advanced single-slice rebinning in cone-beam spiral ct · advanced single-slice rebinning in...

Advanced single-slice rebinning in cone-beam spiral CTMarc Kachelrießa)

Institute of Medical Physics, University of Erlangen–Nurnberg, Germany

Stefan SchallerSiemens AG, Medical Engineering Group, Forchheim, Germany

Willi A. KalenderInstitute of Medical Physics, University of Erlangen–Nurnberg, Germany

~Received 11 August 1999; accepted for publication 12 January 2000!

To achieve higher volume coverage at improved z-resolution in computed tomography ~CT!, sys-tems with a large number of detector rows are demanded. However, handling an increased numberof detector rows, as compared to today’s four-slice scanners, requires to accounting for the conegeometry of the beams. Many so-called cone-beam reconstruction algorithms have been proposedduring the last decade. None met all the requirements of the medical spiral cone-beam CT in regardto the need for high image quality, low patient dose and low reconstruction times. We thereforepropose an approximate cone-beam algorithm which uses virtual reconstruction planes tilted tooptimally fit 180° spiral segments, i.e., the advanced single-slice rebinning ~ASSR! algorithm. Ouralgorithm is a modification of the single-slice rebinning algorithm proposed by Noo et al. @Phys.Med. Biol. 44, 561–570 ~1999!# since we use tilted reconstruction slices instead of transaxial slicesto approximate the spiral path. Theoretical considerations as well as the reconstruction of simulatedphantom data in comparison to the gold standard 180°LI ~single-slice spiral CT! were carried out.Image artifacts, z-resolution as well as noise levels were evaluated for all simulated scanners. Evenfor a high number of detector rows the artifact level in the reconstructed images remains compa-rable to that of 180°LI. Multiplanar reformations of the Defrise phantom show none of the typicalcone-beam artifacts usually appearing when going to larger cone angles. Image noise as well as theshape of the respective slice sensitivity profiles are equivalent to the single-slice spiral reconstruc-tion, z-resolution is slightly decreased. The ASSR has the potential to become a practical tool formedical spiral cone-beam CT. Its computational complexity lies in the order of standard single-sliceCT and it allows to use available 2D backprojection hardware. © 2000 American Association ofPhysicists in Medicine. @S0094-2405~00!00804-X#

Key words: computed tomography ~CT!, spiral CT, multi-slice CT, cone-beam detector systems,3D reconstruction

I. INTRODUCTION

The introduction of multi-row detector systems in 1998 of-fered improved capability of volume scanning. Shorter ex-amination times at a higher z-resolution have becomeavailable.1 Nevertheless, to achieve isotropic submillimeterresolution, these multi-slice scanners, which typically mea-sure four slices simultaneously, still have to use a low tableincrement d per rotation. Covering complete organs, such asthe lung, at a high spatial resolution during a single breath-hold is not always feasible.

Increasing the number of detector rows would remedy thissituation. Moreover, the use of area detectors would moreefficiently use the available x-ray power, since a larger partof the x-ray cone would have to be utilized. Unfortunately,when going to 16, 32, or even more detector rows, one can-not neglect the effect of the increasing cone angle anymore,as it is currently done in four-slice CT. New reconstructionalgorithms that take into account that the measured fan-beams are tilted with respect to the z-axis have to be devel-oped.

The development of the so-called cone-beam reconstruc-

tion algorithms is divided into two parts: the exact and theapproximate algorithms. Exact algorithms try to exactly in-vert the cone-beam transform, either by calculating the Ra-don transform2,3 or by using a filtered backprojectionapproach.4,5 Doing so, they face several problems. In manyalgorithms the object has to be completely covered by thecone-beam for each projection.2,3 Recent approaches over-come this problem by using data combination for truncatedprojections to calculate complete Radon data. This, however,implies that for the case of a spiral trajectory the total spiralscan range has to completely cover the object. The problemof reconstructing an ROI ~region-of-interest; here, a rangealong the z-axis! from a spiral scan extending over the lengthof that ROI only became to be known as the long objectproblem.

Some exact algorithms handle the long object problem byadding two circles at the beginning and the end of the scanpath,6 which is not easily realizable in medical CT since thetable acceleration and deceleration must not exceed certainvalues due to mechanical and patient comfort reasons. Thealgorithm proposed by Schaller et al.7 exactly handles the

754 754Med. Phys. 27 „4…, April 2000 0094-2405Õ2000Õ27„4…Õ754Õ19Õ$17.00 © 2000 Am. Assoc. Phys. Med.

long object problem for the case of a spiral trajectory withoutadding two circles to the trajectory. Nevertheless, it does notallow for a projectionwise, i.e., sequential, processing of themeasured attenuation data ~raw data! which is of high inter-est due to computer memory reasons and reconstructionspeed. Even if this problem was solved, the exact algorithmshave a relatively high computational complexity in commonwhich makes them slow and thus somehow bulky for the usein medical CT.

On the other hand, there are the approximate cone-beamalgorithms.8–12 By not using the Radon space as an interme-diate domain the long object problem is avoided. Further-more, these algorithms are also easier to implement andcomputationally less demanding. Nevertheless, with increas-ing cone angle they introduce severe cone-beam artifacts andthus are acceptable for medical CT only when using a rela-tively low number of detector rows.13

We developed the advanced single-slice rebinning~ASSR! algorithm to potentially overcome most of theseproblems. ASSR is primarily based on Noo’s single-slicerebinning approach.14 However, ASSR uses tilted recon-struction planes instead of transaxial ones. Unlike other simi-lar approaches found in the literature15,16 ASSR is formu-lated as a rebinning algorithm. The measured spiral cone-beam data are rebinned along tilted reconstruction planes toparallel geometry. The tilt angle as well as the attachmentpoints to the spiral trajectory of these virtual reconstructionplanes are optimized using a minimization procedure. Therebinned data sets are reconstructed using a conventional 2Dparallel filtered backprojection to produce tilted images inspatial domain. The stack of tilted images is then resampledwith a z-filtering procedure to give the final reconstructedvolume data on the Cartesian grid.

ASSR is formulated as a rebinning algorithm and thus isable to handle almost any possible detector geometry as longas the focus trajectory is spiral and as long as the completerequired data are sampled. The object of this paper will be topresent practical and theoretical considerations concerningthe ASSR algorithm using a medical CT scanner with cylin-drical area detectors. A set of reconstructed virtual phantomdata corresponding to various scan modes ~varying numberof detector rows and varying table increment! will be pre-sented. Comparisons of the algorithm’s performance ~imagequality, dose, noise, reconstruction speed, etc.! will be madeto single-slice spiral CT with 180°LI reconstruction, thepresent gold standard, since this is the most challenging task.A benchmarking of ASSR and other approximate cone-beamalgorithms will be given in Ref. 13.

This paper is organized as follows. Section II gives a briefintroduction into the notation and an alphabetically sortedlist of all variables used in this paper. In Sec. III the optimalangle g to tilt the reconstruction planes R is derived. Thereconstruction ~i.e., the rebinning procedure! itself is treatedin Sec. IV. It consists of several subsections dealing with thecoordinate systems and the corresponding transformations~Subsection IV A!, the projection of an arbitrary point r ontothe detector plane ~Subsection IV B! as well as how to cal-culate the focus position and the detector position for a given

ray to rebin ~Subsections IV C and IV D!. The length correc-tion factor to correct for the angle between the physicallymeasured ray and the virtual ray used for reconstruction isderived in Subsection IV E. Subsection IV F deals with thevery important problem of how to obtain the image in worldcoordinates, and thus how to avoid another interpolation stepfrom voxels in the tilted planes to voxels in the world’sCartesian coordinate system. The remainder of Sec. IV sum-marizes the required steps to obtain a tilted image and an-swers the question of how many tilted images at what angu-lar increment are needed and how to perform thez-interpolation to finally end up with a reconstructed volume.Results are given in Sec. V and the paper ends with thediscussion, Sec. VI.

II. MATERIALS AND METHODS

The scanner’s rotation angle used in this paper will bedescribed by the projection angle a. This angle will be usedto parametrize the complete spiral trajectory and thus wehave aPR. In the case of a cylindrical detector system weuse the angle b to describe the transaxial position of the raywithin the fan. The corresponding parameters when rebin-ning to parallel geometry are j for the ray’s distance to theorigin and q for its angle. Figure 1 shows the in-plane ge-ometry of the cylindrical detector scanner.

An important concept is the reconstruction position aR .Each planar data set to be rebinned will be centered about acertain angular position aR . The corresponding coordinatein parallel geometry q will count relative to aR and thus wehave q1aR5a1b . The same applies to the local coordi-

FIG. 1. In-plane coordinate system of the cylindrical detector scanner. Therays are described by the angle b within the fan and the rotation angle a ofthe gantry in fan geometry. The corresponding parameters in parallel ~i.e.,rebinned! geometry are j and q. This figure has been drawn assumingaR50.

755 Kachelrieß, Schaller, and Kalender: Advanced single-slice rebinning 755

Medical Physics, Vol. 27, No. 4, April 2000

nate systems: They are given relative to the reconstructionposition aR . Especially the y8 axis coincides with the centralray at a5aR .

The fraction of a 360° rotation in parallel geometry to beutilized for reconstruction will be denoted by f. The require-ment for completeness is f > 1

2 and we will further assumef <1. The data needed for reconstruction thus will rangefrom q5aR2 f p to q5aR1 f p .

Notations and definitions used throughout this paper aregiven below. Most of them depend on the current reconstruc-tion position aR . For convenience this dependency is notexplicitly stated, thus the reader should be aware of the im-plicit dependence on aR . All primed parameters correspondto the local coordinate system, i.e., to the local ~tilted! recon-struction ellipse and reconstruction plane. They depend im-plicitly on aR .

b• c floor function, yields greatest integer lower orequal •

d• e ceil function, yields smallest integer greater orequal •

d(•) Dirac’s delta functionsgn(•) sign function such that x5uxusgn xx∨y maximum function, x∨y5max(x,y)x∧y minimum function, x∧y5min(x,y)a projection angle, aPR

a* attachment angle of the tilted planes; a recon-struction plane centered about aR will be at-tached to the spiral trajectory at a5aR anda5aR6a*

a8 focus position relative to the reconstructionposition aR , a85a2aR

aL8 focus position for the longitudinal approxima-tion to use in rebinning when the rebinned rayhas the parameters q8 and j8

aR projection angle about which the reconstruc-tion is centered

d table increment per 360° rotation, dPR

D detector plane; see Eq. ~2!

cos e length correction factor to account for theangle e between the measured ray and the vir-tual ray used for reconstruction; see Sec. IV E

e(a) elliptical trajectory of the virtual focus; seeEq. ~6!; The ellipse is tilted by the angle g tooptimally match the given focus trajectorys(a) in the interval aP@aR2 f p ,aR1 f p)

f fraction of 360° ~in parallel geometry! to beused for reconstruction; Projection angles qwithin @2 f p , f p) will be used consideringprojections rebinned to parallel geometry; f5

12 is a half-scan reconstruction, f .

12 means

overscan data to reduce interpolation artifactsF fan angle, F52 sin21(RM /RF)F fan-plane5plane which contains the focus for

a given a and a complete fan of rays to beused for reconstruction

g tilt angle of the reconstruction plane R mea-sured with respect to the x-y-plane

o8 origin of the primed coordinate systems; seeEq. ~7!

p pitch value; it is defined as the table incre-ment d divided by the intersection length ofthe collimated cone-beam and the z-axis

p(a ,u ,v) measured projection data at (a ,u ,v)p8(q8,j8) rebinned projection data in parallel geometry

corresponding to the reconstruction plane R inlocal coordinates (q8,j8)

p(q ,j) rebinned projection data transformed to worldcoordinates ~q, j!

r coordinate vector, r5( yz

x

)r(u ,v) world coordinates of detector ~u, v); see Eq.

~2a!

R reconstruction plane, R.e(R); see Eq. ~8!

RùX the line used for reconstruction of a certainray at a certain focus position

RD distance from detector to center of rotation~z-axis!, in our case 435 mm

RF distance from focus to center of rotation ~z-axis!, in our case 570 mm

RM radius of the field of measurement ~FOM!, inour case 250 mm

S slice thickness, as projected onto the axis ofrotation5physical beam width, i.e., z-rangeover which a physical averaging is performedduring the measurement process

s(a) spiral focus trajectory; see Eq. ~1!

q, j world’s beam parameters in parallel geom-etry; they describe a parallel-beam througho81jj(q) with direction h(q). q is givenrelative to aR

q8,j8 local beam parameters in parallel geometry;they describe a parallel-beam throughj8j8(q8) with direction h8(q8)

u, v detector coordinatesuF ,vF detector coordinates for the fan-beam based

approximation of a ray emanating from focusposition a8 through o81j8j8(q8)

u, v, w orthonormal base of the detector coordinatesystem; u'z, viz, and w points toward thesource

j, h, z world’s parallel-beam coordinate system ~or-thonormal! for the rotation angle q; it is givenby rotating x, y and z by q1aR around thez-axis ~i.e., around z!; j5j(q),h5h(q),z5z; see Sec. IV A

j8,h8,z8 local parallel-beam coordinate system ~ortho-normal! for the rotation angle q8; it is givenby rotating x8,y8 and z8 by q8 around thez8-axis ~i.e., around z8); j85j8(q8), h8

5h8(q8), z85z8; see Sec. IV Ax, y, z world’s spatial coordinatesx, y, z orthonormal base of the world coordinate sys-

temx8,y8,z8 orthonormal base of the local coordinate sys-



tem, x8 and y8 lie in R, the origin is o8; seeSec. IV A

X x-ray plane5plane containing both the mea-sured pencil beam and its corresponding vir-tual beam used for reconstruction

z(x ,y) z-coordinate of point rPR as a function of itsx and y coordinates; see Eq. ~14!

zR reconstruction position of the final ~nontilted!image

Dzmean average deviation of the focus from the recon-struction ellipse

(C/W) window setting of the reconstructed images inHU; C is the window center, W the windowwidth

During the rest of the paper we will be free to switch, ifconvenient, from the primed parameter set to unprimed pa-rameters.

Spiral cone-beam data were simulated using the dedicatedsimulation software DRASIM. A virtual scanner with the ge-ometry described in Appendix B was assumed. Our simula-tions were done with 1-mm slice thickness. The only excep-tion is the scanner with square detectors, i.e., a slicethickness of 0.335 mm, which was used to achieve isotropicsampling. Phantom definitions were taken from the world-wide phantom data base at http://www.imp.uni-erlangen.de/forbild

All reconstruction algorithms were implemented on astandard PC with dedicated image reconstruction and evalu-ation software ImpactIR ~VAMP, Mohrendorf, Germany!;reconstruction time is below 5 s per image on a 450 MHzPentium CPU with 256 MB of memory.

III. ADJUSTING THE RECONSTRUCTION PLANE

Our derivations will be based on a flat detector geometryas illustrated in Fig. 2. As will be shown in Appendix A, theflat detector coordinates can be easily transformed to cylin-drical detector coordinates necessary to describe the medicalCT scanner used for the simulations.

We assume the following spiral focus trajectory:

s~a !5S RF sin a

2RF cos a

da/2pD ; aPR. ~1!

For the detector plane we find the parameter representation:

D:r~u ,v !5S 2RD sin a

RD cos a

da/2pD 1uS cos a

sin a

0D 1vS 0

01D , ~2a!

and the normal representation

D:x sin a2y cos a1RD50. ~2b!

What is the optimum tilted plane to be centered about areconstruction position aR such that the focus deviates onlyminimally from that plane when moving from a5aR2 f p toa5aR1 f p whenever f gives the fraction of a full rotationthat is requested for reconstruction? Here we want to point

out that the minimization procedure will be done only withrespect to the central ray. Its angular variable will range froma5aR2 f p to a5aR1 f p . Contributions for noncentralrays, which go beyond this interval, will not be considered inthe optimization procedure for convenience. Since we aregoing to rebin from cone-beam geometry to parallel-beamgeometry we assume f to describe the data range taken inparallel-beam geometry. The angular variable q in parallelgeometry will range from q52 f p to q51 f p . The mini-mal requirement for complete data is f 5

12, i.e., only 180° are

used to reconstruct the tilted plane’s image. Nevertheless,some overscan might be necessary to reduce streaking arti-facts, thus f .

12 might be useful. The data range ~range of a!

required to rebin a range of 2p f of parallel data will beroughly one fan angle larger.

For simplicity we now assume aR50. Let R: x tan g2z50 be the reconstruction plane tilted by the angle g about thecentral ray ~which has the direction 2w that points for a5aR50 along the negative y-axis!. The intersection of Rwith the cylinder described by the spiral focus trajectorygives the following ellipse:

e~a !5RFS sin a

2cos a

tan g sin aD ; aP~2p ,p !.

FIG. 2. Cone-beam coordinate system. d: table increment per 360° rotation,a: rotation angle, RF : distance focus to the center of rotation, RD : distancedetector to the center of rotation, RM : radius of the measurement cylinder.



The tilt angle g should now be chosen such that the meandistance of the focus to that ellipse is minimized:

g5arg ming

S E2 f p

f p

daue~a !2s~a !uqD 1/q

5arg ming

S E0

f p

daue~a !2s~a !uqD 1/q

, ~3!

where qP(0,`# is a parameter to adjust the degree of opti-mization. We will present solutions for the most convenientsetting q51 below. Other cases such as q52 and q5` ~theessential supremum! which have also been considered yieldresults similar to the q51 case. Moreover, ASSR has turnedout to be quite robust with respect to slight changes of the tiltangle, i.e., the reconstructions using q51, q52 or q5`hardly differ from each other.

Since

ue~a !2s~a !u5URF tan g sin a2da

2pU,

we will have the trivial intersection of the plane and thespiral path at a50 and, as long as

1<2p

dRF tan g<

f p

sin f p,

there will be two more intersections at a56a* with theattachment angle a*P(0,f p# . The angle g is then given by

tan g5da*

2pRF sin a*. ~4!

Figure 3 visualizes the intersection of the tube path and thetilted ellipse.

We are going to now show, that the case of three inter-sections must be met in order to minimize Eq. ~3!. We willdo so by ruling out the two cases with only one intersection.

The first situation with one intersection only is given when-ever RF tan g sin fp.(d/2p) f p . The integrand then reducesto

ue~a !2s~a !u5RF tan g sin a2da

2p,

which monotonically decreases ;aP@0,f p# with decreas-ing g until RF tan g sin fp5(d/2p) f p . For the second casewithout intersection, RF tan g,d/2p , we find for the inte-grand

ue~a !2s~a !u5da

2p2RF tan g sin a ,

which monotonically decreases ;aP@0,f p# with increasingg until RF tan g5d/2p . Consequently the case of three inter-sections must be met to further minimize the integrand andthus the integral Eq. ~3!. To find the optimal a* and thus theoptimal g to minimize Eq. ~3! we regard the meanz-deviation

Dzmean~q !

5d

2 f p2 S E0

f p

daU a*

sin a*sin a2aUqD 1/q

5d

2 f p2 S E0

a*daS a*

sin a*sin a2a D q

1Ea*

f p

daS a2a*

sin a*sin a D qD 1/q

.

For the special case q51 it yields

Dzmean~1 !

5d

2 f p2 S a*

sin a* ~11cos f p22 cos a*!

11

2f 2p2

2a*2D .

The minimization of Dzmean(q) for a given f with respect to a*

can be easily calculated by taking the zero value of the de-rivative

]

]a*Dzmean

~q !50,

which for q51 evaluates to

cos a*512~11cos f p !,

which allows us to directly calculate the tilt angle g from Eq.~4!.

We have included Fig. 4 to show the relationship betweenthe range of angles f needed for reconstruction and the opti-mal a* for both q51 and q52 and to show the expectedmean deviation when using the optimized angles. The plotsin Fig. 4~b! confirm that there is hardly any difference be-tween q51 and q52 for small values of f. The expectedmean deviation lies around 1.5% of the table feed regardlessof whether q51 or q52 has been chosen and regardless ofwhether a* is 60°, 67° or something in between.

Proceeding with q51 we find using the optimized settingcos a*5

12(11cos fp) for the mean z-deviation:

FIG. 3. Plot of the functions da/2p corresponding to the spiral trajectory~linear curve! and RF tan g sin a corresponding to the reconstruction ellipse~sinusoidal curve! to demonstrate their intersections at 0 and 6a*. This plotis drawn for a*5p/3 which minimizes the area between the two functions

in the shown interval ~i.e., for f 512).



Dzmean~1 !

5df 2p2

22a*2

4 f p2 . ~5!

Moreover, we want to look at the convenient case of a180° reconstruction ( f 5

12). We then get for a* the three

intersection angles 260°, 0°, and 60° which is quite a niceresult because it uniformly divides the 180° range. This re-sults in Dzmean5d/72, which is only 1.4% of the table feed d.Nevertheless, a very annoying fact is the local maximum ofthe deviation between e(a) and s(a) occurring for a56 f p which is at the edge of the graph in Fig. 3. It mighteasily yield reconstruction artifacts due to imperfect match-ing of opposing views. Thus it should be considered to useoverscan data for reconstruction, i.e., f .

12.

IV. RECONSTRUCTION

We now assume to have a fixed a* and thus the slope

tan g5da*

2pRF sin a*

of the ellipse is also a fixed value.Further on we allow for arbitrary values of the reconstruc-

tion position aR and thus we have to give the generalizedformula for the ellipse:

e~a !5o81RFS sin a

2cos a

tan g sin~a2aR!D ~6!

such that the physical focus position s(a) longitudinally ~inz-direction! corresponds with the virtual focus position e(a)for all aP@aR2 f p ,aR1 f p). The origin o8 of the primedcoordinate system is given by

o85S 0

0

daR

2p

D . ~7!

The general normal representation of the R-plane can easilybe derived from the generalized ellipse formula. The result is

R:x cos aR tan g1y sin aR tan g2z1daR

2p50. ~8!

Since the reconstruction will be centered about aR , the re-construction algorithms can in principle be derived for aR

50 followed by a substitution of a by a8ªa2aR .The aim is to determine which rays of the cone-beam

should be used to synthesize the projection data of the virtualscanner that rotates in R along e(a). Of course these rayswill not lie completely in R except for a5aR6a* and a5aR . Several strategies are possible.

First, one can use complete fans of beams ~i.e., planes F!that emanate from the source and all rays within such a fanshall be taken for reconstruction. In other words, this methodis based on selecting complete fans for each tube position afor a reconstruction centered about aR . These fans will thenbe rebinned to parallel geometry.

The second possibility is based on a ray-by-ray optimiza-tion. For each ray that will be demanded for the parallelrebinning the optimal corresponding tube position and theslope with respect to the reconstruction plane R will be cal-culated.

Now for each of these two possibilities of ray selectionthere are two possibilities of what kind of approximationswill be allowed. The first possible approximation is that allrays will be projected onto R in longitudinal direction ~z-direction! only. The other possibility is an orthogonal projec-tion onto the reconstruction plane.

To summarize, the possible four combinations of approxi-mations are

FL: Fan-beam based, Longitudinal approximationsFO: Fan-beam based, Orthogonal approximationsPL: Parallel-beam based ~ray-by-ray!, Longitudinal ap-

proximationsPO: Parallel-beam based ~ray-by-ray!, Orthogonal approxi-

mations

FIG. 4. Illustration of the optimization procedure. ~a! optimal a* as a func-tion of f. Optimization was done for the cases q51 ~lower graph! and q52 ~upper graph!. For f, around

12, one should use angles between 60° and

67° to attach R to the focus trajectory. ~b! relative z-deviation of the focusfrom the reconstruction plane R as a function of f. The optimized angles a*were used. Thus, four graphs are shown. Top: Dzmean

(1) when minimizing theq52 deviation. Second from top: Dzmean

(1) when minimizing the q51 devia-tion. Second from bottom: Dzmean

(2) when minimizing the q51 deviation.Bottom: Dzmean

(2) when minimizing the q52 deviation. Obviously, when us-ing only a small overscan fraction ~i.e., f lies at or only slightly above

12! one

should choose attachment angles around 65°. The expected mean deviationwill then be about 1.5% regardless of whether q51 or q52 has beenchosen for optimization or for calculation of the deviation.



However, since it has turned out that the differences betweenthese options are negligible we will only present the defini-tions and rebinning equations for the FL method, which isthe most simple one.

Although all following considerations only use elemen-tary geometry, the multitude of different coordinate systemsand the complexity of the ray geometries for a given focusposition a and detectors u and v together with the tiltedplanes under consideration complicate the situation. Thus be-fore starting to discuss the FL reconstruction method we willdo some preparations first.

A. Coordinate systems

The global j-h-z system is given by rotating the x-y-zaxes by q1aR about the z-axis. The base vectors are

j~q !5S cos~aR1q !

sin~aR1q !

0D , h~q !5S 2sin~aR1q !

cos~aR1q !

0D ,

z~q !5S 001D 5z.

We define the local ~i.e., corresponding to a given recon-struction position aR) tilted coordinate system with base(x8,y8,z8) to have both the x8- and the y8-axis lying in R.The y8-axis shall coincide with the central ray at projectiona5aR . Thus we have

x85S cos aR cos g

sin aR cos g

sin gD , y85S 2sin aR

cos aR

0D ,

z85S 2cos aR sin g

2sin aR sin g

cos gD .

Rotating this system by q8 about the z8-axis yields the localparallel geometry j8-h8-z8 system with the base vectors

j8~q8!5S cos aR cos q8 cos g2sin aR sin q8

sin aR cos q8 cos g1cos aR sin q8

cos q8 sin gD ,

h8~q8!5S 2cos aR sin q8 cos g2sin aR cos q8

2sin aR sin q8 cos g1cos aR cos q8

2sin q8 sin gD ,

z8~q8!5S 2cos aR sin g

2sin aR sin g

cos gD 5z8.

All coordinates are given with respect to the world coor-dinate system’s base ~x, y, z!. The center of the above givencoordinate system is located at the primed origin o8. Allprimed and thus local coordinate systems reduce to their cor-responding world coordinate systems ~unprimed identifiers!whenever aR and g are set to zero.

It will turn out that the transformation between local rayparameters (q8,j8) and global ray parameters ~q, j! is quiteimportant. It is given from the longitudinal projection ~alongz! of a ray from local to world coordinates. To be moreprecise: For a given ray with the parameters (q8,j8) we arelooking for the parameters ~q, j! that the corresponding linewould yield after having been projected into the plane z50. Mathematically this yields the term

P~o81j8j8~q8!1Rh8~q8!!5jj~q !1Rh~q !,

with the projection operator

P5S 1 0 0

0 1 0

0 0 0D .

This allows us to derive the desired transformation rules. Wewill just state the results:

cos q5cos q8

Acos2 q81cos2 g sin2 q8,

sin q5sin q8 cos g

Acos2 q81cos2 g sin2 q8, ~9!

j5j8 cos g

Acos2 q81cos2 g sin2 q8.

To calculate the primed parameters as a function of theunprimed ones we need the inverse transform of Eq. ~9!:

cos q85cos q cos g

Asin2 q1cos2 g cos2 q,

sin q85sin q

Asin2 q1cos2 g cos2 q, ~10!

j85j

Asin2 q1cos2 g cos2 q.

For convenience, Fig. 5 gives a view onto the reconstructionplane R and the primed coordinates. Further we want to givea useful relationship that directly becomes evident from Eqs.~9! and ~10!:

Acos2 q81cos2 g sin2 q8Asin2 q1cos2 g cos2 q5cos g .

B. Projections onto the detector plane

It will be necessary to know the projection of a givenpoint r from the focus location s(a) onto the detector. Thecalculation is uninstructive, we will simply state the result indetector coordinates u and v:

u5r~2x cos a2y sin a !,~11!

v5rS da

2p2z D

with



r5RD1RF

x sin a2y cos a2RF,

where x, y, and z denote the components of r.

C. Focus position for the longitudinal approximation

The plane containing the measured ray used for the ap-proximation and its corresponding virtual ray will be denotedby X. The condition that must be satisfied for the longitudinalapproximation is: The measured and virtual ray must lie in aplane parallel to the z-axis, i.e., Xiz ~longitudinal approxi-mation!.

Since h8(q8)iX and o81j8j8(q8)PX the plane X isfully determined:

X:~h8~q8!3z!•~r2~o81j8j8~q8!!!50.

Its intersection

Xùs~R!:~h8~q8!3z!•~s~a !2~o81j8j8~q8!!!50

with the spiral focus trajectory determines the desired focusposition a5a81aR :

RF~cos q8 sin a82sin q8 cos a8 cos g !2j8 cos g50.

Instead of solving for a8 we use Eq. ~9! and get

RF sin~a82q !2j50,

which is advantageous, since it is already stated in worldcoordinates. The familiar result

aL8~q ,j !ªa85q1sin21j

RF

determines the focus position to be used to rebin the ray ~q,j! using the longitudinal approximation.

Further on we may state that the range of a needed for thereconstruction is

aP@aR2 f p212F ,aR1 f p1

12F# ,

where we have used F52 sin21 (RM /RF) for the fan angle

and jP@212RM , 1

2RM# and qP@2 f p , f p).

D. Point of intersection for the fan-beam basedapproximation

The fan to be used for a certain focus position s(a) inter-sects the reconstruction plane R at a line through its origino8. The average deviation of the plane F that contains the fanand the plane R is thus minimized. F is defined by the threepoints

s~a !, e~a212p !, and e~a1

12p !;

FùR is the line

FùR:o81t1~e~a112p !2e~a2

12p !! with t1PR

~12a!

FIG. 5. Plane R and ellipse e(a). The shaded area depicts the FOM. Thelocal coordinate system x8-y8-z8 is shown as well as the rotated parallelcoordinate system j8-h8-z8. The halfaxes of the ellipse are RF /cos g andRF , respectively. The x-ray ~solid line L! emanating from the source aboveR penetrates the reconstruction plane at LùR and hits the detector below R~dashed line L!. The focus appears to be located outside the ellipse since thisfigure’s view is along 2z8. The dotted line depicts that ray of the focuswhich intersects the plane R at the origin o8 of the primed coordinate sys-tems. The line that depicts the detector is the intersection FùD of thecurrent fan-beam with the detector plane.

FIG. 6. Parametric plot of the detector occupation (u ,v) as a function of ~q, j! for q ranging from q5212p ~top line! to q5

12p ~bottom line! in steps of

1/10p and jP@2RM ,RM# . Geometric parameters as given in Appendix B have been chosen: RF5570 mm, RD5435 mm, RM5250 mm. Table increment

was chosen as d572 mm ~and thus Dzmean51 mm) to yield a mean deviation of below 0.5 mm when projected onto the edge of the FOM. f 512, a*

560°, g'1.4°. ~a! flat detector. Requested detector size: 981 mm392 mm. The longitudinal extention of 92 mm corresponds to an effective detector sizeof 52 mm at the center of rotation and thus a pitch of 1.38. ~b! cylindrical detector. Requested detector size: 52°383 mm5912 mm383 mm. The longitudinalextension of 83 mm corresponds to an effective detector size of 47 mm at the center of rotation ~see Appendix A!.



through the points e(a612p) which cuts the ellipse into

halves. This line of intersection is not necessarily perpen-dicular to o82s(a). The x-ray plane X is defined by thepoint s(a) and a line through o81j8j8(q8) with directionh8(q8). Obviously the intersection XùFùR is given byintersecting FùR with the line

XùR:o81j8j81t2h8 with t2PR. ~12b!

This means solving

o81t1~e~a112p !2e~a2

12p !!5o81j8j81t2h8

for at least one of the free parameters t1 and t2 . Subtractingo8 and multiplying by j8 directly yields t1 which, insertedinto Eq. ~12a!, gives the point of intersection

XùFùR:

o81j8 cos g

cos q8 cos a81cos g sin q8 sin a8 S cos a

sin a

tan g cos a8D .

At the detector this results in the projection at

u5RD1RF

RF

j8 cos g

cos a8 cos q81sin a8 sin q8 cos g,

v5RD1RF

RFS j8 cos a8 sin g

cos a8 cos q81sin a8 sin q8 cos g2d

a8

2pD ,

which is even nicer when going to world coordinates:

uF~q ,j ,a8!ªu5RD1RF

RF

j

cos~a82q !,

vF~q ,j ,a8!ªv5RD1RF

RFS j cos a8 tan g

cos~a82q !2d

a8

2pD .

As one can easily see, the resulting contributions are linesv(u) at the detector with

v~u !5u cos a8 tan g2RD1RF

RFd

a8

2p.

As an example, Fig. 6 depicts the corresponding curves atthe detector when using uF and vF together with aL8(q ,j) asdescribed in Sec. IV C. The angle a8 is not fixed in thisfigure; it rather is a function of j and q: a85aL8(q ,j). ThusFig. 6 depicts the detector for varying source and detectorlocation. Moreover, due to the high aspect ratio of the detec-tor it seems as if the line for q50 was horizontal which isnot really the case!

E. Length correction

Since the virtual rays that are going to be used for recon-struction and their corresponding measured rays differ by asmall angle « we must apply a length correction to yield thecorrect virtual projection values from the measured attenua-tion.

So calculate the cosine of the angle «5«(a8,q ,u ,v) be-tween the measured and the virtual ray

cos «5~r~u ,v !2s~a !!

ur~u ,v !2s~a !u•h8~q !5

S u cos a2~RD1RF!sin a

u sin a1~RD1RF!cos a

v

DAu2

1v21~RD1RF!2

•

S 2sin~q1aR!cos g

cos~q1aR!cos g

2sin q sin gD

Asin2 q1cos2 g cos2 q

5u sin~a82q !cos g1~RD1RF!cos~a82q !cos g2v sin q sin g

Au21v

21~RD1RF!2Asin2 q1cos2 g cos2 q

~where again a85a2aR) and weight the measured datawith this value.

F. How to reconstruct in world coordinates

The parallel projection data we have calculated up to here,for aR fixed, correspond to the following projection throughthe object function f (x ,y ,z):

p8~q8,j8!5E dx8 dy8 f ~x ,y ,z !d~x8 cos q8

1y8 sin q82j8!uz5z~x ,y ! ~13!

with xx1yy1zz5o81x8x81y8y8 and thus

z~x ,y !5x cos aR tan g1y sin aR tan g1daR

2p, ~14!

which follows from Eq. ~8!.It is easy to reconstruct the object function in local coor-

dinates, because the object function f 8(x8,y8,0) in local co-ordinates can be reconstructed directly from p8(q8,j8).Nevertheless, if one had reconstructed the object in localcoordinates another interpolation step from voxels at discretelocations in x8, y8, z8 to discrete voxel positions in x, y, zwould be necessary to transform the function to the worldcoordinate system and to account for the corresponding dis-tortions.

We will avoid this step by proceeding with Eq. ~13!. Us-



ing the transformations given in Sec. IV A to transform Eq.~13! ~i.e., j8, q8, x8, and y8) to world coordinates we get

p8~q8,j8!5E dx dy

cos gAsin2 q1cos2 g cos2 q f ~x ,y ,z !

3d~x cos~aR1q !

1y sin~aR1q !2j !uz5z~x ,y ! .

The cos g in the denominator results from the Jacobian](x8,y8)/](x ,y). Defining

p~q ,j !ªcos g

Asin2 q1cos2 g cos2 qp8~q8,j8!

5E dx dy f ~x ,y ,z !d~x cos~aR1q !

1y sin~aR1q !2j !uz5z~x ,y ! ~15!

yields the desired result since this function can be recon-structed with standard methods such as filtered backprojec-tion or Fourier reconstruction eventually using available re-construction hardware. The result will have the correct x andy coordinates. The only interpolation remaining to be done isthe one in z.

G. Putting everything together

The complete rebinning procedure to gain the parallel rawdata for one of the tilted slices R that can readily be recon-structed to yield the corresponding image plane in worldcoordinates consists of several steps:

• Choose aR .

• For each qP@2 f p , f p) and each jP@212RM , 1

2RM#

—Calculate the relative focus position aL8 as described inSec. IV C.

—From aL8 calculate the detector coordinates uF and vF

which were derived in Sec. IV D.—Assign

p~q ,j !ªcos g

Asin2 q1cos2 g cos2 q

3cos « p~aL81aR ,uF ,vF!.

The cos « factor @«5«(aL8 ,q ,uF ,vF)# is the length cor-rection to account for the angle between the measured~physical! ray and the virtual ray used for reconstruction ashas been derived in Sec. IV E. The square root factor derivedin Eq. ~15! may be interpreted as a length correction whengoing from the local to the global coordinate system.

—If f .12 do not forget to account for the overscan by

proper overscan weighting.

• Reconstruct the rebinned rawdata set p(q ,j) to yieldthe corresponding object function f (x ,y ,z) with z5z(x ,y) given in Eq. ~14!.

The described rebinning process must account for the dis-crete nature of the projection data. In principle this will leadto a trilinear interpolation between neighboring measureddata points in a, u and v . Of course more sophisticated in-terpolation functions are possible. We will see below a casewhere special care has to be taken when interpolating be-tween adjacent detector rows (v-direction!.

For performance reasons we suggest to build lookuptables for all necessary values of aL8 , uF and vF togetherwith the corresponding trilinear interpolation weights, thelength correction factors and the overscan weighting.

The remaining problem is the interpolation between vox-els in z-direction. This is only possible if a certain number ofneighboring images have been reconstructed. This naturallyleads to the question of what values of aR to use to gain thecomplete volume without losing information.

H. Where to reconstruct

We now consider two successive reconstruction positionsaR1 and aR2 to derive a condition for their maximal spacingDaRªaR22aR1 .

This condition can be found by looking at the two corre-sponding reconstruction ellipses. Their maximal distance inz-direction will be a measure of the lower limit of the achiev-able z-resolution. However, since we want to reconstructwithin the FOM only, we are not going to regard the recon-struction ellipses themselves but rather the corresponding el-lipses at the edge of the FOM, i.e., the intersection of thereconstruction plane R and the measurement cylinder of ra-dius RM . The maximal distance of two such ellipses will bedenoted by Demax and it can easily be checked that

Demax5Ud DaR

2p12RM tan g sin

1

2DaRU. ~16!

This maximal value occurs for a512(aR11aR2)1p . Obvi-

ously, the deviation Demax can be made arbitrarily small byreducing the reconstruction increment DaR .

To estimate the achievable z-resolution and thus a reason-able voxel size Dz we have to take into account that therebinning procedure already introduces an error in z since thefocus trajectory has an average deviation of Dzmean from thereconstruction ellipse ~see Sec. III!.

Moreover, the finite detector width of the detector inz-direction has to be taken into account. We will assume thatthe mean beam width ~averaged over the FOM! is knownand we will denote it by S. If the focal spot is of infinitesimalsize, then S simply denotes the finite detector width projectedonto the center of rotation. In general the focal spot is offinite size and thus has to be taken into account. If it iscomparable to the detector width then the beam would benondivergent and S would stand for some average of thefocal spot size and the detector width. In 2D CT one wouldcall S the collimated slice width.

At this point we want to be quite pragmatic. We assumethat these three sources of error can be approximately takeninto account by convolving the object function with threerectangle functions of the corresponding widths



dDaR

2p12RM tan g sin

1

2DaR ,

RM

RFDzmean , and S .

As one can see, we are going to regard the error at the edgeof the FOM which is the region of lowest achievablez-resolution. Moreover, we have left away the absolute valuefunction since we have sgn tan g5sgn d and we simply as-sume DaR.0 and d.0 without loss of generality.

It can be shown that the FWHM ~full width at half-maximum! of the convolution of three rectangle functions ofwidths a, b, and c is given by

a∨b∨c

as long as 2(a∨b∨c).a1b1c . In other words, the FWHMis given by the maximum of the widths as long as this maxi-mum exceeds the sum of the two smaller widths ~for othercases the situation becomes more complicated!. Since ourreconstruction shall not decrease the system’s inherentz-resolution, described by S, we have to demand

dDaR

2p12RM tan g sin

1

2DaR1

RM

RFDzmean<S .

Using the formula Eq. ~4! for tan g and Eq. ~5! for Dzmean

we finally find ~assuming DaR to be positive!

1

2DaR1

RM

RF

a*

sin a*sin

1

2DaR<

p

dS2

RM

RF

f 2p222a*2

4 f p.

~17!

As an example let us consider again the geometric setupwhich has already been used to produce Fig. 6: RF

5570 mm, RM5250 mm, d572 mm. Using f 512 and a*

513p yields a reconstruction increment of DaR'1.8° when

assuming a slice thickness of S51 mm. Thus 200 recon-structions per rotation or 2.8 reconstructions per slice thick-ness are necessary.

Equation ~17! can be used to determine the maximal pos-sible table increment since the right hand side of Eq. ~17!must never become zero or negative which means

d<RF

RM

4 f p2

f 2p222a*2 S . ~18!

For our specific geometrical settings we find d<164S . Nev-ertheless with increasing relative table increment d/S thenumber of reconstructions per slice thickness increases aswell. This is depicted in Fig. 7: As long as d<100S abouttwo to five reconstructions per slice thickness—a numberthat is also recommended for conventional spiral CT1,17—arerequired for optimal image quality. For table incrementshigher than that, the number of necessary backprojectionsincreases drastically.

I. z-Interpolation in the Image Domain

Up to here we have a certain number of reconstructedimages which correspond to f (x ,y ,z(x ,y)) as describedabove. What remains to be done is the z-interpolation ofthose tilted image planes onto a Cartesian grid. We do notwant to keep all those tilted images in memory before start-ing this interpolation step. Let us thus have a closer look atthe intersection of a tilted plane R with the edge of the FOM~parametrized by x5RM sin f and y5RM cos f):

z~RM sin f ,RM cos f !5RM cos aR tan g sin f

1RM sin aR tan g cos f1daR

2p

5RM tan g sin~aR1f !1daR

2p,

where we have used Eq. ~14! for the z-coordinate. Now for agiven z-position zR for the current slice to interpolate alltilted images with

aRP

2p

d~zR1RM tan g @21,1# !

need to be kept in memory. This equation can be generalizedif advanced filters in z-direction shall be used that requireadditional room for interpolation, let’s say z:

aRP

2p

d~zR1~ z∨RM tan g !@21,1# !.

This requires to hold

d2~ z∨RM tan g !

dDaR/2p eadjacent tilted images in memory.

For the example above, this makes 11 slices before theimage interpolation for a given z-position can start ~here zwas assumed to be zero!.

The object function f (x ,y ,z) is up to now known only forvalues z5z(x ,y) as given from Eq. ~14! for discrete aR

FIG. 7. Number of necessary 2D backprojections per slice thickness S as afunction of the relative table increment d/S . As long as the table incrementlies below 100S approximately two to five reconstructions per slice thick-ness are required. This is roughly the number suggested for conventionalspiral CT as well. Using larger table increments drastically increases thenumber of backprojections and thus the reconstruction time. RF5570 mm,

RM5250 mm, f 512, a*5

13p .



PDaRZ. We use the following weighting equation to obtainthe transaxial image f (x ,y) at the position zR :

f ~x ,y !ª(aR

L~z2zR! f ~x ,y ,z !

(aRL~z2zR! U

z5z~x ,y !

.

The denominator properly normalizes the interpolated objectfunction. L(•) denotes the triangular weighting function. Itswidth ~5 half of the base line! is chosen as follows. FromEq. ~16! we know the maximum distance of two adjacentreconstruction ellipses projected onto the edge of the FOM.At r5Ax2

1y2 this distance corresponds to

Ud DaR

2p12r tan g sin

1

2DaRU

and the interpolation at ~x,y! is done using a triangular filterof ~half! width

Ud DaR

2p12r tan g sin

1

2DaRU∨ z .

This ensures the filter to cover the data gaps in thez-direction and a minimal width of z is achieved as well.

V. RESULTS

In order to give some more explicit examples of how touse the theoretical results derived in the sections above todetermine possible scan geometries and/or scan parameters,we have discussed a possible setup for a medical CT scanner

~Appendix B!. The simulations performed below correspondto the medical CT scanner using scan parameters which arewithin the limits derived in Appendix B.

To evaluate the ASSR algorithm we have simulated vari-ous geometries which are given in Table I. These geometriesvary only in z-direction but not in the transaxial direction,i.e., the number of detector channels, their spacing, etc. re-mained constant.

It turned out that ASSR is very robust with respect to thekind of approximation ~longitudinal, orthogonal, or fan-beambased and parallel-beam based! used. There are no visibledifferences between these options. Moreover, the variationsintroduced by varying the attachment angle a* ~at least inthe range from 55° to 70°! are negligible as well. Thereforeall of the images presented below are reconstructed using theFL ~fan-beam based, longitudinal! approximation, an attach-ment angle of 60° and f 552%. The main reason for thesefacts are ~a! the scan settings simulated are within the errorrange Eq. ~18!, i.e., d,164S , which obviously ensures goodresults for any of the four possible approximation methodsand ~b! the integration along the detector introduces artifactsof more dominant magnitude due to imperfect transitionsbetween detector rows.

Obviously fact ~b! deserves more attention. We have thusprepared Fig. 8 which shows a reconstructed slice for threedifferent reconstruction increments DaR . As DaR decreases~at constant scan parameters and z-filtering width! the artifactlevel decreases also. A first conclusion one can draw is to useDaR as low as reasonably acceptable regarding the compu-

FIG. 8. Influence of DaR and the lineinterpolation technique on the artifactlevel. Images two and three were re-constructed with half and a quarter ofthe first image’s reconstruction incre-ment. For the last two images a qua-dratic interpolation between detectorrows was used instead of the linear in-terpolation. Parameters: ASSR32, zR

5214 mm, z51 mm.

TABLE I. Table of the simulated geometries. We have included a single-slice scan with 180°LI reconstruction and a multi-slice reconstruction for comparison.The number M of detector rows corresponds to the rows required by our algorithm and not to the ~larger! number of simulated detector rows. The noise valuess were measured in a circular ROI ~diameter 25 mm! located in the center of the ventricle at z50 as the standard deviation of the CT values. The scannerASSR36 was added to have isotropic sampling of the volume. Its slice thickness S was selected equal to the detector size in b direction ~values projected ontothe center of rotation!. Common parameters: attachment angle a*560°, f 552%.

Name d S M s g Cone Comment

180°LI1.5 1.5 mm 1 mm 1 8.1 HU — 0° 180° linear interpolation180°MFI6 6 mm 1 mm 4 9.0 HU — 0.30° 180° multi-slice filtered interpolationASSR6 6 mm 1 mm 4 7.6 HU 0.12° 0.30° advanced single-slice rebinningASSR12 12 mm 1 mm 8 8.5 HU 0.23° 0.70° advanced single-slice rebinningASSR16 16 mm 1 mm 12 7.9 HU 0.31° 1.11° advanced single-slice rebinningASSR32 32 mm 1 mm 21 8.7 HU 0.62° 2.01° advanced single-slice rebinningASSR64 64 mm 1 mm 40 8.5 HU 1.24° 3.92° advanced single-slice rebinningASSR96 96 mm 1 mm 60 8.2 HU 1.86° 5.93° advanced single-slice rebinningASSR36 36 mm 335 mm 66 9.4 HU 0.70° 2.19° isotropic ASSR



tational demands. Then the streaking artifacts which areemanating in this case from the inner ear and the frontalsinus will be averaged away since more integrations along agiven projection contribute to each z-interpolated image. Itmust be mentioned that the z-interpolation itself and the filterwidth z remained unchanged when decreasing DaR . De-creasing the reconstruction increment thus only increases thenumber of data points contributing to the z-filter.

In addition, Fig. 8 also includes two images that werereconstructed using quadratic ~convolution with a triangleand a rectangle function! instead of linear ~convolution witha triangle function! weights for interpolation between thedetector rows. This quadratic interpolation smoothes the im-perfect transitions between the rows and thus effectively re-duces the streaking artifacts. Nevertheless, one must not for-get that a loss in z-resolution is introduced by quadraticweighting as well ~see Table II!.

Our implementation of the z-interpolation procedure takesthe required filter width z as a freely selectable parameter. Atriangular filter as wide as the maximum of both, spacing andz , is used in z-direction ~Sec. IV I!. The behavior for varyingz can be seen in Table II as well. For small required widths,the slice profiles are hardly changing since the spacing of thetilted images is the dominating factor for z-filtering. Never-theless, for z>1 mm the FWHM of the slice sensitivity pro-file increases as does z .

The most interesting result of Table II is that the figuresgiven there apply to all simulated ASSR scanners. Obviouslythe shape and the dimensions of the SSPs are neither affectedby the number of detector rows nor by the simulated tableincrement. Thus the ASSR algorithm is very robust with re-spect to the simulated scan mode. The slice profile qualityindex SPQI17 is the same ~at least up to 61%! for the cone-beam and the conventional spiral scanners. This indicatesthat the shapes of all profiles are very similar. Since plottingthe profiles confirms this indication, i.e., there is no signifi-cant difference in the SSPs’ shape ~except for a scaling alongthe abscissa!, we omit the corresponding plots.

However, as it is well known that SSPs might changewhen going off-center we have additionally investigated off-center profiles. Raw data for the following four off-centerdelta objects were generated and reconstructed:

d~y6r !d~x !d~z ! North, South,

d~x6r !d~y !d~z ! East and West.

The distance r to the isocenter was chosen to be r512RM

5125 mm. The North/South peaks lie at a distance of 5 mmfrom the a.p. edge of the head phantom whereas the East/West peaks are located 29 mm off the lateral boundary of thehead at z50. For all simulated geometries it turned out thatthe SSP variations are lower than what is known from180°LI. This is clearly confirmed by Fig. 9 which shows theSSP obtained at the isocenter ~solid line! together with thefour off-center SSPs ~dashed and dotted lines!. We have cho-sen to depict ASSR64 since it represents the behavior of theother geometries very well. Obviously the deviations of thedashed and dotted lines of ASSR64 @Fig. 9~b!# are smallerthan for 180°LI @Fig. 9~a!#. Including additional z-filtering@Fig. 9~c!# improves the off-center behavior even more. Withrespect to off-center SSP variations ASSR is obviously per-forming better than the standard z-interpolation 180 °LI.

A comprehensive comparison of the simulated scanmodes between ASSR and a spiral single-slice scanner withp51.5 is given in Figs. 10 and 11. We have taken care toselect the presented slices in an unprejudiced way: For zR

5225 mm the single-slice scanner performs better, for zR

522 mm the ASSR algorithm is superior whereas the per-formance at zR5230 mm and zR50 mm seems to be bal-anced between the conventional and the cone-beam geom-etry. Nevertheless, as it can be seen from the example withd564 mm, the results depend on the absolute angular posi-tion of the tube: ASSR64 shows severe artifacts at zR

5230 mm but a simulation of the same scanner with thestart angle increased by 90°, ASSR64190°, shows far lessartifacts in the respective slice. All scans, except forASSR64190°, were oriented so as to have the same tubeposition at z50 which, in other words means that all scantrajectories intersect at z50.

Moreover, it can be seen from these figures that the arti-fact content and the artifact level of the reconstructionsASSR6 through ASSR96 does not increase drastically withincreasing table increment. Concentrating upon z50 as thebest position for comparisons ~since there the tube positions

TABLE II. Figures of merit ~Ref. 17! of the slice sensitivity profiles for the simulated scanners for five different filter widths z for all simulated cone-beamscanners. The values are given in multiples of the slice thickness S to allow to include the isotropic scanner ASSR36. Since all simulated scanners ~ASSR6,ASSR12, ASSR16, ASSR32, ASSR64, ASSR96, ASSR36! yield the same figures of merit up to the given precision, the tables do not distinguish betweenthem! The figures of merit of the single-slice scanner 180°LI1.5 yield FWHM51.1S , FWTM51.9S , SPQI581% and the descriptors of the four-slice scannerwith multi-slice filtered interpolation 180°MFI6 yield FWHM51.4S , FWTM52.3S , SPQI581%.

Linear line weighting Quadratic line weighting

z FWHM FWTM SPQI z FWHM FWTM SPQI

0.0S 1.3S 2.3S 80% 0.0S 1.5S 2.6S 79%0.5S 1.4S 2.4S 80% 0.5S 1.5S 2.7S 79%1.0S 1.6S 2.8S 80% 1.0S 1.8S 3.1S 79%1.5S 2.0S 3.5S 80% 1.5S 2.1S 3.7S 80%2.0S 2.5S 4.2S 80% 2.0S 2.5S 4.4S 79%



of all scans are the same! and ignoring ASSR64190° onecan see almost no significant deterioration of the image qual-ity except for the scan with the highest table increment~ASSR96! which shows a very slight HU enhancementaround the right inner ear. The multiplanar reformations atthe bottom of the figures confirm these results for the com-plete reconstructed volume: no significant image deteriora-

tion when going to larger cone angles. However, regardingthe MPRs only, 180°LI performs better than the other algo-rithms.

Comparing the ASSR reconstructions to the single-slice180°LI reconstruction shows no severe drawbacks for theapproximate cone-beam reconstruction. The artifact contentin the images is comparable or only slightly increased ascompared to the single-slice scan. An impressive fact is thatthe reconstruction of the 431 mm scanner with a table in-crement of d56 mm yields less artifacts when using the ad-vanced single-slice rebinning algorithm ~ASSR6! than whenusing the standard multi-slice filtering algorithm 180°MFI.Even in the case of four simultaneously measured sliced itmight pay out to use ASSR instead of typical multi-slicez-interpolation algorithms.

To show the influence of the additional filter on the imagequality, especially the artifact level, we have performed re-constructions of the head phantom with z50 mm, 1 mm, and2 mm. In Fig. 12 the influence of z becomes quite clear:z-filtering reduces the artifacts almost completely. It must beemphasized that we have artificially increased the recon-struction increment DaR to produce visible differences in theprintout of Fig. 12; using the optimal reconstruction incre-ment as derived above, almost no artifacts would be visibleeven for z50 and the concept of additional z-filtering wouldnot be necessary in the absence of noise.

The performance of ASSR in reconstructing the Defrise–Phantom ~Feldkamp–Killer! is demonstrated in Fig. 13. Thephantom consists of a stack of ellipsoids with rx5ry

580 mm and rz57.5 mm ~half-axes! centered at x5y50and z5Z•24 mm. Obviously none of the typical cone-beamartifacts are introduced here, except for a HU-value deviationfor the ASSR96 scanner. Regarding this figure it seems as ifASSR16 through ASSR64 would perform better than thegold standard 180°LI: The edges of the phantom show noartificial CT-value enhancement as compared to the single-slice scanner. A profile along the z-axis @Fig. 13~b!# makesclear that there are hardly any deviations between the differ-ent reconstructions and scan modalities. This is the reasonwhy we have only shown two exemplary profiles: the one for180°LI and the profile of ASSR96, which is the worst per-forming scanner regarding the CT-value constancy. The off-center profile @Fig. 13~c!#, however, clearly confirms the ob-served HU deviations of ASSR96. Nevertheless, the shape ofthe phantom is not significantly distorted by ASSR96. Again,the off-center profiles of the other algorithms are too similarto the 180°LI profile to be depicted in the plot.

To complete our evaluations we have further done a com-parison to the original single-slice rebinning algorithm14

which uses nontilted slices and a ray-by-ray optimization.We have achieved this using ASSR with zero tilt angle g50° and the PL optimization ~of course, for nontilted slicesthe PL approach is equivalent to the PO approach!. We willfurther denote these settings by SSR ~single-slice rebinning!.The reconstructed slices at zR5230 mm, 225 mm, and 0mm using SSR12 through SSR96 ~i.e., table increments d512 mm...96 mm) are given in Fig. 14. Obviously the arti-fact content is increased as compared to ASSR. For low table

FIG. 9. Center ~solid line! and off-center ~dashed and dotted lines! SSPs forthe standard algorithm 180°LI and ASSR64. The profile variations ~devia-tions from the center profile! are reduced by ASSR. Additional z-filteringyields, not surprisingly, even nicer profiles. The horizontal grid lines havebeen placed at one tenth of the maximum value, at half the maximum valueand at the maximum value, respectively. ~a! 180°LI variations. ~b! ASSR64variations with z50. ~c! AASR64 variations with z51 mm.



increment values this becomes evident from the slices at zR

5230 mm and zR5225 mm.Moreover, SSR shows a behavior which is also known

from other approximate cone-beam algorithms:13 With in-creasing cone angle the artifact level increases drastically.Especially for d564 mm and d596 mm the image qualitybecomes unacceptable. Severe cone-beam artifacts appearespecially around the petrous bone, the frontal sinus, and itssurrounding bones. Thus even for the relatively small coneangles as used in this paper, it pays out to use the advancedapproach ASSR, i.e., tilted reconstruction planes.

VI. DISCUSSION

The advanced single-slice rebinning ASSR has the poten-tial to become a practical tool for 3D cone-beam reconstruc-tion in medical spiral CT. Both its performance and the im-age quality are nearly comparable to the gold standard spiral

single-slice CT. It has been shown that ASSR outperformsother known approximate cone-beam algorithms, not onlyregarding image quality, but also computation time.13 More-over, ASSR can make use of available backprojection hard-ware since the backprojection procedure is in 2D. A veryimportant fact is that ASSR does not drastically increase theartifact level when going from small cone angles, such asASSR16, to moderate cone angles as in ASSR96. Of course,there is an upper limit for the number of detector lines andfor the table increment of the spiral scan as has been thor-oughly discussed above and in Appendix B. Nevertheless,this is no restriction for medical CT since cone-beam scan-ners with much more than 60 or 70 detector rows are un-likely to be built.

Regarding image noise no significant differences betweenthe standard z-interpolation algorithms and the single-slicerebinning have been found ~Table I!. The reason is that all

FIG. 10. Comparison between differ-ent scanners using the head phantom.The scan parameters are given inTable I. The top four lines are recon-structed at the z-positions zR

5230 mm, 225 mm, 22 mm, 0 mm.The bottom two lines are coronal andsagittal MPRs containing the axis ofrotation.



FIG. 11. Comparison between differ-ent scanners using the head phantom.The scan parameters are given inTable I. The top four lines are recon-structed at the z-positions zR

5230 mm, 225 mm, 22 mm, 0 mm.The bottom two lines are coronal andsagittal MPRs containing the axis ofrotation.

FIG. 12. Influence of z on the artifactlevel. We have used a quite sparse re-construction increment DaR ~the opti-mal value was doubled! to emphasizethe artifacts. As can be clearly seen,the additional z-filtering drastically re-duces the artifacts emanating from theright inner ear. Parameters: ASSR96,zR50, linear line weights.



algorithms have the same amount of data contributing to theimage ~assuming z50): 180°LI can be regarded as an inter-polation between two half-scans with equally distributed in-terpolation weights in @0, 1# and the same applies to ASSR inthe case of z50 where z-interpolation is done between thetwo closest available tilted slices ~which were reconstructedfrom two rebinned half-scans as well!. Answering the ques-tion of patient dose is a question of collimator design. SinceASSR at its present state cannot make use of redundant data~in the sense of artificially increasing the required number of

detector rows!, it is advantageous to collimate out unuseddetector areas ~e.g., see Fig. 6!. In this case the noise valuesof Table I show that no difference in patient dose will beobserved when using ASSR. However, the inability of ASSRto make use of overlapping data is a disadvantage for medi-cal applications which often require to use overlapping pitchvalues in order to accumulate dose and thus reduce imagenoise. Nevertheless, image noise can be reduced by increas-ing the tube current or by simply increasing the rotation timeof the scanner. Increasing the tube current may in some casesbe limited due to technical reasons ~tube load! but decreasingthe rotation speed will be equivalent or superior—regardingimage noise and temporal resolution—to using redundantdata.

Since the ASSR reconstruction time is dominated by the2D backprojection, and since it has been shown at the end ofSec. IV H that ASSR approximately requires as many back-projections per slice thickness as recommended for standardspiral z-interpolation algorithms, there is only a slight de-crease in reconstruction speed as compared to 180°LI ~as-suming a reconstruction increment of one-third of the slicethickness!. The decrease in reconstruction speed mainly re-sults from the rebinning procedure and from the z-filtering.In contrast to the 180°LI reconstruction speed of ASSR isnot dependent on the user selectable reconstruction incre-ment: Tilted images are precomputed at increments of DaR

and the final z-filtering step to yield the image at zR is nottime consuming.

A very important fact of ASSR not mentioned explicitlyabove is its high temporal resolution which lies in the orderof t rot/2 since only about half a rotation is needed to recon-struct a tilted image ~regarding the central ray!. Exact cone-beam algorithms, in contrast, require a significantly largerrange of scan data ~in many cases the complete data set con-tributes to each voxel! to reconstruct a single slice whichresults in a loss of temporal resolution. Thus in case of pa-tient motion, patient breathing or cardiac motion, the ad-vanced single-slice algorithms will perform superior to theexact algorithms since fewer motion artifacts will show up inthe images.

Moreover, it is possible to apply methods to reduce pa-tient dose at constant image quality by adaptive filtering inthe raw data domain and to reduce metal artifacts using asimilar technique. Respective results are shownelsewhere.18,19

Since ASSR is formulated as a rebinning algorithm it isable to handle arbitrary detector geometries besides the cy-lindrical and flat horizontally aligned detectors discussedabove. Moreover, all kinds of ~known! geometrical misalign-ments can be included in the rebinning equations which isespecially important for micro CT applications where mostof the misalignments are unavoidable and not easy to correctmechanically. Realizing these supplements to ASSR isstraightforward and thus will not be explicitly stated in thispaper ~the manifoldness of possible detector geometries andpossible misalignments would not allow for a brief formula-tion of these extensions!.

Summarizing, we think that ASSR offers a multitude of

FIG. 13. ~a! MPRs ~plane x50) of the Defrise–Phantom ~0/30!. The rangedepicted is 281 mm<z<181 mm at a voxel size of 0.5 mm. ~b! axialprofiles ~line x5y50) of 180°LI1.5 and ASSR96. ~c! axial profiles off-center ~line x50, y550 mm! of 180°LI1.5 and ASSR96. The profilesf (0,0,z) and f (0,50 mm,z) show the reconstructed object function along thez-axis for the 180°LI scan ~solid line! and for ASSR96 ~dashed line!. Thecorresponding profiles for the other scans are too similar to depict theirdifferences in the plot.



advantages, including reconstruction speed, image quality,flexibility with respect to possible scanner geometries andmisalignments, and robustness. Thus ASSR is a very prom-ising candidate for future medical and nonmedical spiralcone-beam CT applications.

ACKNOWLEDGMENTS

This work was supported by the ‘‘FORBILD’’ Grant No.AZ 286/98 ‘‘Bayerische Forschungsstiftung, Munchen, Ger-many.’’ The CT-simulation software DRASIM was madeavailable to us by Dr. Karl Stierstorfer, Siemens MedicalEngineering, Forchheim, Germany.

APPENDIX A: CYLINDRICAL DETECTOR

Here we are going to give the coordinate transformationwhen going from the flat (u ,v)-detector to a cylindrical de-tector centered about the focus with coordinates (b ,b) whereb is the angle of the ray with respect to the central ray ~bcounts along the negative u-axis! and b is the vertical posi-tion at the cylindrical detector and thus directly correspondsto the flat detector’s v:

D:r~b ,b !5S RF sin a

2RF cos a

da/2pD 1~RD1RF!S 2sin~a1b !

cos~a1b !

0D

1bS 001D .

Using Eq. ~11! we can find the corresponding transforma-

tion from cylindrical coordinates (b ,b) to flat coordinates(u ,v):

u52~RD1RF!tan b ,

v5b

cos b.

The inverse transform is

b52tan21u

RD1RF,

b5v cos b5v~RD1RF!

A~RD1RF!21u2

.

APPENDIX B: A POSSIBLE SETUP FOR AMEDICAL CT SCANNER

We here assume to have a cone-beam scanner with anin-plane geometry equivalent to a typical modern CT scanner~Siemens SOMATOM Volume Zoom!. This means that wehave the following values fixed:

RF5570 mm, RD5435 mm, RM5250 mm.

Our virtual scanner has a cylindrical area detector and a fanangle of F552°. The in-plane resolution will be determinedby the size b and by the temporal integration a of the de-tectors. Since this is not object of this paper, we will onlyconsider the influence of the detector size b in z-direction onimage quality and assume to have reasonable values for band a .

FIG. 14. Noo’s original single-slice re-binning algorithm for comparison. Thethree rows correspond to zR

5230 mm, 225 mm, and 0 mm. Theimages were reconstructed usingASSR with a tilt angle of g50° andthe ray-by-ray optimization PL ~or PO,which is equivalent to PL in the caseof nontilted reconstruction planes!.These settings are equivalent to theoriginal algorithm ~Ref. 14!. z51 mm.



Now, what is the optimal number Nb of detector lines andwhat is their optimal size b to yield a scanner with highvolume scanning capabilities and a high z-resolution togetherwith the reconstruction scheme presented in this paper?

First, let us regard the beam profile at the center of rota-tion. It is made up of the focal spot size f in the z-direction~we assume a rectangular intensity distribution! and of thedetector size b ~where we assume a 100% geometrical effi-ciency and a rectangular sensitivity profile in the z-direction!.Both effects together yield a trapezoidal profile at the centerof rotation with a FWHM of

Db5bRF∨ f RD

RD1RF. ~B1a!

Assuming that a focal spot size of f < b can be realized ~as itis the case in today’s scanners!, the equation will be domi-nated by the detector element size and thus below we will befree to use

Db5 bRF

RD1RF~B1b!

for convenience.We know from Sec. IV H that we must demand

Dzmean<RF

RMDb

in order not to decrease the theoretical achievable spatialresolution.

From Fig. 4 we find that the mean z-deviation lies below2% of the table increment per 360° rotation. Making use ofthis fact and the above equation we can write

0.023dmax5RF

RMDb .

For a given table increment d<dmax we must demand forthe detector size Nbb in the z-direction

dp1F

2p5Nbb

RF

RD1RF,

which means that the detector must at least cover a range of180° plus the fan angle in order to allow for full rebinning~see rectangular areas in Fig. 6!. ~Tilting the detector by acertain angle about the central ray will almost completelyremove the effect of the fan angle F in the above equationand reduce the number of required detector lines by a factorof 1.5. Detector designs like this can be used for academicreasons but since the detector is dependent on the table in-crement and on the table direction it is far from optimal for amulti-purpose CT-scanner.!

Now using d5dmax and Eq. ~B1b!, we find

Nb5 dp1F

2p

RF

0.023RMe574.

The optimal size b of a detector line cannot be determined

from these equations, since all formulas scale according to bin the z-direction. The detector size rather has to be deter-mined from the requirements of the scanner, e.g. by demand-ing a certain spatial resolution Db in the z-direction.

a!Send correspondence and reprint requests to: Dr. Marc Kachelrieß, Insti-tute of Medical Physics, University of Erlangen–Nurnberg, Kranken-hausstraße 12, D-91054 Erlangen, Germany, Tel.: 49-9131-8522957, Fax:49-9131-8522824, electronic mail: [email protected]

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