Extended parallel backprojection for standard three ... · Extended parallel backprojection for...

Extended parallel backprojection for standard three-dimensionaland phase-correlated four-dimensional axial and spiral cone-beamCT with arbitrary pitch, arbitrary cone-angle, and 100% dose usage

Marc Kachelrieß,a) Michael Knaup, and Willi A. KalenderInstitute of Medical Physics, University of Erlangen—Nu¨rnberg, Germany

~Received 12 September 2003; revised 7 April 2004; accepted for publication 7 April 2004;published 27 May 2004!

We have developed a new approximate Feldkamp-type algorithm that we call the extended parallelbackprojection~EPBP!. Its main features are a phase-weighted backprojection and a voxel-by-voxel180° normalization. The first feature ensures three-dimensional~3-D! and 4-D capabilities with oneand the same algorithm; the second ensures 100% detector usage~each ray is accounted for!. Thealgorithm was evaluated using simulated data of a thorax phantom and a cardiac motion phantomfor scanners with up to 256 slices. Axial~circle and sequence! and spiral scan trajectories wereinvestigated. The standard reconstructions~EPBPStd! are of high quality, even for as many as 256slices. The cardiac reconstructions~EPBPCI! are of high quality as well and show no significantdeterioration of objects even far off the center of rotation. Since EPBPCI uses the cardio interpo-lation ~CI! phase weighting the temporal resolution is equivalent to that of the well-establishedsingle-slice and multislice cardiac approaches 180°CI, 180°MCI, and ASSRCI, respectively, andlies in the order of 50 to 100 ms for rotation times between 0.4 and 0.5 s. EPBP appears to fulfillall required demands. Especially the phase-correlated EPBP reconstruction of cardiac multiplecircle scan data is of high interest, e.g., for dynamic perfusion studies of the heart. ©2004American Association of Physicists in Medicine.@DOI: 10.1118/1.1755569#

Key words: Cone-beam CT~CBCT!, cardiac imaging, 4-D reconstruction, image quality

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I. INTRODUCTION

The ongoing development of medical cone-beam~CBCT! scanners requires providing cone-beam reconsttion algorithms adequate for medical purposes. These mbe capable of handling arbitrary pitch standard and phacorrelated reconstruction for circular, sequential and sptrajectories.~Flexible pitch selection in the range from abo0.3 to about 1.5 is used for dynamic studies to havecontrol over the table speed, and it is used for cardiac CTadopt the amount of data overlap to the patient’s heart rOne further adjusts the pitch to compensate for tube polimits: low pitch values allow us to accumulate dose bincrease the total scan time. Finding the best compromrequires flexibility in pitch selection; two or three distinpitch values are certainly not enough.! Due to the increasingdose awareness, it is further of importance to ensure 10dose usage, which means that each measured ray contrito the reconstructed images.

As far as we know, three types of image reconstructalgorithms can be found in today’s 16-slice medical CT scners: z-interpolation, single-slice rebinning, and Feldkamtype image reconstruction. To reconstruct stationary obje~standard reconstruction! manufacturers use Feldkamp-basalgorithms1–6 or they use the so-called single-slice rebinnialgorithms such as the advanced single-slice rebinning arithm ~ASSR!.7–12The reconstruction of periodically-movinobjects such as the beating heart is done with phacorrelated algorithms. There, two-dimensionz-interpolation algorithms dedicated for four-slice CT a

1623 Med. Phys. 31 „6…, June 2004 0094-2405 Õ2004Õ31„6

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adopted and used to reconstruct cardiac data for scanwith more than four slices.

However, there are several restrictions to theseproaches that may inhibit their use in scanners with signcantly more than 16 slices. It is known that the Feldkamtype reconstruction of spiral data is of acceptable quaonly if the direction of convolution corresponds to the diretion of the spiral tangent.13 Whereas this fact can be accounted for, a severe drawback is the use of a detector wdow similar to the Tam window14 that results in pitch-dependent dose efficiency values that are alwsignificantly below 100%. The ASSR algorithm yields gooimage quality for scanners with up to 64 slices7 and has beengeneralized for arbitrary pitch spiral CT with 100% dousage.15 Its generalization to cardiac scanning ASSRCI~car-dio interpolation! yields good images only for as little as 1to 32 slices.15

Besides these restrictions, the development of wide cangle medical CT scanners with large area detectors~64slices or more! will likely draw attention toward the circularscan trajectory. Especially cardiac CT is likely to push tdevelopment of reconstruction algorithms for circle scaAssume a scanner withM5256 slices and a slice thicknesof S50.75 mm. The total collimated width ofMS5192 mm is sufficient to cover the heart completely andcircle scan will be the trajectory of choice. This scan mowould allow us to increase temporal resolution simplyincreasing scan overlap, which directly corresponds to

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1624 Kachelriess, Knaup, and Kalender: EPBP for CBCT 1624

scan time or the number of rotations used for a circle sca16

Even more important, the circle scan is the trajectoryquired for dynamic studies such as perfusimeasurements.17,18

To accommodate these demands, we developedevaluated the extended parallel backprojection~EPBP!,which is a new class of standard and phase-correlated imreconstruction algorithms that handle circular scans, seqtial scans and spiral scans with arbitrary pitch and 10dose usage.19,20EPBP is a Feldkamp-type algorithm that peforms azimuthal, longitudinal, and radial rebinning to parlel geometry before an extended 3-D cone-beam backprotion with voxel-dependent weights generates the fivolume.

Rebinning to parallel geometry using a slanted detec~longitudinal rebinning! has been discussed in the literatubefore. An example are the PI and the PI-SLANalgorithms.21,22However, these do not achieve a full detecutilization nor do these approaches allow for phacorrelated reconstruction. Detector utilization itself hascome an issue recently. For example, Ref. 23 proposes aweighting scheme to improve the detector usage and toduce image noise; 100% dose usage is not achieved, hever. Also recently introduced was the extended cardreconstruction.24,25 It uses only a simple illumination window and avoids longitudinal rebinning~therefore the direc-tion of convolution may not be optimal!. The extended cardiac rebinning is evaluated for 16 slice scanners only. Effoin circular or sequential image reconstructions for mediCT can be found in Refs. 26, 27. This group uses a pararebinning T-FDK extension of the original Feldkamp algrithm and achieves promising image quality. However, nther cardiac imaging is considered nor full dose usageguaranteed due to the truncation of the detector to a virrectangular flat panel detector.

Note that EPBP belongs to the class of approximate cobeam image reconstruction algorithms and could theorcally be outperformed by analytically exact reconstructalgorithms. However, even when considering recent brethroughs in exact cone-beam reconstruction28 these analyti-cal methods are far away from providing a solution to tissues full dose usage and phase-correlated imaging.13

In this paper we specifies EPBPStd and EPBPCI. Iorganized as follows. A nomenclature section introdumost symbols. Then, the extended parallel backprojectiongorithm EPBP is defined. Numerous simulation and msurement results demonstrate the capabilities of EPBP cpared to the gold-standard four-slice spiral algorith180°MFI and 180°MCI29 and ASSR. Due to the focus omedical CT imaging, we restrict our derivations and simutions to cylindrical detector systems that are centered abthe focal spot and aligned parallel to the axis of rotation. Tis no loss of generality since the respective rebinning eqtions can be equally well formulated and implementedother detector types such as flat panel detectors.

Medical Physics, Vol. 31, No. 6, June 2004

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II. MATERIALS AND METHODS

The projection or rotation angle is calleda. It will be usedto parametrize the complete circular, sequential or spiraljectory. The angle within the fan is given byb, the ray’sposition along the detector’sz-axis is denoted asb. The par-allel ray geometry is parametrized byj for the ray’s~signed!distance to the origin andq for its angle, and we usel for theparallel detector’s longitudinal component. These notatioconform to those in Ref. 9.

The cardiac phase as a function of the view angle isnoted asc(a) and is to be taken modulo 1. It parametrizthe motion with respect to the motion synchronization peaIn the case of ECG correlation, these peaks are taken asR peaks of the ECG. We implicitly assume a correct handlof the modulo properties ofc:

b•c floor function, yields greatest integer lower oequal

d•e ceil function, yields smallest integer greaterequal

d~•! Dirac’s delta functionsgn(•) sign function such thatx5uxusgnxa,b,b ray parametrization for the cylindrical

detectoraR projection angle about which the reconstru

tion is centeredB detector boundary,2B<b<B, with B

5MSRFD/2RF

cose length correction factor, cos2 e5RFD2 /(b21RFD

2 )d,d table increment per rotation~spiral! or between

successive rotations~sequence!, d5(0,0,d)F fan angle,F52 sin21(RM /RF), in our case 52°G cone angle,G52 tan21(B/RFD), in our case up

to 19.1°M number of simultaneously acquired slices,

our case 4,...,256o,o origin of the scan trajectory,o5(0,0,o)p sequential or spiral pitch,p5d/MSp0(a,b,b) measured projection datap1(q,b,b) azimuthally rebinned projection datap2(q,b,l ) azimuthally and longitudinally rebinned datap3(q,j,l ) data rebinned to parallel geometryQ parameter used to quantify image qualityr coordinate vector,r5(x,y,z)RD distance from the detector to the center of

rotation ~z axis!; in our case 470 mmRF distance from focus to center of rotation~z

axis!; in our case 570 mmRFD distance from focus to detector,RFD5RF

1RD ; in our case 1040 mmRM radius of the field of measurement~FOM!; in

our case 250 mmS slice thickness; in our case 0.75 mms~a! focus trajectoryw(q), w(q) 180°-complete weight function and 180°-

normalized weight function~C/W! display window center and width of the recon

structed images

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Cone-beam data were simulated using a dedicated xsimulation tool ~ImpactSim, VAMP GmbH, Mo¨hrendorf,Germany!. The scanner geometry was chosen correspondto a modern medical CT scanner~Sensation 16, SiemenMedical Solutions, Forchheim, Germany!: 1160 projectionsper rotation and 672 channels per slice were simulaPhantom definitions were taken from the phantom data bat http://www.imp.uni-erlangen.de/forbild except for the cdiac motion phantom that is defined in Ref. 30. For simution, a 232 subsampling was used for the focal spot for tdetector; integration time was simulated by subsamplingangular direction with two samples. The starting locationthe spiral wasz15272 mm2 1

2d for the thorax phantom andz15272 mm2ud/pu for the cardio phantom. For the circlscan we usedz572 mm andz50 for the thorax and thecardio phantom, respectively. The thorax sequence sstarted atz5272 mm. The angular position for the firsview wasa150 for all simulations. Poisson noise was addto the simulated data by undoing the log, adding noise,performing the log again. The number of incident quawere chosen to get a realistic noise value in the order oHU within a 20 cm water phantom. Simulations were rstricted to monochromatic beams and scatter was switcoff to avoid introducing nonlinearities.

Simulated and measured raw data were reconstructeding ~a! the four-slicez-interpolation algorithms 180°MFI and180°MCI, ~b! the single-slice rebinning algorithms ASSRSand ASSRCI, and~c! the EPBPStd and EPBPCI algorithmdescribed in this paper. All reconstruction algorithms weimplemented on a standard PC with dedicated image restruction and evaluation software~ImpactIR, VAMP GmbH,Mohrendorf, Germany!.

III. NOISE AND RESOLUTION MEASUREMENTS

To quantify image noise and spatial resolution we haperformed reconstructions of a simulated 20 cm water phtom that has two delta objects embedded. Both delta objare located at a 3 cmdistance from the isocenter. The firstthe point

d~x!d~y230 mm!d~z!

and the second object is the line

d~x!d~y130 mm!.

The data were simulated and reconstructed using the sparameters and algorithms as for the thorax and cardiotion phantom data. A voxel size of 0.1 mm, which is webeyond the expected resolution, was used to fully capturepoint spread function.

The reconstructed water phantom images were usequantify noise and resolution as follows. The in-plane Pwas computed by computing 60 radial profiles within 18through the delta object and by averaging these profilesall radial angles and for all reconstructedz-positions. Thecomputation included a bilinear interpolation between neiboring pixels. The axial PSF, also known as the slice setivity profile ~SSP!, was determined from a longitudinal pro


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file through the delta point. In-plane resolutionDr andlongitudinal resolutionDz ~effective slice thickness! are de-fined as the full width at half-maximum~FWHM! of thein-plane PSF and the SSP, respectively. Image noises wascomputed as the standard deviation of the pixel valuescuboid region of interest that extends over all reconstrucslices and over an transaxial area of 1 cm by 1 cm centerex530 mm andy50.

Further, a quality factor to relativize variations in resoltion and/or image noise and/or pitch value was computed

Q2}p

Dr3Dzs2 ; ~1!

the constant of proportionality was chosen to obtainQ5100% for the gold-standard four-slice scan withp51 re-constructed with 180°MFI.

The rationale for the definition ofQ is as follows. For agiven dose and a given image reconstruction, the algoriimage noise variances2 is inverse proportional to the size othez-resolution element and to the third power of the sizethe in-plane resolution element, i.e.,Dr3Dzs25const.31

Lower values of this constant imply higher image qualiand vice versa. Further, s2 is inverse proportional to theapplied dose, which, in turn, is proportional to 1/p ~if thetube current is the same for all measurements!, thereby di-viding the variance by the pitch factor compensates fordifferences in dose.

IV. GEOMETRY

Figure 1~a! shows a view from the operator’s viewpoinonto the CT gantry. The coordinate system is attached topatient table. Thez axis is defined to be the axis of rotationThe y axis is defined to point vertically upward~for zerogantry tilt! and thex axis is given by the orthonormalityrelation xÃy5z. Moving the table into the gantry yieldincreasingz positions.

A. Fan beam

The fan-beam geometry of medical CT scanners useslindrical detectors of radiusRFD . The focal spot rotates aradiusRF about the isocenter~z axis! and the patient tableadvances byd5(0,0,d) during one rotation. We useaPRto denote the gantry rotation angle~projection angle!, bP@21/2F,1/2F# to denote the angle within the fan andb asthe longitudinal detector coordinate. Leto5(0,0,o) be thesource’sz position for a50. Then, the source location iparametrized as follows:

s~a!5RFS sina2cosa

0D 1d

a

2p1o. ~2!

The coordinate vector of the detector (a,b,b) is given as

r ~a,b,b!5s~a!1RFDS 2sin~a1b!

cos~a1b!

0D 1bS 0

01D . ~3!


FIG. 1. ~a! Gantry as viewed from the operator. The coordinate system is attached to the table. The direction of increasinga ~counterclockwise rotation! isindicated.~b! Coordinate system of a cylindrical detector scanner projected into thex-y-plane. The rays are described by the angleb within the fan and therotation anglea of the gantry in fan geometry. The corresponding parameters in parallel~i.e., rebinned! geometry arej andq. ~c! Side view onto the raygeometry of a scanner withM54 slices. The midplane is the dotted line.

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2B. Parallel beam

For image reconstruction it is convenient to convertdata to in-plane parallel geometry. This means thatz-component of the rays is temporarily ignored, which cresponds to projecting the scan into thex-y plane. The par-allel rays are parametrized by their distancej to the center ofrotation and by their angleq with respect to the negativeyaxis. The ray itself is given by the expressionjj(q)1Rh(q). Thereby, we have introduced the unit vectorjpointing from the isocenter to the ray~q, j! and the ray’sdirection vectorh:

j5j~q!5S cosqsinq D , h5h~q!5S 2sinq

cosq D .

This definition was chosen to have the central rays forfan beam (b50) and for the parallel beam (j50) coincid-ing as long asa5q. The normal form of the ray is thefamiliar expressionx cosq1ysinq2j50.

Please note that

j5x cosq1y sinq,~4!

h5y cosq2x sinq,

allow us to represent a given pointr5(x,y) as r5jj1hhin the rotated~j, h! coordinate system.

C. Point projection

For 3-D backprojection as it is performed in EPBP, fexample, the perspective projection of a pointr5(x,y,z) inthe spatial domain from the focal spots~a! onto the cylindri-cal detector is of interest, i.e., we are interested in its (b,b)coordinates.


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Transaxially, the one-to-one correspondence betweeray’s fan-beam coordinates and its parallel beam coordinis simple:

q5a1b,j52RF sinb, as well as

a5q1arcsinj/RF ,b52arcsinj/RF, ~5!

describe the same ray. Therefore, we are free to use theallel beam coordinates when deriving the point projectcoordinates.

For the point r we can easily state thatj5x cosq1ysinq is its radial detector coordinate. From~5! we imme-diately find the point’sb coordinate.

The longitudinal detector coordinate must be compuusing the intersection theorem. The transaxial distanceD ofthe respective voxelr5(r cosw,r sinw,z) to the sources isgiven as

D25~RF sina2x!21~RF cosa1y!2

5RF222RFr sin~a2w!1r 2.

Alternatively @cf. Figs. 2 or 1~b!#, we can obtain the relation

D5RF cosb1h5ARF22j21h,

with j and h given by ~4!. Now, computeb by scaling theaxial distancez2d(a/2p)2o of the voxel’s z coordinateand the source’sz coordinate fromD to RFD :

b5RFD

D S z2da

2p2oD . ~6a!

And, by representing sinb by the quotient of the opposite leand the hypotenuse~see Fig. 2!, we get another useful representation ofj:

j52RF sinb5RF

x cosa1y sina

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V. DEFINITION OF EPBP

EPBP is an approximate 3D cone-beam image reconsttion algorithm concept suitable for various kinds of trajecries; in this paper we restrict our investigations to the cirlar, sequential, and spiral trajectory. In contrast tozinterpolation and single-slice rebinning algorithms, a tr3-D backprojection of shift-invariant filtered cone-beam dis performed; therefore one might consider EPBP to be ofFeldkamp type.

EPBP image reconstruction consists of the followisteps:

~1! Azimuthal rebinning:p0(a,b,b)cose→p1(q,b,b).~2! Longitudinal rebinning:p1(q,b,b)→p2(q,b,l ).~3! Radial rebinning:p2(q,b,l )→p3(q,j,l ).~4! Convolution:p3(q,j,l )→ p(q,j,l ).~5! Weighting and backprojection:p(q,j,l )→ f (x,y,z).

Since our data are discretized, rebinning and backprotion always requires some kind of interpolation. We perfodestination-driven rebinning and backprojection withsimple linear interpolation on the source side. In view of thigh image quality we achieve with this standard approacis beyond the scope of this paper to test more sophisticinterpolation techniques.

A. EPBP rebinning

The general coordinate transform and the backward traform between a ray in fan-beam coordinates~a, b! and aparallel-beam ray~q, j! are given by~5!. Medical cone-beamdata are measured in the fan-beam domain. Wep0(a,b,b) to denote the acquired~logarithmic! attenuation

FIG. 2. In-plane illustration of the derivation of Eq.~6!.


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data. The variableb denotes the longitudinal detector coodinate @cf. Eqs. ~2! and ~3!#. Figure 3 illustrates that thefan-beam data can be converted into parallel beam dataresorting or rebinning procedure. This transaxial rebinningusually decomposed into an azimuthal and a radial rebinnstep. In between both steps, EPBP performs a longitudrebinning.

1. Azimuthal rebinning

The conversion of fan-beam data~a, b! to fan-parallel-beam data~q, b! is called azimuthal rebinning and corresponds to switching from the independent variablea to thenew independent variableq:

p1~q,b,b!5p0~a,b,b!.

On the rhs,a5a(q,b) is a function of the destination variablesq andb according to Eq.~5!. For the cone-beam algorithms a length correction factor cose defined by cos2 e5RFD

2 /(b21RFD2 ) is multiplied to the data during azimutha

rebinning to account for the obliqueness for the rays wrespect to thex-y plane.

2. Longitudinal rebinning

Until recently, many Feldkamp-type algorithms peformed convolution along the direction of constantb just asit is the case with classicalz-interpolation algorithms. How-ever, severe image artifacts were observed, and theserithms were found to be inacceptable for medical CT. It wthe success of ASSR and the theory behind exact imageconstruction that indicated that convolution should ratherperformed in the direction of the spiral tangent.13

For performance reasons it is necessary to perform a rby-row convolution. Consequently the fan-parallel data hato be converted into an adequate format. This is the reawhy longitudinal rebinning is required. To derive the rebining formula, we regard the tangent direction,

FIG. 3. Azimuthal and radial rebinning converts fan-beam into parallel-bedata. Longitudinal coordinates are not influenced.

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which is the optimal direction of convolution. The relatioship betweenb andj when moving an arbitrary object poinalong the directionds and regarding its projection onto thdetector can be derived using~6!:

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In the last step, the components ofds are inserted. Now,define a new longitudinal variablel as

b5 l 1lj, with l5db

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such thatdl/dj50 in the direction ofds. Then,

p2~q,b,l !5p1~q,b,b!,

with b5b(b,l )5 l 1lj5 l 2lRF sinb is the longitudinalrebinning that switches tol as the new independent variabl

The convolution step that precedes the backprojectionquires access to alljP@2RM ,RM#. To allow for longitudi-nal rebinning without losing parts of the original detectareabP@2B,B# we want to providel within the rangelP(B1uluRM)@21,1#. According to b5 l 1lj this, how-ever, requires access tobP(B12uluRM)@21,1#. We pro-vide this area by extending the detector by a simple extralation that repeats the outermost detector rows—only th100% dose usage can be achieved. This extrapolated arused exclusively for convolution. Backprojection then rspects the original detector area. Figure 4 illustrates the sation. Our procedure is justified because the main weighthe reconstruction kernel lies at its central element that nereaches extrapolated areas. We further have evaluatedinfluence of extrapolation on the final images by switchiextrapolation on and off and found no negative effects.

3. Radial rebinning

Data are then converted to parallel geometry via radrebinning,

p3~q,j,l !5p2~q,b,l !.

Here,b5b(j) is a function ofj according to Eq.~5!.

FIG. 4. Longitudinal rebinning includes an extrapolative extension ofdetector~gray! to allow for convolution. The extrapolated areas~white! willbe used exclusively for convolution and will not be used during backprotion.


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B. Convolution

The remaining steps are convolution and weighted baprojection. Standard convolution kernels for parallel dasuch as the Shepp–Logan kernel can be used for EPBP.volution yields convolved data,

p~q,j,l !5p3~q,j,l !* k~j!.

As already mentioned, the convolution exceeds the origdetector limits and uses data of the extended detector.

C. Weighting

EPBP uses a voxel-dependent weight function~a! to per-form phase-correlation and~b! to normalize the back-projected contributions to 180°.

The number of simultaneously scanned slices ranged f4 to 256 in our simulations. Since submillimeter slice thicnesses are predominately used in today’s scanners andour clinical 16-slice scanner allows for 0.75 mm slices,choseS50.75 mm for all simulations and measuremenThese parameters result in the following cone angles~seeTable I!.

With our implementation of EPBP, three kinds of sctrajectories are possible: the circular scan, the sequenceand the spiral trajectory. For the circular scan and forsequence scan~5multiple circles!, it is possible to analyti-cally calculate thez range that is reconstructable within thFOM. This is best done using the results of Sec. III C. Droping the da/2p term of ~6a! we find that uzu<DB/RFD .Now, regard the point (RM cosw,RM sinw) at the edge of theFOM. For that point we findD25RF

222RFRM sin(a2w)1RM

2 , whose value obviously changes while the scannertates about the point we fixed. If we want the point toexposed during 360°, one easily finds that the infimum vaof D is RF2RM , which is reached whenever the sine evaates to one. This meansuzu<1/2MS(12RM /RF)51/2MS(12sin 1/2F) or, equivalently, d<MS(12sin 1/2F) and p<12sin 1/2F'0.56. For data completeness, however, an exposure range ofp1F ~in fan-beamcoordinates! is sufficient. Choosing21/2(p1F)21/2p<a2w<1/2(p1F)21/2p yieldsD>RF cos 1/2F. In thatcaseuzu<1/2MS cos 1/2F andp<cos 1/2F'0.90 follows.

In cardiac CT, another degree of freedom is introducedis the relation of the patient’s local heart ratef H to the scan-ner’s rotation timet rot . It has been shown~see Refs. 16, 30and 31! that the decisive parameter is the productf Ht rot .Thus, our cardiac results and image quality are not a funcof f H and t rot but rather only a function off Ht rot . For con-venience, we fixedt rot50.5 s while varying the heart rate ia range from 60 to 120 min21. Conversions to other rotation

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TABLE I. Cone angles.

M 4 16 32 64 128 256

G 0.302° 1.21° 2.41° 4.82° 9.63° 19.1

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FIG. 5. Gold standard four-slice image reconstruction applied to four-slice data. The cardio data were simulated withp50.375. Ticks indicate the location othe axial slices and the MPRs~0 HU/300 HU!.

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timest rot8 , such as the frequently used 0.375 s and the 0.4rotation time, can be easily done by solving the relatf Ht rot5 f H8 t rot8 .

It is of help to introduce the visibility rangeV of the voxelr of interest:V(r ) is the set of all view angles under whichris measured. The visibility range can be computed usingpoint projection equations of Sec. IV C. For the spiral trajetory, no closed analytical solution can be given andV mustbe determined numerically. Note thatV is not an interval buta union of several disjunct intervals, in general.

Given V phase weighting can be performed by defining


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weight functionw(q) that is zero in the complement ofVand that is zero or positive inV. Several phase-weightingapproaches are found in the literature, ranging from singphase or partial scan methods~such as the cardio delta approach described in Ref. 16! to bi-phase or even multi-phasapproaches. In our case we use the cardio interpolation~CI!algorithm that is a multi-phase approach. It combines mtiple allowed data ranges of subsequent motion cyclesusing adaptive triangular weights for each data range.16,30

The CI’s technique of adaptively optimizing the temporwindow width and thereby the temporal resolution led to t

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FIG. 6. Standard and cardio reconstructions of the thorax phantom withp50.375 andf H5100 min21, various algorithms, and various cone angles. Axial slat z5132.75 mm~0 HU/300 HU!.


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FIG. 7. Spiral scan withM564 slices with two pitch values. Thorax phantom, standard reconstruction. Ticks indicate the location of the axial slicesMPRs ~0 HU/300 HU!.

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insted.CT

development of similar approaches that are topic of ongoresearch.24

Since phase-correlation can always be formulated aphase-weighting algorithm EPBP is able to cope with aphase-correlated approach. If one desires to perform a sdard reconstruction, a weight function that is constant witV and zero outside can be used.

A valid weight function has been found as soon as it ffills the 180° condition,

(kPZ

w~q1kp!.0, ;q;

otherwise no CT images of sufficient quality can be pvided.

To avoid data combination artifacts, we further multipthe weight function obtained so far by trapezoidals covereach of the disjunct intervals ofV. Thereby, a 7.2° transitionrange on each side of the trapeze is used in our implemetion.

Sincew must be 180° complete, we further normalize tprojection weight to get a 180°-normalized weight as flows:

w~q!5w~q!

(kw~q1kp!.

The proper normalization ofw can be seen from the following identities:


g

ayn-

n

-

-

g

ta-

-

(k52`

`

w~q1kp!51 and E2`

`

dqw~q!5p.

D. Backprojection

Backprojection means evaluating

f ~x,y,z!5E dq p~q,j,l !w~q!,

with

j5j~x,y,q!5x cosq1y sinq,

l 5b~x,y,z,a!2lj,

a5q2b5q1arcsinj/RF .

This yields the voxel value at~x, y, z!. This procedure musbe repeated for each voxel locationr . Consequently, theweight function implicitly depends onr , too: w(q)5w(r ,q). Note that phase-correlation and the full dose uage both are hidden in the definition of the weight functiwhich, in turn, depends on the voxel-dependent visibilrangeV.

VI. SIMULATIONS AND MEASUREMENTS

To evaluate our new approach and to benchmark it agathe other algorithms, various simulations were performWe restricted ourselves to the geometry of a medical


FIG. 8. Spiral scan withM5256 slices of the thorax phantom. Standard reconstruction. Ticks indicate the location of the axial slices and the MPRs~0 HU/300HU!.

dcto

se

ape

tesn

in

.

drd intan-note

beofandd atIithlur-

re-cts.

ted.arethe

scanner withRF5570 mm, with 1160 views per rotation ana 52° fan angle that is transaxially covered by 672 deteelements.

Summarizing, the following parameter set has been ufor evaluation:

MP$4,16,32,64,128,256%,

pP$0.375,0.5,0.9,1.0,1.5%,

f HP$60,80,100,120% min21,

trajectoryP$circle,sequence,spiral%,

algorithmsP$180°MFI,180°MCI,ASSRStd,ASSRCI,

EPBPCI,EPBPStd%;

some of these combinations are shown in Sec. VII. Wepologize for not being able to show a comprehensive ovview of all parameter combinations simulated and evaluaThe limited space of the paper does not allow us to doNevertheless, we will also discuss some results that arereproduced here.

To quantify our results we have further computed theplane point spread function~PSF!, the slice sensitivity pro-files ~SSP! that constitute the axial PSF and image noise


r

d

-r-d.o.ot

-

VII. RESULTS

The four-slice algorithms 180°MFI and 180°MCI applieto four slice data have been chosen as the gold standathat paper; the other algorithms must compare to that sdard. Gold standard images are shown in Fig. 5. Pleasethat only the 180°MFI reconstruction withp51 ~top right! isused to normalize theQ factor of Eq.~1!. Therefore it is theonly figure whereQ exactly equals 100%.

A. Results for the spiral trajectory

To give a first overview over the image quality that canachieved with the variety of algorithms a compilationaxial images is presented in Fig. 6. It shows standardcardio reconstructions of the thorax phantom reconstructez5132.75 mm for p50.375. The gold-standard 180°MFachieves high image quality in the case of four slices. W16 slices, however, the humerus starts suffering from bring artifacts. For higher pitch reconstructions~not shownhere!, geometrical distortions become visible. The ASSRconstruction of the same 16 slice data show no such artifaWhen moving to 128 slices or more~not shown! the ASSR’simage quality becomes severely degraded, as expecEPBP, in contrast, results in much better images. Thereno geometrical distortions and no artifacts produced by

ruction.


FIG. 9. Spiral scan (p50.375) withf H5120 min21 of the thorax phantom with an increasing number of simultaneously scanned slices. Cardio reconstTicks indicate the location of the axial slices and the MPRs~0 HU/300 HU!.

owth

C

bn

ina

w

a

y

ng

totial

ionstchan-

ntshey

arti-ithts

high contrast bone structures for EPBP. Evidently, the lpitch EPBP reconstructions appear to be equivalent togold-standard reconstructions.

For the cardiac images of Fig. 6, advantages of ASSRover 180° MCI at 16 slices are not apparent. WithM532slices, clear improvements of EPBPCI over ASSRCI canseen. Even for as many as 256 slices the EPBPCI doessuffer from distortion artifacts. However, slight artifactsthe tissue-equivalent parts of the phantom are visible. Incases, the artifact behavior varies with the heart rate andthe chosen reconstruction phase~not shown!. This is becausethe allowed data ranges are a function of the heart ratereconstruction phase.

To give a more comprehensive overview, MPR displa


e

I

eot

llith

nd

s

are needed in addition to the axial images. The followisequence of figures shows the planesx50 mm ~sagittal!, y50 mm ~coronal!, andz50 mm ~axial! of the reconstructedthorax phantom. The remaining fourth quadrant is useddisplay the quantitative parameters for image noise, sparesolution, and the quality indexQ.

Figure 7 shows ASSR and EPBP standard reconstructof the thorax phantom for a 64-slice scanner and two pivalues. Axial as well as MPR views are presented. The phtom’s ribs are made up of inclined hollow cylinder segmeand can be seen in the axial and in the coronal images. Tare of special interest since they tend to easily producefacts. For example, artifacts around the ribs are visible wASSRStd for thep51.0 scan. For EPBPStd, no rib artifac


FIG. 10. Reconstructions of the cardiac motion phantom scanned withM564 slices at 60 min21 with various algorithms~0 HU/500 HU!.

anpte

nsth-

icefor

are visible. The sagittal images show the sternum, the heand the spine. The vertebra, and similarily the transitiobetween the thorax and the shoulder segments, tend toduce high-frequency streak-like artifacts that are orien


rt,sro-d

horizontally. These are of little concern since realistic scainclude scatter and physical objects that do not exhibit maematically perfect alignment. A comparison to the 256-slreconstructions of Fig. 8 shows still excellent images


FIG. 11. Reconstructions of the cardiac motion phantom scanned withM564 slices at 100 min21 with various algorithms~0 HU/500 HU!.

stteitae con-

EPBP whereas the ribs in the ASSR reconstructions areverely degraded. Again, the low pitch images are of bequality than the high pitch images. Regarding the quanttive values for these two figures, we find that image nois


e-r-

is

increased with ASSR and theQ is below 100%. EPBPachieves remarkably highQ, which is mainly due to thelower image noise and due to the improvedz resolution.

The next series of figures shows phase-correlated re

truction.


FIG. 12. Sequence scan withp50.375 andp50.9 of the thorax phantom with an increasing number of simultaneously scanned slices. Standard reconsTicks indicate the location of the axial slices and the MPRs~0 HU/300 HU!.

hios

zetylay

enSthBosa

of

e-lgis

, ft t

show

in

o-es.bednceoth

ottanceis

r iseastss

ent

lgo-ontoord-

structions of the thorax phantom~which is stationary! and ofthe cardiac motion phantom. Figures of merit given for tthorax phantom apply to the reconstructions of the motphantom as well: the scan and reconstruction parameteridentical in both cases.

The CI reconstructions of Fig. 9 have been synchroniusing a heart rate of 120 min21. ASSRCI shows significanblurring in regions off the rotation axis. Artifacts mainlmanifest in the inclined ribs but also the sagittal dispshows blurring. The vertebrae are not delineated very wand the MPRs appear smoothed. Even the humerus is sigcantly degraded. This is confirmed by the FWHM of the Sthat lies at 1.4 mm for the 64-slice scanner. In contrast toadvanced single-slice rebinning reconstruction, EPachieves good delineation of all structures, even for ththat are far away from the rotation center. This applies forcone angles simulated, i.e., up to 256 slices.

The cardiac motion phantom is shown with various algrithms and three reconstruction phases in Figs. 10 and 11the heart rates 60 and 100 min21, respectively. We used thspiral 64-slice scanner withp50.375 to produce these images. The top row is reconstructed using the four-slice arithms, the middle row is ASSR, and the bottom rowEPBP. The first column shows a standard reconstructioncomparison, whereas the remaining three columns depicmotion phantom in the slow~0%!, medium~30%!, and highmotion ~50%! phase, respectively.


enare

d

llifi-PePell

-or

o-

orhe

Inspecting the axial images, we find only little variationbetween the three algorithms. Here, EPBP tends to sslightly less motion artifacts. This can be seen especiallythe slow motion phase~0% of S–S! of the 60 min21 and ofthe 100 min21 reconstructions~Figs. 10 and 11!.

The 120 min21 case that is not shown here exhibits mtion artifacts for all algorithms and reconstructed phasThis behavior, which has already been thoroughly descriand analyzed in Refs. 16, 30, 32, is due to the resonabehavior of the heart motion and the scanner rotation: btake place at a frequency of 120 min21. At that frequency,the duration of the slow motion phase~20% of the cardiaccycle! is as low as 100 ms. The cardio interpolation is nable to improve its temporal resolution by collecting dafrom adjacent motion cycles since these are in resonawith the rotation angle. Therefore, CI’s temporal resolutiononly 250 ms~half of the rotation time! and motion will evenbe seen in the slow motion phase. The same behavioexpected for the 60 min21 case, which is also in resonancwith the scanner. Here, however, the slow motion phase l200 ms, which is only insignificantly lower than the 250 mtemporal resolution, and motion artifacts are not appar~see Fig. 10!. For the 80 min21 ~not shown! and 100 min21

reconstructions there is no resonance behavior and the arithm’s temporal resolution is high enough to freeze motiin the slow motion phase completely. One possibilityavoid resonance cases is adjusting the rotation time acc

e axial


FIG. 13. Cardiac circle scan~1/0.375 rotations! with M5256 slices at an increasing heart rate. EPBPCI reconstruction. Ticks indicate the location of thslices and the MPRs~0 HU/300 HU!.

inti

gohalu

truesx-y

inir

e

a

echo

on-ar-heameolu-

on-cts

ud-

caleenne

tan-o-

-ion:

di-theth

ing to the patient’s~expected! heart rate, as recommendedRefs. 16, 30, 32. Some manufacturers provide respecproduct implementations.

More significant advantages of EPBP over the other alrithms can be found in the sagittal and coronar MPRs tshow the 2 mm objects. Here, EPBPCI exhibits far less bring in thez direction than 180°MCI and ASSRCI.

B. Results for the sequence trajectory

We have used EPBPStd to perform sequence reconstions of the thorax phantom for two different pitch valuand for M ranging from 16 to 256. Figure 12 shows eamples with 32 and 256 slices. In all cases, image qualitsatisfactory. Images are slightly better for thep50.375 scanand artifacts in the MPR tend to increase for an increasnumber of slices. Compared to the corresponding spscans and EPBPStd reconstructions~e.g., thep50.375 re-construction of Fig. 7!, we find the sequence trajectory to bcompetetive regarding image quality.

In the sequence mode, EPBPCI combines allowed dsegments in the order of 1/p adjacent rotations~dependingon thez position! to become phase selective and to achievhigh temporal resolution. Since the sequence cardio smode seems to have no practical importance, we do not srespective results, however.


ve

-t

r-

c-

is

gal

ta

aanw

C. Results for the circular trajectory

Figures 13 and 14 show EPBPStd and EPBPCI recstructions for circle scans of the thorax phantom and of cdio motion phantom, respectively. Since for EPBPCI tnumber of rotations has been chosen to be 1/0.375 the samount of data overlap and hence the same temporal restion as in the spiral case can be expected.

From Fig. 14 we can see that the circle scan EPBP recstruction behaves very well regarding the motion artifaand the sharpness of the structures in thez direction. It is thistrajectory that has the potential to perform dynamic CT sties of the heart in the near future.

D. Patient data

To demonstrate EPBP’s capabilities of processing clinidata reconstructions of standard 12-slice CT data have bperformed. Figure 15 shows axial slices and MPRs of opatient reconstructed with ASSR and EPBP in both the sdard and cardio interpolation modes. The latter uses kymgram data for synchronization.33 To illustrate the phase selectivity, two phases have been chosen for reconstruct0% and 50% of K–K.

EPBP achieves very high image quality with these mecal data. The standard reconstructions are blurred due toheart motion. Apparently, ASSRStd suffers from smoo

on-th

tialtody

truc-

6:mens.BPis

andthe

r ef-ely,

al-ev-tructothction

vesrs.with

5 to

or

ttereso-edBPe tow-ofnoteadly

uc-sssly

tionthelies

ortsthatiralcts.uslynda-

mhi


FIG. 14. The 256-slice cardiac circle scans of the cardiac motion phantovarious heart rates. Reconstruction: EPBPCI at the slow, medium, andmotion phase~0 HU/500 HU!.


horizontal streaks that are not visible in the EPBPStd recstruction. The CI reconstructions are of high quality for boreconstruction algorithms. Due to EPBP’s higher sparesolution, image noise is slightly increased comparedASSR. This confirms our observations of the phantom stuand can be compensated for by choosing smother reconstion kernels.

Similar results are obtained with the patient in Fig. 1EPBP achieves excellent image quality that is of the saquality as today’s state-of-the-art cardiac reconstructioThe reader should note that this good performance of EPfor as few as 12 slices is far from evident. Since EPBPdesigned to be used with scanners with far more slicessince EPBP performs complicated oblique integrations ondetector area, one might expect discretization and bordefects to become dominating for small detectors. Fortunatthis is not the case.

VIII. DISCUSSION

The extended parallel backprojection is a generalizedgorithm that combines and improves the capabilities of seral reconstruction approaches. EPBP is able to reconsdata from circular, sequential, and spiral trajectories in bthe standard mode and the phase-correlated reconstrumode.

It has been shown that the proposed algorithm achievery high image quality for a huge range of parameteEPBP performance has been demonstrated for scannersas many as 256 slices, for pitch values ranging from 0.371.5, and for heart rates between 52 and 120 min21. Furtheron, EPBP copes well with data stemming from today’s 12-16-slice algorithms.

Our quantitative results indicate that EPBP makes beuse of the available data than ASSR: it achieves better rlution and lower image noise. This may be mainly attributto the fact that the degree of approximation used with EPis less than for ASSR. Thereby, EPBP is even comparablthe gold standard 180°MFI and 180°MCI, respectively. Hoever, the fact that EPBP’sQ happens to be larger than that180°MFI must be taken with care: the quality factor doestake into account the complete shape of the point sprfunction; it rather accounts for one figure of merit, namethe full width at half maximum.

As a general result, we further observed that reconstrtions of low pitch data result in better image quality and leartifacts than reconstructions of high pitch data. Obvioudue to the data redundancy~scan overlap! cone-beam arti-facts are somehow averaged and appear less dominant.

For cardiac reconstructions the same temporal resoluand the same motion artifact behavior as known fromolder algorithms has been observed with EPBP. This appeven for the cardiac circle scan where no profound repcan be found in the literature, up to now. We have seenthe cardiac circle scan is comparable to the cardiac spscan except for few cases with increased motion artifaOur statements are valid for as many as 256 simultaneoscanned slices and probably more. Here, the recomme

atgh


FIG. 15. Patient example, 52 min21 ~0 HU/1000 HU!.



FIG. 16. Patient example, 70 min21 ~0 HU/700 HU!.


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tion to use variable rotation times~as a function of the hearrate! can be put forward again. For the cardiac sequence sone might even control the interscan delay as a functionheart rate to optimize the position of the allowed data wdows.

Currently, the only drawback is the reduction of recostruction speed by a factor 2–5~depending on the pitchvalue! compared to thez-interpolation approaches. Since oimplementation is not yet optimized and since computatiospeed will increase according to Moore’s law we find threconstruction speed is not really an obstacle.

Altogether, our results indicate that EPBP is an imareconstruction algorithm that handles the current and fudemands of medical CT. It allows for arbitrary pitch spirand sequential scans with or without phase correlation.gardless of the cone-angle, EPBP ensures 100% dose uand shows only minor cone-beam artifacts. Of particularterest is the cardiac circle reconstruction that even allowsdynamic studies of the heart—especially when considethat commercial 64-slice CT systems have recently becoavailable.

a!Send correspondence and reprint requests to PD Dr. Marc KacheInstitute of Medical Physics~IMP!, University of Erlangen—Nu¨rnberg,Henkestraße 91, D-91052 Erlangen. Telephone: 49-9131-8522957;49-9131-8522824; electronic mail: [email protected]

1L. A. Feldkamp, L. C. Davis, and J. W. Kress, ‘‘Practical cone-bealgorithm,’’ J. Opt. Soc. Am.1, 612–619~1984!.

2X.-H. Yan and R. M. Leahy, ‘‘Derivation and analysis of a filtered bacprojection algorithm for cone beam projection data,’’ IEEE Trans. MImaging10, 462–472~1991!.

3X. Yan and R. M. Leahy, ‘‘Cone beam tomography with circular, ellipcal, and spiral orbits,’’ Phys. Med. Biol.37, 493–506~1992!.

4G. Wang, T.-H. Lin, P. Cheng, and D. M. Shinozaki, ‘‘A general conbeam reconstruction algorithm,’’ IEEE Trans. Med. Imaging12, 486–496~1993!.

5Th. Kohler, C. Bontus, M. Grass, and J. Timmer, ‘‘Artifact analysisapproximate helical cone-beam CT reconstruction algorithms,’’ MPhys.29, 51–64~2002!.

6H. Turbell, ‘‘Cone-beam reconstruction using filtered backprojection,’’‘‘Linko ping studies in Science and Technology~Dissertation No. 672!,’’Linkoping Universitet, 2001.

7M. Kachelrieß, S. Schaller, and W. A. Kalender, ‘‘Advanced single-slrebinning in cone-beam spiral CT,’’ Med. Phys.27, 754–772~2000!.

8H. Bruder, M. Kachelrieß, S. Schaller, K. Stierstorfer, and T. Flo‘‘Single-slice rebinning reconstruction in spiral cone-beam computedmography,’’ IEEE Trans. Med. Imaging19, 873–887~2000!.

9M. Kachelrieß, T. Fuchs, S. Schaller, and W. A. Kalender, ‘‘Advancsingle-slice rebinning for tilted spiral cone-beam CT,’’ Med. Phys.28,1033–1041~2001!.

10S. Schaller, K. Stierstorfer, H. Bruder, M. Kachelrieß, and T. Flo‘‘Novel approximate approach for high-quality image reconstructionhelical cone beam CT at arbitrary pitch,’’SPIE Medical Imaging Confer-ence Proc., Vol. 4322, pp. 113–127, 2001.

11K. Stierstorfer, T. Flohr, and H. Bruder, ‘‘Segmented multiple plareconstruction—a novel approximate reconstruction for multi-slice spCT,’’ Phys. Med. Biol.47, 2571–2851~2002!.


anf

-

-

lt

erele-age-rge

ß,

x:-

.

.

,-

,

l

12L. Chen, Y. Liang, and D. Heuscher, ‘‘General surface reconstructioncone-beam multislice spiral computed tomography,’’ Med. Phys.30,2804–2821~2003!.

13K. Sourbelle, ‘‘Performance evaluation of exact and approximate cobeam algorithms in spiral computed tomography,’’ inBerichte aus demInstitut fur Medizinische Physik~Shaker-Verlag, Berlin, 2002!.

14K. C. Tam, S. Samarasekera, and F. Sauer, ‘‘Exact cone-beam CT wspiral scan,’’ Phys. Med. Biol.43, 1015–1024~1998!.

15M. Kachelrieß, T. Fuchs, R. Lapp, D.-A. Sennst, S. Schaller, and WKalender, ‘‘Image to volume weighting generalized ASSR for arbitrapitch 3D and phase-correlated 4D spiral cone-beam CT reconstructiProceedings of the 6th International Meeting on Fully 3D Image Recstruction, November 2001, pp. 179–182.

16M. Kachelrieß and W. A. Kalender, ‘‘Electrocardiogram-correlated imareconstruction from subsecond spiral computed tomography scans oheart,’’ Med. Phys.25, 2417–2431~1998!.

17S. Ulzheimer, L. Muresan, M. Kachelrieß, W. Ro¨mer, S. Achenbach, andW. A. Kalender, ‘‘Considerations on the assessment of myocardial pesion with multi-slice spiral CT,’’ Radiology221, 458 ~2001!.

18M. Kachelrieß, ‘‘Phase-correlated dynamic CT,’’IEEE InternationalSymposium on Biomedical Imaging, 2004, pp. 616–619.

19M. Kachelrieß and W. A. Kalender, ‘‘Extended parallel backprojectioncardiac cone-beam CT for up to 128 slices,’’ Radiology225, 310 ~2002!.

20M. Kachelrieß and W. A. Kalender, ‘‘Extended parallel backprojecti~EPBP! for arbitrary cone angle and arbitrary pitch 3D and phascorrelated 4D CT reconstruction,’’Proceedings of the 7th InternationaMeeting on Fully 3D Image Reconstruction, June 2003, pp. Tu-Am2-3.

21H. Turbell and P. E. Danielsson, ‘‘An improved PI-method foreconstructing helical cone-beam projections,’’Conf. Rec. IEEE MIC,1999.

22H. Turbell, ‘‘Three-dimensional image reconstruction in circular and hlical computed tomography,’’ in ‘‘Linko¨ping Studies in Science and Technology ~Thesis No. 760!,’’ Linko ping Universitet, 1999.

23K. Taguchi, B. Chiang, and M. D. Silver, ‘‘New weighting scheme fcone-beam helical CT to reduce the image noise,’’ in Ref. 20,Mo-Am2-1.

24R. Manzke, M. Grass, T. Nielsen, G. Shechter, and D. Hawkes, ‘‘Adtive temporal resolution optimization in helical cardiac cone beamreconstruction,’’ Med. Phys.30, 3072–3080~2003!.

25M. Grass, R. Manzke, T. Nielsen, P. Koken, R. Proksa, M. Natanzon,G. Shechter, ‘‘Helical cardiac cone-beam reconstruction using retrostive ECG gating,’’ Phys. Med. Biol.48, 3069–3084~2003!.

26M. Grass, Th. Ko¨hler, and R. Proksa, ‘‘3D cone-beam CT reconstructifor circular trajectories,’’ Phys. Med. Biol.45, 329–347~2000!.

27Th. Kohler, R. Proksa, and M. Grass, ‘‘A fast and efficient methodsequential cone-beam tomography,’’ Med. Phys.28, 2318–2327~2001!.

28A. Katsevich, ‘‘Analysis of an exact inversion algorithm for spiral conbeam CT,’’ Phys. Med. Biol.47, 2583–2597~2002!.

29W. A. Kalender,Computed Tomography~Wiley, New York, 2004!.30M. Kachelrieß, S. Ulzheimer, and W. A. Kalender, ‘‘ECG-correlated im

age reconstruction from subsecond multi-slice spiral CT scans ofheart,’’ Med. Phys.27, 1881–1902~2000!.

31R. A. Brooks and G. DiChiro, ‘‘Statistical limitations in X-ray reconstruction tomography,’’ Med. Phys.3, 237–240~1976!.

32M. Kachelrieß, S. Ulzheimer, and W. A. Kalender, ‘‘ECG-correlated imaging of the heart with subsecond multislice CT,’’ IEEE Trans. MeImaging19, 888–901~2000!.

33M. Kachelrieß, D.-A. Sennst, W. Maxlmoser, and W. A. Kalender, ‘‘Kmogram detection and kymogram-correlated image reconstruction fsubsecond spiral computed tomography scans of the heart,’’ Med. P29, 1489–1503~2002!.

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