that imparted by existing radiographic techniques.This is because phase-sensitive methods, such as MIR, canfunction well at high x-ray energies, where absorption con-trast and radiation dose are low.

Towards this end, we have recently proposed and investi-gated a new imaging method called MIR �Refs. 1 and 2� thatproduces a comprehensive description of an object’s absorp-tion, refraction, and ultrasmall-angle scattering properties.Although the three images produced in MIR provide a richdescription of the object’s physical properties, they are 2Dimages that represent projections of the 3D distributions ofobject properties. This can render it difficult to spatially cor-relate the detailed information regarding the object propertiesthat is contained in the images.

FIG. 9. CT-MIR images of a human talus bone and CT-DEI images of a human talus bone slice 45 �with slicethickness 50 �m�.

FIG. 10. CT-MIR images of a human talus bone sliceNo. 49 �with slice thickness 50 �m�.


In this work, we have developed and investigated a to-mographic implementation of MIR, which we refer to asCT-MIR. An explicit imaging model for CT-MIR was pro-vided, which revealed that the parametric images computedin MIR were related to the corresponding 3D object proper-ties via an x-ray transform. The CT-MIR method utilizes thethree parametric images produced in MIR as the raw projec-tion data from which volumetric images of the object’s at-tenuation, refractive index gradient, and ultra small-anglescattering properties can be reconstructed.

The CT-MIR method was implemented experimentallyand employed for reconstruction of the absorption,refractive-index gradient, and ultrasmall-angle scatteringproperties of a phantom object and a human talus bone. Itwas demonstrated that, in addition to producing an image ofthe ultrasmall-angle scattering properties of the object that isnot produced in CT-DEI, the CT-MIR method produces im-ages of the refractive index gradient and attenuation proper-ties that are more accurate then those produced by CT-DEI.This can be explained by the fact that the CT-DEI imagingmodel does not account for the effects of ultrasmall-anglescattering that are present in many biological tissues.

In future clinical implementations of CT-MIR, the mea-surement data will likely be photon limited. To address thisreality, we investigated the CT-MIR method under photon-limited conditions and utilized a regularized iterative recon-struction method for reconstruction of CT-MIR images fromexperimental data sets that contain enhanced Poisson noiselevels. Despite the high noise levels in the data, most of theobject features were still discernable in the reconstructednoisy images. The iterative reconstruction method was alsofound to produce CT-MIR images that possessed weaker ar-tifacts and better apparent separation of certain object struc-tures from the background than those reconstructed by theFBP algorithm. These results strongly suggest that photon-limited implementations of CT-MIR may be viable for bio-medical imaging applications.

Because CT-MIR is in a preliminary stage of its develop-ment, there remain numerous important and interesting as-pects of the method to explore. We are currently investigat-ing the diagnostic utility of CT-MIR for several importantmedical imaging tasks that include breast cancer imagingand visualization of articular cartilage. An important topic offuture research is the task-based assessment and optimizationof CT-MIR reconstruction algorithms.

ACKNOWLEDGMENTS

This research was supported by NIH/NIAMS Grant No.AR48292 and NIH/NCI Grant No. CA111976.

APPENDIX

Below we provide a heuristic derivation of Eq. �20� basedon a semidiscrete representation of the ultrasmall-angle scat-tering properties of the object. The object will be describedby a 3D array of voxels that each have dimensions �p �i.e.,the volume of each voxel is �p3�. Associated with a voxel
centered at location ri will be a voxel angular impulse re-

sponse function f l�� ;ri�. Here, the superscript l is employedto distinguish the voxel impulse response from the objectimpulse response employed in Eq. �12�. Note that f l�� ;ri� isindependent of the view angle � because the ultrasmall-angle scattering properties of each voxel are assumed to beisotropic.

Consider a probing beam I0�� that is incident on a voxelcentered at location ri= �x ,y ,z�. After traversing the voxel,the angular intensity spectrum of the beam is given by

I0�� * f l�w�yr,�;ri��;ri� , �A1�

where w�yr ,� ;ri� is a weight factor that represents the con-tribution of the voxel to the spreading of the transmittedbeam due to ultrasmall-angle scattering. The weight factorcan be defined as

w�yr,�;ri� = ��yr,�;ri��− 13 , �A2�

where ��yr ,� ;ri� denotes the path length �in the �y ,z� plane�through the voxel centered at location ri for a beam that ismeasured at detector location �x ,yr� at view angle �. Notethat a beam that travels along the diagonal of the voxel hasthe largest possible path length with �=�p�2.

Let Ml�ri� and Mlw�yr ,� ;ri� denote the second central

moments of f l�� ;ri� and f l�w�yr ,� ;ri�� ;ri�, respectively. Itcan be shown readily that Ml�ri� and Ml

w�yr ,� ;ri� are relatedas

Mlw�yr,�;ri� =

1

�w�yr,�;ri��3 Ml�ri� = ��yr,�;ri�Ml�ri� .

�A3�

If I0��= I0�� then Mlw�yr ,� ;ri� describes the second cen-

tral moment of the beam after traveling through the voxel.Equation �A3� states that the angular spreading of the beam�i.e., Ml

w�yr ,� ;ri�� increases linearly with the voxel pathlength. This linear behavior is consistent with, and requiredby, the MIR imaging model described by Eq. �3� �or equiva-lently Eq. �12��. We have confirmed the linearity of the MIRimage parameters theoretically and experimentally in a sepa-rate work that is summarized in a technical report.16

Let N�x ,yr ,�� denote the number of voxels that a probingbeam intersects when traveling through �the discretized rep-resentation of� the object and is measured at detector location�x ,yr� at view angle �. According to Eqs. �12� and �A1�, themeasured angular intensity spectrum is given by

y��;x,yr,�� = R�� * fN��;x,yr,�� , �A4�

where fN�� ;x ,yr ,�� is defined as

fN��;x,yr,�� = f l�w�yr,�;r1��;r1� * ¯

* f l�w�yr,�;rN��;rN� , �A5�

where the convolution is with respect to � and the coordi-nates r1¯rN denote the centers of the N=N�x ,yr ,�� voxelsthat are intersected by the probing beam as it travels to de-tector location �x ,yr�. We will represent these voxel locations
by the set


L�x,yr,�� r1 ¯ rN� .

Let MN�x ,yr ,�� denote the second central moment offN�� ;x ,yr ,�� that can be expressed as

MN�x,yr,�� = �r�i�L�x,yr,�� Mlw�yr,�;ri� ,

or, equivalently,

MN�x,yr,�� = �r�i�L�x,yr,�� yr,�;ri�Ml�ri� . �A6�

Equation �A6� indicates that the second central moment offN�� ;x ,yr ,�� is a weighted sum of the second central mo-ments of the voxel impulse responses f l�� ;ri� that corre-spond to voxels that are intersected by a beam that is mea-sured at detector location �x ,yr�. Equation �11� can beemployed to determine an estimate of MN�x ,yr ,�� from dis-cretely sampled measurement data. Therefore, Eq. �A6� rep-resents a discrete tomographic imaging model in the form ofa system of linear equations that can be solved �e.g., byalgebraic reconstruction algorithms� for determination of thesecond central moments of the local impulse response func-tions f l�� ;ri�.

A continuous version of the tomographic imaging modelcan be obtained from Eq. �A6� as a limiting case where thediscretization of the object becomes increasingly fine. IfMl�ri� are viewed as spatial samples of a function Ml�r� thatis integrable in the sense of Riemann, then in the limit as thevoxel size �p→0 �which implies �L�x ,yr ,�� → and �→0� one obtains the x-ray transform

M�x,yr,�� = �−

Ml�r�dzr, �A7�

where M�x ,yr ,�� is defined as the second central moment ofthe object impulse response function f�� ;x ,yr�.

a�Electronic mail: [email protected]. N. Wernick, O. Wirjadi, D. Chapman, O. Oltulu, Z. Zhong, and Y.Yang, “Preliminary investigation of a multiple-image radiographymethod,” Proc. IEEE/NIH International Symposium on Biomedical Imag-ing, pp. 129–132, 2002.

2M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y.Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman,“Multiple-image radiography,” Phys. Med. Biol. 48, 3875–3895 �2003�.

3
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Hartwig, and W. Thomlinson, “A method to extract quantitative informa-tion in analyzer-based x-ray phase contrast imaging,” Appl. Phys. Lett.82, 3421–3423 �2003�.

4D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N.Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction en-hanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 �1997�.

5J. Mollenhauer, A. E. Aurich, Z. Zhong, C. Muehleman, A. A. Cole, M.Hasnah, O. Oltulu, K. E. Kuettner, A. Margulis, and L. D. Chapman,“Diffraction-enhanced x-ray imaging of articular cartilage,” OsteoarthritisCartilage 10�3�, 163–171 �2002�.

6E. D. Pisano, R. E. Johnston, D. Chapman, J. Geradts, M. V. Iacocca, C.A. Livasy, D. B. Washburn, D. E. Sayers, Z. Zhong, M. Z. Kiss, and W.Thomlinson, “Human breast cancer specimens: Diffraction-enhanced im-aging with histologic correlation-improved conspicuity of lesion detailcompared with digital radiography,” Radiology 214, 895–901 �2000�.

7E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J.Baruchel, and J. Härtwig, “Quantitative comparison between two phasecontrast techniques: Diffraction enhanced imaging and phase propagationimaging,” Phys. Med. Biol. 50�4�, 709–724 �2005�.

8S. Majumdar, A. S. Issever, A. Burghardt, J. Lotz, F. Arfelli, L. Rigon, G.Heitner, and R. Menk, “Diffraction enhanced imaging of articular carti-lage and comparison with micro-computed tomography of the underlyingbone structure,” Eur. Radiol. 14�8�, 1440–1448 �2004�.

9Y. I. Nesterets, T. E. Gureyev, D. Paganin, K. M. Pavlov, and S. W.Wilkins, “Quantitative diffraction-enhanced x-ray imaging of weak ob-jects,” J. Phys. D 37�8�, 1262–1274 �2004�.

10U. Bonse and M. Hart, Appl. Phys. Lett. 7, 238 �1965�.11L. Rigon, H. -J. Besch, F. Arfelli, R.-H. Menk, G. Heitner, and H. P.

Besch, “A new DEI algorithm capable of investigating sub-pixel struc-tures,” J. Phys. D 36, 107–112 �2003�.

12M. N. Wernick, J. G. Brankov, D. Chapman, Y. Yang, C. Muehleman, Z.Zhong, and M. A. Anastasio, “Multiple-image radiography and computedtomography,” Proc. SPIE 5535, 369–379 �2004�.

13J. G. Brankov, M. N. Wernick, D. Chapman, Z. Zhong, C. Muehleman, J.Li, and M. Anastasio, “Multiple-image computed tomography,” Proc.IEEE/NIH International Symposium on Biomedical Imaging, 2004.

14F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, D. Chapman, I. Orion, andW. C. Thomlinson, “Computed tomography of x-ray index of refractionusing the diffraction enhanced imaging method,” Phys. Med. Biol. 45,933–946 �2000�.

15S. Fiedler, A. Bravin, J. Keyrilainen, M. Fernandez, P. Suortti, W. Thom-linson, M. Tenhunen, P. Virkkunen, and M.-L. Karjalainen-Lindsberg,“Imaging lobular breast carcinoma: Comparison of synchrotron radiationCT-DEI technique with clinical CT, mammography and histology,” Phys.Med. Biol. 49, 175–188 �2004�.

16G. Khelashvili, J. G. Brankov, D. Chapman, M. A. Anastasio, Y. Yang, Z.Zhong, and M. N. Wernick, “A physical model of multiple-image radiog-raphy,” Phys. Med. Biol. 51, 221–236 �2006�.

17D. S. Lalush and M. N. Wernick, “Iterative image reconstruction,” Emis-sion Tomography: The Fundamentals of PET and SPECT �Academic, SanDiego, 2004�.

18J. Fessler, “Penalized weighted least-squares reconstruction for positron
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