A cone-beam reconstruction algorithm for circle-plus-arc data-acquisition geometry

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 9, SEPTEMBER 1999 815

A Cone-Beam Reconstruction Algorithm forCircle-Plus-Arc Data-Acquisition Geometry

Xiaohui Wang,Member, IEEE,and Ruola Ning*

Abstract—In cone-beam computerized tomography (CT), pro-jections acquired with the focal spot constrained on a planar orbitcannot provide a complete set of data to reconstruct the objectfunction exactly. There are severe distortions in the reconstructednoncentral transverse planes when the cone angle is large. In thiswork, a new method is proposed which can obtain a completeset of data by acquiring cone-beam projections along a circle-plus-arc orbit. A reconstruction algorithm using this circle-plus-arc orbit is developed, based on the Radon transform andGrangeat’s formula. This algorithm first transforms the cone-beam projection data of an object to the first derivative ofthe three-dimensional (3-D) Radon transform, using Grangeat’sformula, and then reconstructs the object using the inverseRadon transform. In order to reduce interpolation errors, newrebinning equations have been derived accurately, which allowsone-dimensional (1-D) interpolation to be used in the rebinningprocess instead of 3-D interpolation. A noise-free Defrise phantomand a Poisson noise-added Shepp–Logan phantom were simulatedand reconstructed for algorithm validation. The results fromthe computer simulation indicate that the new cone-beam data-acquisition scheme can provide a complete set of projection dataand the image reconstruction algorithm can achieve exact recon-struction. Potentially, the algorithm can be applied in practicefor both a standard CT gantry-based volume tomographic imag-ing system and a C-arm-based cone-beam tomographic imagingsystem, with little mechanical modification required.

Index Terms—Circle-plus-arc orbit, cone-beam computerizedtomography, cone-beam reconstruction algorithm, Radon trans-form.

I. INTRODUCTION

I N the past two decades, X-ray computerized tomography(CT) has revolutionized the diagnostic imaging and nonde-

structive test imaging techniques [1]. However, the currentCT technology is limited by the relatively low longitudi-nal resolution and the relatively slow volume scan speed.One method utilized to address this problem is the use ofcone-beam tomography, where the single circular orbit iswidely adopted, due to its minimal mechanical complexityand hardware modification of the current CT scanner [2].Feldkamp’s algorithm, which targets the single circle data-

Manuscript received April 7, 1997; revised August 4, 1999. This work wassupported in part by the National Institutes of Health under Grant 5R01 HL48603. The Associate Editor responsible for coordinating the review of thispaper and recommending its publication was C. Crawford.Asterisk indicatescorresponding author.

X. Wang was with Department of Radiology and the Department ofElectrical and Computer Engineering, University of Rochester, Rochester, NY.He is now with the Health Imaging Research Laboratory, the Eastman KodakCompany, Rochester, NY 14650-2033 USA (e-mail: [email protected]).

*R. Ning is with the Department of Radiology and the Department ofElectrical and Computer Engineering, the University of Rochester, Rochester,NY 14642 USA (e-mail: [email protected]).

Publisher Item Identifier S 0278-0062(99)09057-6.

acquisition geometry, has been commonly accepted becauseof its excellent computation efficiency [3]. However, thisalgorithm is mathematically approximate and is limited to thereconstruction of small objects. When imaging large objects,such as the human brain or chest, the cone angle of X-raybeams can span from 10to 20 or larger, in which caseFeldkamp’s algorithm degrades severely in reconstruction ac-curacy at locations far away from the central transverse plane.For exact three-dimensional (3-D) cone-beam reconstructions,Tuy showed that each projection plane passing through theobject should intersect the orbit of the focal point [4]. Thisrequirement is referred to as Tuy’s data sufficiency condition.The single circular orbit, in which the focal point always lieson a circle, does not satisfy this condition because planesparallel to the circle do not intersect the orbit.

To address the problem of Feldkamp’s algorithm in cone-beam tomography, nonplanar data-acquisition orbits, as wellas their mathematical fundamentals, have been proposed bya number of researchers [4]–[16]. Among them, the mostrepresentative are Tuy, Smith, and Grangeat. Tuy developedan analytically exact reconstruction formula [4], which wasnumerically implemented later by Zeng and Gullberg using anorbit consisting of a circle and two orthogonal lines [5]. How-ever, the implementation was approximate. The approachesdeveloped by Smith [6] and Grangeat [7] differ from Tuy’sin that they both use an intermediate function between thecone-beam projection and the 3-D Radon transform of theobject function. Smith’s function is defined as the Hilberttransform of the first derivative of the Radon transform,while Grangeat’s function is simply the first derivative ofthe Radon transform. To implement Smith’s algorithm, manyissues need to be addressed. For example, the algorithm needsto be approximated in a discrete format. Smith and Chendiscussed those issues and made some recommendations inone of their later publications [8], in which a simulationorbit was used which can be considered as two periods of asinusoid that have been wrapped around a cylindrical surfaceto form a close 3-D curve. Zeng and Gullberg modifiedSmith’s function and provided a practical implementationof the reconstruction algorithm, using an orthogonal circle-and-line orbit [10]. However, this still is an approximateimplementation.

Kudo and Saito showed that both Smith’s and Grangeat’sfunctions can be described by a single mathematical expres-sion, namely, the Smith–Grangeat inversion formula [11].They further obtained a new reconstruction algorithm byreformulating the Smith–Grangeat formula. The algorithmwas validated by a number of orbit examples, such as dual

0278–0062/99$10.00 1999 IEEE

816 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 9, SEPTEMBER 1999

orthogonal circles, the helical orbit, and the orthogonal circle-and-line orbit. They stated that the algorithm is approximateif it is based upon Smith’s function, while it is exact if basedupon Grangeat’s function. Using Grangeat’s function for 3-Dreconstruction usually involves a rebinning process, as Wenget al. did for the helical orbit [13]. To improve the geometricmapping and interpolation accuracy in the rebinning process,Eriksson and Danielsson proposed a linogram method for thehelical orbit [14]. Defrise and Clack proposed their recon-struction algorithm, using shift-variant filtering to eliminatethe rebinning process when applying Grangeat’s formula [15].They validated the algorithm using a dual-orthogonal-circleorbit. In practice, the cone-beam orbit is usually spatiallysampled and is composed of a discrete vertex set. Nooet al.recently analyzed the general discrete vertex set and proposedthree rebinning algorithms which can be used for a varietyof orbits [16]. A number of efficient algorithms based onGrangeat’s function have been developed. For example, Nooet al. [16] used Feldkamp’s algorithm to process projectionsof the circular orbit, while using shift-variant filtering for thesupplementary orbit and another, by Axelsson and Danielsson,used the direct Fourier method to reduce the computation from

to [17], [18]. Noo’s method can be usedto process many “circle-plus” orbits. Grangeat’s algorithm wasextended by Kudo and Saito for partially measured cone-beamprojection data [19], [20]. The new extension requires a lessrestrictive data sufficiency condition than Tuy’s.

A majority of the aforementioned works are designedto target single photon emission computerized tomogra-phy (SPECT). For cone-beam X-ray volume tomographicreconstruction systems, including the X-ray volume CTsystem and the newly commercialized C-arm-based cone-beam tomographic imaging system, considerations have tobe taken, not only for the exactness of the reconstructionalgorithm, but also for the practical feasibility of collectingthe projection data when using a specific orbit. One would liketo build a cone-beam CT scanner with a small modification tothe current spiral CT gantry. To this end, the helical orbit andthe orthogonal circle-and-line orbit are preferable. However,when using these orbits, either the patient or the whole gantryneeds to be translated during the data acquisition. Movingthe patient is sometimes undesirable, especially when thepatient cannot be moved during an image guided interventionalprocedure requiring intraoperative imaging. On the other hand,moving the whole gantry while keeping the patient stillcan significantly increase the complexity of the cone-beamCT. For the C-arm-based cone-beam tomographic imagingsystem, which is specially designed for imaging guidedinterventional procedures requiring intraoperative imagingin which moving the patient during tomographic scanning isprohibited, neither the helical orbit nor the orthogonal circle-and-line orbit is practical for clinical applications. In addition,the reconstruction algorithms currently used for the C-arm-based system are either approximate or iterative [21], [22].

In this work, an analytic and exact reconstruction algo-rithm suitable for both the cone-beam volume CT and theC-arm-based tomographic imaging system is presented. Thisalgorithm utilizes a cone-beam data-acquisition orbit which

Fig. 1. Geometry of the circle-plus-arc orbit. The object is limited within thesphere of radiusR, which is concentric with the circle-plus-arc orbit. minis the minimum cone angle of the beam and�min is the minimum spanningangle of the arc orbit.

is the combination of a circle and a small orthogonal arc(Fig. 1). The arc orbit is introduced to provide focal pointsfor the projection planes that do not intersect the circularorbit. By doing this, Tuy’s data sufficiency condition canbe satisfied. When this method is applied to cone-beam CTin practice, it corresponds to (1), acquiring one set of two-dimensional (2-D) projections (namely, circle projections)while rotating the X-ray tube and an area detector on astandard CT gantry, then (2) fixing the X-ray tube and detectoron the gantry and acquiring another set of 2-D projections(namely, arc projections) while tilting the gantry by a smallangle. This can be accomplished by most clinically used CTgantries now available. Similarily, for a C-arm-based cone-beam tomographic imaging system, one can acquire the circleprojections by rotating the C-arm on a circular orbit, thenacquiring the arc projection by moving the C-arm in anorthogonal arc orbit. In both cases, the scan scheme does notinvolve either patient or gantry translation.

The derivation of the algorithm in this work is basedon the circle-plus-arc data-acquisition geometry, 3-D Radontransform and Grangeat’s formula. While the approaches ofDefrise and Clack [15] or Nooet al. [16] can be appropriate,a rebinning process is used, due to its exactness and simplicity.To reduce the interpolation errors in the rebinning process, newrebinning equations have been derived exactly for both thecircular orbit and the arc orbit. The equations for the circularorbit are independent of the scan angle, which provides twounique advantages: 1) one-dimensional (1-D) interpolation canbe used instead of 3-D trilinear interpolation and 2) reducedcomputation. A noise-free Defrise phantom and a Poissonnoise-added Shepp–Logan phantom were simulated to validatethe new algorithm. In addition, a preliminary investigationof the impact on image quality, resulting from reduced arclength and the arc sampling rate, was also conducted. Theimplementation of the circle-plus-arc data-acquisition geome-try on our newly-built cone-beam volume CT scanner will beaccomplished in the future.

II. THEORY

A. Circle-Plus-Arc Orbit and Tuy’s Data Sufficiency Condition

The circle-plus-arc orbit is shown in Fig. 1. First, the planeof the arc orbit is perpendicular to the circular orbit and the two

WANG AND NING: CONE-BEAM RECONSTRUCTION ALGORITHM 817

orbits intersect at the center of the arc. For the simplicity ofthe mathematical derivation, the two orbits are assumed to beconcentric at point and have the same radius. Second, it isassumed that the object function has a finite boundary,i.e.,

when for some finite positive (1)

where . Third, the cone beam originating from anypoint on the orbit should fully cover the object, i.e.,

(2)

where is the minimum required cone angle. Fourth, anextreme situation of the circle-plus-arc orbit occurs when itis constructing two orthogonal circles if the arc extends to awhole circle, in which case the size of the object should besmall enough to be enclosed by the orbit, i.e., the radius of thesphere that constrains the object has to satisfy the inequalitygiven by

(3)

Fifth, to satisfy Tuy’s data sufficiency condition, the arc orbitshould supply focal points to planes that do not intersectthe circular orbit. Among them, the outermost plane is 1)tangential to both the circular orbit and the sphere of radius,which constrains the object and 2) perpendicular to the planeof the arc orbit. If the minimum arc spanning angle is ,then from the geometry in Fig. 1, the following inequalityshould be satisfied:

(4)

Inequalities (2)–(4) guarantee that any plane that intersectsthe object will also intersect either the circular orbit or the arcorbit, which therefore fulfills the data sufficiency condition. Animportant conclusion can be drawn that the minimum spanningangle of the arc orbit should be twice as large as the minimumcone angle if the cone beam can contain the object completely.

B. Cone-Beam Projections and the 3-D Radon Transform

In this section, the cone-beam projection and 3-D Radontransform of the object function will be expressed in termsof the coordinate systems defined in this work. The cone-beam geometry is shown in Fig. 2. In the absolute 3-D spatialdomain, point is the origin of the coordinate system and

is the position vector of the cone-beam focal point.This convention will be used throughout the rest of this work.The virtual detector plane is defined to be perpendicular to

vector and always contains point . Due to this, one canuse to uniquely represent the detector plane. Pointisany point in the detector plane and is the unit directional

vector of , i.e.,

(5)

Fig. 2. The cone-beam projection is a collection of integrals of the objectfunctionf(~x) along lines originating from focal pointS. PointO is the originof the coordinate system. The virtual detector plane� contains pointO andOS is its normal vector.A is any point in the virtual detector plane.

Fig. 3. The Radon transform is the plane integral of the object functionf(~x). The Radon plane is defined as�( ;̂ �) where^ is its normal vector and� is its distance from the origin of the coordinate.

The cone-beam projection of the object function canbe expressed as

(6)

where is also called the directional vector along the ray ofthe line integral.

The Radon transform of the object is defined as the planeintegral of . In the 3-D Cartesian space (Fig. 3), anyRadon plane can be defined by a unit vectorand a scalar

where

(7)

is the normal vector of plane and is the distance fromthis plane to the origin of the coordinate system. The Radontransform of is given by

(8)

where the function constrains the 3-D integral within plane. The object function can be exactly reconstructed


Fig. 4. Cone-beam geometry. Radon plane�( ;̂ �) intersects thevw plane(also the virtual detector plane) at lineD1D2. The p axis is perpendiculartoD1D2 and the angle between thep axis and thev axis is�. The distancefrom the lineD1D2 to the center of the coordinate isp.

by using the inverse 3-D Radon transform [23]

(9)

if is known for every on set

(10)

Therefore, in cone-beam tomography, one can reconstruct theobject function from its cone-beam projection data if therelationship is established between the projection and the 3-DRadon transform.

C. Transformation of Cone-Beam Projections to Radon Space

Grangeat’s formula establishes the relationship between thecone-beam projection of the object function and the firstderivative of its 3-D Radon transform [7]. This formula willbe introduced here, using the coordinate systems of this work(Fig. 4). A local Cartesian coordinate system isdefined for the virtual detector plane with theaxis being

coincident with , plus the axis and axis in plane .The Radon plane , where the plane integral takes place,goes through focal point and intersects the detector plane

at line . Another local Cartesian coordinate systemis defined with the rotation of the axis and the

axis about the axis by an angle such that theaxis is perpendicular to line . The distance from the

origin of the coordinates to line is . Let point beanywhere on line with as its coordinate in thelocal system, then , the directional vector of the line

integral along in (6), can be uniquely determined byand . For this reason, the cone-beam projection along line

can be described by Now Grangeat’s formula

can be written as

(11)

where

(12a)

(12b)

The process based on (11) is usually computationally ex-pensive. For each point in the Radon space, an integraland a derivative need to be calculated. The computationtime for a medical image of a useful size, such as 512pixels for example, is prohibitive in practice. However, bysimply rewriting (11) in terms of the local coordinateand swapping the order of the integral and derivative, thecomputation can be significantly reduced [7]. Appendix Ashows the derivation of the new equation, which is written

(13)

where is the weighted cone-beam projection, i.e.,

(14)

Since the partial derivatives andon the right-hand side of (13) need to be calculated onlyonce for the whole reconstruction process, the computationcomplexity is greatly reduced. In this work, these partialderivatives are calculated by convoluting (using FFT) a 1-D derivative filter with for a fixed and afixed respectively. To reduce the reconstruction noise,the derivative filter is multiplied by a Hamming window in thefrequency domain. As Weng did [13], a line integral algorithm,based on linear interpolation between pixels, is applied to (13)for the integral calculations [24].

D. Rebinning to the Radon Domain

The rebinning process finds the transform functions amongthe parameters on the left-hand side and the right-hand sideof (13). For each point in the Radon space, thesetransform functions calculate the corresponding, the focalpoint position; , the direction of the integration; and,the distance from the line of integration to the origin ofthe coordinates. Depending on the requirement of efficiencyand accuracy, this process can be accomplished in two ways:1) forward mapping from discrete values of and to


Fig. 5. Projection data from the circular orbit. Focal pointS moves alongthe circular orbit. VectorOS is also theu axis. The angle between theuaxis and thex axis is �c.

discrete values of and with extrapolation in the Radondomain and 2) backward mapping from discrete values of

and to discrete value of and continous values ofand with interpolation in the domain. Wengetal. used the forward mapping method for their work [13].While their process reduces the computation load for thehelical orbit, it is not suitable for the circle-plus-arc geometry.This is because the discrete-to-discrete extrapolation may notreach every point in the Radon space and there may bediscontinuities (holes) which can be further exaggerated bythe second derivative operation during the reconstruction.Therefore, severe artifact is expected in the reconstructedimages for the forward mapping method. On the contrary,there are no discontinuities for the backward mapping method.Due to this, it is used in this work and and are alllinearly quantized into 256 levels in domain

Correspondingly,a new set of rebinning equations have been derived for thecircular orbit and the arc orbit, respectively.

1) Rebinning from the Circular Orbit:The circular orbit ispositioned in the plane and is centered at the origin ofthe coordinate system (Fig. 5). Theaxis and theaxis are defined as the rotation of theaxis and the axisabout the axis, respectively. The rotation angle is. Theintersection point of the axis and the circular orbit is thecone-beam focal point . As the focal point moves along thecircular orbit, a set of cone-beam projections are acquired. TheRadon plane intersects the detector plane (plane) at line

. Another local coordinate system, , is definedthe same as that in Fig. 4 and so are the definitions ofand

. Any Radon plane that intersects the circular orbit has twointersection points, except when the Radon plane is tangentialto the circular orbit. Either intersection point represents acorresponding focal point position. In order to improve thequality of the reconstructed images, both projections from thetwo focal points are used. First, the two points are named

and , respectively, and the position arrangement foris counterclockwise if observed above the

circular orbit. Second, the angle between and the axis

is and that between and the axis is . Then,for a given point in the Radon space, andcan be calculated directly from the coordinates ofand ,respectively.

As derived in Appendix B, and can be solved exactlyfor a given for

(15a)

for

for

(15b)

and for

(16a)

for

for

(16b)

The above two rebinning equations provide two advantages: 1)reduced computation for and since they are only anddependent, therefore once calculated, their values can be usedfor any and 2) reduced interpolation error. In practice,

and are discrete variables. For a given point inthe Radon space, the scan angles and of the cone-beam focal spot on the circular orbit can be obtained exactly.However, the actual scan angles are only available as discretenumbers, since cone-beam projections are acquired at a limitednumber of positions. Therefore, interpolation is necessary forthe rebinning, which is usually a 3-D interpolation ( and

). From (15) and (16), one can notice that since bothandare independent of the cone-beam scan angle, 1-D linear

interpolation of the cone-beam projections in thedimensionis enough for the rebinning, which can reduce the computationand possibly the interpolation error as well.

From (28) of Appendix B, one can find the region wherethe projection data from the circular orbit can contribute tothe Radon space

for

i.e.,

or

where (17)

which is the mathematical proof that shows that a singlecircular orbit does not satisfy Tuy’s data sufficiency condition.

2) Rebinning from the Arc Orbit:Being concentric withthe circular orbit, the arc orbit is placed in the plane withits center on the axis (Fig. 6). The local coordinatesystem is obtained by the rotation of theaxis and the axisabout the axis by an angle of . The intersection betweenthe arc orbit and the axis is the cone-beam focal point.The readers are referred to Fig. 4 for the definition of the local

coordinate system and the definitions ofand .


Fig. 6. Projection data from the arc orbit. Focal pointS moves along thearc orbit. VectorOS is also theu axis. The angle between theu axis andthe x axis is �a.

The arc orbit is able to provide projection data to the Radontransform in the region where the circular orbit does not. Frominequality (17), the complementary region is

(18)

As derived in Appendix C, and can be solved exactly fora given

(19a)

for

for

(19b)

E. Reconstruction of Object Function

After the first derivative of the Radon transformis obtained from the rebinning process, calculation of the sec-ond derivative can be accomplished by convolutingwith a 1-D derivative filter. Again, to reduce the reconstructionnoise, the filter is multiplied by a Hamming window in the fre-quency domain. The object function can then be reconstructedby using backprojection, as indicated in (9).

III. COMPUTER SIMULATION

A. Phantom and Orbit

Since the Defrise phantom (Fig. 7) can effectively demon-strate the artifact caused by the incompleteness of projectiondata acquired with a single circle cone-beam geometry, it hasbeen widely used for evaluating the performance of analyticcone-beam reconstruction algorithms [5], [13], [18]. In thiswork, the Defrise phantom consists of seven ellipsoids, eachseparated by a distance of 30.0 mm along theaxis. Theellipsoids are of the same size, and the lengths of their semi-major axes are 100.0, 100.0, and 7.5 mm in the anddirections, respectively. A sphere of radius mm,centering at the origin of the coordinates, can therefore containthis phantom. The density of the object is assumed 1.0 insidethe ellipsoids and 0.0 outside.

Fig. 7. Geometry of the Defrise phantom and the circle-plus-arc orbit. Thecircular orbit lies in thexy plane and the arc orbit in thexz plane. Thexaxis goes through the center of the arc.

Because two steps of derivative operations need to becalculated during the reconstruction, noise in the projectiondata can be severely exaggerated in the reconstructed image.To investigate the noise sensitivity issue for practical applica-tions, a standard Shepp–Logan phantom was simulated, withPoisson noise being added to the projection data. An entranceX-ray photon fluence of 400 000 photons/pixel/projection wasassumed. This photon fluence is approximately equivalentto 1.5-mR X-ray radiation around 50-keV effective photonenergy. The X-ray attenuation in the background region insidethe phantom is assumed being equal to that of water.

The circle-plus-arc orbit used in the simulation is shown inFig. 7. The circular orbit lies in the plane with an equationgiven by

(20)

and the arc orbit lies in the plane with the equation

for (21)

where mm is the radius of both orbits. A squaredetector plane of 300.0 mm on each side is assumed to have256 256 pixels. This geometrical setup of the orbit and thedetector yields a cone angle of 21.8, which can completelycover the Defrise phantom and the Shepp–Logan phantom.Tuy’s data sufficiency condition is also satisfied with thissetup. A total of 256 equiangular projections are simulatedfor the circular orbit [25]. Since during the rebinning processRadon planes corresponding to the circular orbit can find twofocal points, this in fact increases the sampling rate of thecircle. However, some Radon planes corresponding to the arcorbit can only find one focal point on the arc, in order toachieve comparable sampling rate on the arc a total of 65equiangular projections are simulated for the arc orbit.

B. 3-D Image Reconstruction

The procedure to reconstruct the 3-D images of the phan-toms from their cone-beam projections, acquired from thecircle-plus-arc orbit, is described as follows:

1) preweighting cone-beam projections (14);2) calculating partial derivatives of the preweighted pro-

jections (13);3) rebinning to the first derivative of Radon transform [(13),

(15), (16) and (19)];


(a) (b)

(c) (d)

Fig. 8. Reconstructed images of the central sagittal plane. (a) Perfect Radoninversion. (b) Feldkamp. (c) Grangeat: single circular orbit. (d) Grangeat:circle-plus-arc orbit.

(a) (b)

(c) (d)

Fig. 9. Profiles along the central horizontal lines of the images in Fig. 8. (a)Perfect Radon inversion. (b) Feldkamp. (c) Grangeat: single circular orbit. (d)Grangeat: circle-plus-arc orbit.

4) calculating the second derivative of Radon transform;5) reconstructing the images using inverse Radon transform

(9).

IV. RESULTS

The simulation results are shown in Figs. 8–12. First, aperfect reconstruction of the Defrise phantom is shown in

(a) (b)

(c) (d)

Fig. 10. Reconstructed images using a circle plus a half-sized arc orbit. (a)Central sagittal plane. (b) Noncentral transverse planez = 90 mm. (c) and(d) Profiles along the central horizontal lines of (a) and (b), respectively.

(a) (b)

(c) (d)

Fig. 11. Reconstructed images using a circle-plus-arc orbit. The samplingrate on the arc is reduced to one quarter of the normal case. (a) Central sagittalplane. (b) Noncentral transverse planez = 90 mm. (c) and (d) Profiles alongthe central horizontal lines of (a) and (b), respectively.

Fig. 8(a). This reconstruction was achieved by 1) calculatingthe plane integrals of the object function directly, in orderto bypass the cone-beam projection process and 2) applying thedirect inverse Radon transform. Reconstruction results fromFeldkamp’s algorithm and Grangeat’s formula, using the single


(a) (b)

(c) (d)

Fig. 12. (a) Perfect slice atz = �38:5 mm, directly calculated from theShepp–Logan phantom data. (b) Reconstructed image of the Shepp–Loganphantom with Poisson noise-added projection data. (c) and (d) Profiles alongthe dotted lines in (a) and (b), respectively.

circular orbit are shown in Fig. 8(b) and (c), respectively. Theresult from the algorithm of this work is shown in Fig. 8(d).The corresponding line profile of each image in Fig. 8 isshown in Fig. 9. Comparison of these images and their profilesshows that Feldkamp’s algorithm and Grangeat’s formula bothintroduce severe artifact in the noncentral transverse planeswhen the single circular orbit is used. The comparison alsoshows that our new algorithm can provide exact reconstructionand that it causes no image blurring or distortion.

To show the contribution of the arc orbit, which supplementsthe circular orbit to satisfy Tuy’s data sufficiency condition, areconstruction was obtained using projections from the circularorbit and from the arc orbit of only half the minimum requiredsize. As shown in Fig. 10, a small dc shift (9%) can beobserved in both the sagittal and transverse planes: the planeat mm and that at mm, respectively. No obviousartifact can be found, although reconstruction accuracy isslightly reduced.

The arc sampling rate (number of projections obtainedfrom the arc orbit), was also investigated (Fig. 11). Seven-teen equiangular projections from the arc orbit were used toreconstruct the phantom, which is about half the sampling rateused on the circle. The reconstructed images and their profilesclearly demonstrate that no obvious artifact (except% dcshift) can be observed, due to the reduction of the arc samplingrate.

A transverse slice of the Shepp–Logan phantom atmm is in Fig. 12(a) for reference, which is directly

calculated from the phantom data. The same slice recon-structed from cone-beam projection data with added Poissonnoise is shown in Fig. 12(b) for comparison. Both the images

and their corresponding profiles display good reconstructionquality, which indicates that the analytic cone-beam imagereconstruction algorithm is noise stable.

The simulations were accomplished on an ULTRA SPARC-1 SUN workstation using C++ language. It took approximate270 min to calculate the Radon cube of 256voxels in therebinning process and about 56 minutes to reconstruct animage plane of 256 pixels.

V. DISCUSSION

An analytic cone-beam reconstruction algorithm using acircle-plus-arc data-acquisition geometry has been developed,based on the Radon transform and Grangeat’s formula. Thealgorithm is validated by computer simulations with projectiondata of a noise-free Defrise phantom and a Poisson noise-added Shepp–Logan phantom. The simulation results indicatethat the circle-plus-arc data-acquisition geometry does providea complete set of data, and the algorithm can achieve exact3-D reconstruction, even in cases where Feldkamp’s algorithmfails.

The results from the investigation of the reduced arc lengthand arc sampling rate indicate that no obvious artifact isobserved when compared to Feldkamp’s algorithm, and onlya minor reduction of reconstruction accuracy is introduced.The practical significance of these results is that the data-acquisition time on the arc orbit can be significantly reducedby decreasing the arc length or arc sampling rate, withoutintroducing unacceptable reconstruction errors.

In practice, the new data-acquisition scheme and its re-construction algorithm can be applied to both a cone-beamvolume CT scanner that uses a standard CT gantry and a2-D detector, such as a flat-panel X-ray imager, and a C-arm-based cone-beam tomographic imaging system. To use it onthe standard CT gantry, the circle projection data are obtainedby rotating the X-ray tube and the detector, then acquiringthe arc projections while tilting the gantry by a small angle

15 – 30 with the X-ray tube and the detector being fixedon the gantry. This scan method is practically feasible ona standard CT gantry without introducing much mechanicalcomplexity, and can provide exact 3-D reconstructions ofan object with a diameter of 25–40 cm. On the other hand,for the C-arm-based cone-beam volume tomographic imagingsystem, the circle-plus-arc data-acquisition geometry can beeasily achieved by rotating the C-arm on a circular orbit for180 plus a cone angle, to acquire the circle projections [26]and then moving the C-arm for 15–30 in the orthogonalplane, to acquire the arc projections.

Compared to the helical orbit and the circle-plus-line orbit,the circle-plus-arc orbit does not need to move the patientduring tomographic scanning. With the development of moreand more tomographic imaging guided procedures, the circle-plus-arc cone-beam tomographic data-acquisition orbit canpotentially have an advantage over the helical orbit and thecircle-plus-line orbit for the design of a cone-beam volumeCT scanner that will be used for imaging guided procedures.However, to determine which geometry among the helicalorbit, the circle-plus-line orbit, and the circle-plus-arc orbit


should be chosen for the future design of a diagnostic cone-beam volume CT scanner, phantom or even dynamic studiesshould be performed. The comparison of these three differentorbits should include mechanical feasibility and mechanicalcomplexity of each orbit, and image quality resulting fromeach data-acquisition scheme. The comparison study is beyondthe scope of this paper. Currently we are constructing aflat-panel detector-based volume CT scanner and planing toconduct the phantom comparison studies of the three cone-beam orbits in the future.

APPENDIX A

Grangeat’s formula, as expressed in Eq. (11), can be rewrit-ten by swapping the integral and the derivative in order toimprove computation efficiency. First let

(22)

With the transforms in (12) from the coordinates to thecoordinates, by swapping the integral with the derivative,

(11) can be reformulated as

(23)

APPENDIX B

The rebinning equations for the projection data from thecircular orbit are given here. With reference to Fig. 5, theRadon plane which contains line and focal point canbe described in the coordinate system as

(24)

Since the transform from the absolute coordinate tothe local coordinate is given by

(25)

the Radon plane represented by (24) can be rewritten usingthe absolute coordinate in terms of parametersand as

(26)

Comparing (26) with the following representation of thisRadon plane in terms of and

(27)

it can be shown that

(28a)

(28b)

(28c)

(28d)

The solutions to the four subequations in (28) are for

(29a)

for

for

(29b)

and for

(30a)

for

for

(30b)

APPENDIX C

With reference to Fig. 6, the rebinning equations for thearc orbit are derived as follows. The arc orbit comes with therotation of focal point about the axis by an angle and

is defined as the axis. The transform between the localcoordinate and the absolute coordinate can

be described as

(31)

Again, the Radon plane represented by (24) can be rewrittenin terms of the absolute coordinate as

(32)


Comparison with (27), which is the representation of theRadon plane in terms of parameters and , yields thefollowing results:

(33a)

(33b)

(33c)

(33d)

Therefore, the solution to the above equations for a givenis

(34a)

for

for

(34b)

ACKNOWLEDGMENT

The authors wish to thank Dr. T. Morris, A. Norder and D.Hartley for proofreading this manuscript.

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A cone-beam reconstruction algorithm for circle-plus-arc data-acquisition geometry