Flying Focal Spot (FFS) in Cone-Beam CT Marc Kachelrieß, Member, IEEE, Michael Knaup, Christian Penßel, and Willi A. Kalender

1238 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 3, JUNE 2006 Flying Focal Spot (FFS) in Cone-Beam CT Marc Kachelrieß, Member, IEEE, Michael Knaup, Christian Penßel, and Willi A. Kalender Abstract In the beginning of 2004 medical spiral CT scanners that acquire up to 64 slices simultaneously became available. Most manufacturers use a straightforward acquisition principle, namely an x ray focus rotating on a circular path and an opposing cylindrical detector whose rotational center coincides with the x-ray focus. The 64-slice scanner available to us, a Somatom Sensation 64 spiral cone-beam CT scanner (Siemens, Medical Solutions, Forchheim, Germany), makes use of a flying focal spot (FFS) that allows for view-by-view deflections of the focal spot in the rotation direction ( FFS) and in the -direction ( FFS) with the goal of reducing aliasing artifacts. The FFS feature doubles the sampling density in the radial direction (channel direction, FFS) and in the longitudinal direction (detector row direction or -direction, FFS). The cost of increased radial and azimuthal sampling is a two- or four-fold reduction of azimuthal sampling (angular sampling). To compensate for the potential reduction of azimuthal sampling the scanner simply increases the number of detector read-outs (readings) per rotation by a factor two or four. Then, up to four detector readings contribute to what we define as one view or one projection. A significant reduction of in-plane aliasing and of aliasing in the -direction can be expected. Especially the latter is of importance to spiral CT scans where aliasing is known to produce so-called windmill artifacts. We have derived and analyzed the optimal focal spot deflection values and as they would ideally occur in our scanner. Based upon these we show how image reconstruction can be performed in general. A simulation study showing reconstructions of mathematical phantoms further provides evidence that image quality can be significantly improved with the FFS. Aliasing artifacts, that manifest as streaks emerging from high-contrast objects, and windmill artifacts are reduced by almost an order of magnitude with the FFS compared to a simulation without FFS. Patient images acquired with our 64-slice cone-beam CT scanner support these results. Index Terms Computed tomography, cone-beam CT, image quality, image reconstruction, spiral-ct. I. INTRODUCTION CORRECT sampling requires to satisfy the Nyquist condition: at least two sample points should be taken per full width at half maximum (FWHM) of the detector point spread function (PSF). 1 In many cases this situation is not easy to achieve. In CT the spacing of the detector samples is slightly larger than the active width of the detector pixels far from sampling the active detector area twice. One workaround is the quarter Manuscript received November 9, 2004; revised March 11, 2006. The authors are with Institute of Medical Physics (IMP), University of Erlangen-Nürnberg, D-91052 Erlangen, Germany (e-mail: marc.kachelriess@imp.uni-erlangen.de). Digital Object Identifier 10.1109/TNS.2006.874076 1 Actually, the sample spacing must be 1=2b or less in order to allow to exactly recover a band-limited function whose support in frequency domain lies in [0b; b]. In real systems that rarely deal with band-limited but rather with essentially band-limited functions one can only recover approximations to the true function. In our case a good approximation to the sampling theorem is to acquire at least two samples per spatial resolution element. detector offset. If no special care is taken and the detector is aligned symmetrically with respect to the central ray a 360 scan will measure each ray twice with respect to its radial position (i.e. with its distance to the origin). In that case the data are redundant. Shifting the detector array (channel direction) by one quarter of the detector sampling distance pays out since opposing rays interlace and, by combining opposing views, one effectively doubles the sampling [1], [2]. However, for cone-beam scans and for spiral scans this kind of data redundancy is not really available since opposing rays do not exist; they rather differ by their tilt-angle with respect to the rotation axis and by their -position. Further, the quarter shift does not improve the sampling in the detector row direction (longitudinal or -direction). An alternative solution is deflecting the focal spot between adjacent detector read-outs (readings) as it is done in our tube (Fig. 1) [3]. This flying focal spot (FFS) can be used to double the sampling density in both directions regardless of the coneangle and the spiral trajectory [4]. Recently, a manufacturer-dependent evaluation of the image quality achieved with the flying focal spot was presented by the engineers [5]. The present work, that was carried out in early 2004 and submitted, is independent thereof insofar as all evaluations, implementations and simulations were performed without knowing manufacturer details. Of course, the actual scan data that we use to demonstrate the potential of the FFS was acquired with the scanner provided by the manufacturer. Aliasing artifacts due to inadequate longitudinal sampling manifest themselves as so-called windmill artifacts [6]. These are streaks emerging from high-contrast objects in a star-like pattern. The streaks are caused by interpolation errors in the longitudinal direction (between adjacent detector rows or slices): every time the interpolation partners change, say from the pair of slices and to the pair of slices and, a new windmill segment is introduced. Thereby, the number of streaks is a function of the number of detector rows that are simultaneously read out during the spiral CT scan, and a function of the spiral table increment per rotation. In contrast to alternative attempts to suppress these spiral artifacts, such as the use of preferred spiral pitch values, conjugate cone-beam backprojection algorithms, or iterative image reconstruction [7], [8], where image quality improvements are achieved only close to the center of rotation the flying focal spot technique realized in our scanner provides true high density sampling and is neither restricted to special pitch values nor to certain image regions. This paper provides an analysis of the flying focal spot geometry and of the achievable image quality. We specify and optimize the FFS deflection values, we clarify the geometry, we introduce the subfan approximation and we demonstrate the effect of the FFS in a simulation study and in phantom and patient measurements. Our focus lies on the removal of spiral windmill artifacts. Additionally, noise and spatial resolution are studied. 0018-9499/$20.00 2006 IEEE

KACHELRIEß et al.: FLYING FOCAL SPOT (FFS) IN CONE-BEAM CT 1239 Fig. 1. A rotating envelope tube with focal spot deflection capability. Due to the anode angle the periodic motion of the electron beam, whose position is controlled by the magnetic field B, translates into a focal spot deflection 6@z in the z-direction and in a change 6@R of the focal spot s distance to the isocenter. II. SOURCE AND DETECTOR GEOMETRY To define the focal spot positions we avoid introducing a local tube coordinate system. We rather use the fact that the vertex positions can be described by small corrections in the view angle and in the -position of a circular or spiral scan. With being the distance of the undeflected focal spot to the isocenter, being the table increment per rotation,, and being the view angle let us define the spiral source trajectory as the scanner s rotation axis a focal spot deflection in the -direction will effectively change by where is the anode angle (in our case 7 ). The focal spot s radial motion is significant and this side-effect must be accounted for during image reconstruction. Note that the variation in due to is of second order in and therefore negligible. The detector is a cylindrical detector whose symmetry axis coincides with. We parameterize it as Circular scans are a special case of spiral scans and can be obtained by setting. The vector is the source trajectory vector of a spiral CT scanner without FFS. Note that the detector is centered about while the focal spot can be electromagnetically deflected to the positions. The new parameters and are the focal spot deflection angle and deflection length that are used to improve the in-plane and the axial sampling properties of CT. These deflection parameters are of very small magnitude; using the small angle approximation (first order Taylor series in the two deflection parameters) is justified and will be implicitly used below when deriving the ideal values for and. The remaining parameter is not independent (see Fig. 1). It accounts for small variations in the radius of the deflected focal spot to the isocenter. Since the rotation axis of the x-ray tube is parallel to where is the distance focal spot to detector, is the angle within the fan and is the detector s longitudinal coordinate. We also define the distance isocenter to detector such that holds. Note that the flying focal spot deflection parameters do not enter the definition of the detector coordinates. To treat sampling we must go into discretization. We assume equidistant sampling in all three coordinates and let, and denote the sample spacing. The angle is the angular increment of the gantry between two read-outs. The angle is the transaxial angle between two detector elements as seen from ; the actual distance between two detector elements (in the transaxial direction) is then given by. The sampling distance is the longitudinal distance between two detector rows. It is related to the nominal slice thickness by scaling to the isocenter:. The samples are located at for the gantry angle, at for the angle within the fan and at for the longitudinal detector coordinate. Thereby and denote the starting values and and count the readings, channels and detector rows, respectively.

1240 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 3, JUNE 2006 Fig. 2. Geometry of the FFS deflection (gantry rotation not shown). Physically, the detectors are spaced by R 1 ^ which makes R 1 ^ at the isocenter. To halve this sampling distance the focal spot deflection 2R @ between two adjacent detector read-outs must be chosen such that it corresponds to (1)=(2)R 1 ^ at the isocenter. III. FLYING FOCAL SPOT DEFLECTION Now, the optimal focal spot deflection parameters are derived. To do so, we note that both effects can be separated and consider the FFS first, and then the FFS. Also note that azimuthal sampling (number of acquired angular positions during one rotation) itself is not an issue since it can be, and actually is, kept constant by simply increasing the number of detector read-outs per rotation by a factor of 2 if either the FFS or the FFS are used or by a factor of 4 if both FFS are active. A. In the fan or channel direction the CT detector elements are arranged on a circle of radius centered at. Their angular spacing yields a physical distance of. In the isocenter this corresponds to a sampling distance of. The aim of the FFS is to reduce this sampling distance by a factor of two. This is achieved by deflecting the focal spot with a deflection parameter of alternating sign such that between two focal spot positions yields 0.85 mm approximately. Be aware of the fact that the FFS improves the radial sampling which means that only the distance of the rays to the origin has to be considered and not their angle (which would be the azimuthal sampling). This is the reason why Fig. 2 does not show the angular increment of the detector between the two readings acquired. B. FFS The longitudinal sampling distance ( -direction) is given by ; scaled to the isocenter this gives, the value is also called the nominal slice thickness (although it rather corresponds to the longitudinal sampling distance in multi-slice CT). The FFS aims at doubling the longitudinal sampling density. This is achieved by deflecting the focal spot with a deflection parameter of alternating sign such that is the focal spot position of reading. The situation is illustrated in Fig. 2. The alternating sign yields a total deflection difference of which corresponds to a spacing of of the two focal spot positions. After scaling to the isocenter this becomes and should equal (half the sampling distance of the original fan). This is the case when one chooses is the focal spot position of reading. The situation is illustrated in Fig. 3. The rays shall be longitudinally separated by when intersecting the isocenter. Neglecting the table increment per reading which is about three orders of magnitude smaller than we find by simple geometric scaling that should be just the same distance which means The numerical value given applies to our scanner where mm, mm. The distance The numerical value applies to our scanner where mm, mm, mm. Obviously, both focal spot positions are separated by about 0.66 mm in the -direction. Due

KACHELRIEß et al.: FLYING FOCAL SPOT (FFS) IN CONE-BEAM CT 1241 Fig. 3. Geometry of the zffs deflection. to the anode angle we get mm which means that the radial separation of both focal spots is 5.4 mm. C. Both FFSs If both the FFS and the FFS are switched on simultaneously the deflection values and remain the same as derived in the previous sections. However, the focal spot itself will jump in a rectangular fashion relative to the anode The symbols indicate that the sign of the deflection values depends on. The order in which the focal spot positions are visited depends on the implementation; for our scanner applies. D. View Definition The question arises what to define as a projection (or view). Up to here, we have learned to know the readings which comprise a set of simultaneous detector read-outs. Without FFS one reading makes up what is usually called a projection. With the FFS switched on up to four readings collect new sample positions. Therefore we define the view or projection to be the collection of adjacent readings that make up a total FFS cycle. Thus, the view consists of two readings if either the FFS or the FFS is switched on and it consists of four readings if both flying focal spots are used. IV. RECONSTRUCTION Three types of image reconstruction algorithms can be found in today s 16- and 64-slice clinical CT scanners: -interpolation, single-slice rebinning and Feldkamp-type image reconstruction. To reconstruct stationary objects (standard reconstruction) manufacturers use Feldkamp-based algorithms [9] [14] or they use so-called single-slice rebinning algorithms such as the advanced single-slice rebinning algorithm (ASSR) [15] [20]. The algorithms can be grouped into two categories. Either they are based on some fan-beam filtered backprojection or they perform a rebinning of the data to fan-parallel or parallel geometry. According to this classification image reconstruction from FFS data can either use the subfan approximation or rebinning. A. Subfan Approximation To be able to perform fan-beam filtered backprojection it is necessary to bring the data into a format that allows for shift-invariant filtering; only a certain class of geometries, including the fan-beam, allows to do so [21]. In this paper s context this means that the complicated geometry that results from the focal spot deflection must be approximated by true fan beams where the focal spot lies in the center of the detector arc. 1) -Subfan Approximation: Since the -focal spot deflection is only a tiny angle one can approximate each of the two deflected subfans by an ideal subfan whose focal spot coincides with the rotational center of the detector cylinder. Thus, we try to replace the deflected system by the ideal fan-beam system-that is a system where the focal spot is fixed and centered relative to the detector arc Regarding Fig. 2 and using the small angle approximation we find that yields the desired compensation, and

1242 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 3, JUNE 2006 Fig. 4. The penumbra cannot be used for imaging. Therefore the pre-patient collimators must open wide enough. Two dose profiles offset by a distance O result when using the zffs. The offset O and the penumbra width P are geometrically related to R 1, to@z, to the focal spot size F, and to the collimator distance D. results. If the FFS is not used implies. Note that the minus sign is introduced since a clockwise deflection of the focal spot can be approximately compensated by a counterclockwise deflection of the detector channels. 2) -Subfan Approximation: For the FFS a similar approximation can be established. Instead of assuming a shifted focal spot during reconstruction one may rather assume a shifted -axis, i.e. a detector array that is shifted by. The deflected system shall be replaced with the ideal system Regarding Fig. 3 we find that desired compensation and yields the results. If the FFS is not used implies. The minus sign shows that a forward deflection of the focal spot can be approximately compensated by a backward deflection of the detector rows. B. Rebinning An alternative to using the subfan approximation is rebinning. This means that the data are resampled to correspond to some more convenient geometry. During the resampling step one can exactly account for the geometry of each individual ray and no approximation is required. Note that rebinning always involves some kind of interpolation along the ray coordinates, and, however. In many cases rebinning to parallel beam geometry is performed, including the Feldkamp-type extended parallel backprojection (EPBP, see [22]) that is used in this paper. It can be split into azimuthal, longitudinal and radial rebinning. Azimuthal rebinning means resorting the rays into non-equidistant parallel geometry (fan-parallel geometry) by performing a resampling in the -direction. Radial rebinning is the step to convert the fan-parallel data to parallel data; the latter are characterized by equidistant parallel rays. During azimuthal and radial rebinning one can exactly account for the FFS geometry. Many popular reconstruction algorithms, such as spiral -interpolation (single-slice CT), spiral -filtering (up to 4-slice CT), ASSR-type algorithms and EPBP (cone-beam spiral CT) perform a resampling of the data where only the longitudinal coordinate is involved. Since this means taking into account the focal spot s -position longitudinal rebinning is the step to account for. V. DOSE ISSUES The FFS requires to open the pre-patient collimator a bit wider than it is necessary without a flying focal spot: the collimator s mechanical inertia prohibits to follow the movement of the flying focal spot with such high frequency (in the order of 10 khz). Basic geometric considerations help to get an idea of the magnitude of this additionally required collimator opening that directly converts to a patient dose increase. A schematic illustration is given in Fig. 4. It basically shows the trapezoidal dose profile at the axis of rotation. The area of the dose profile that can be used for imaging is the collimated slices area. Other parts of the dose profile, such as the penumbra and the offset, are relevant for the patient dose but irrelevant for imaging.

KACHELRIEß et al.: FLYING FOCAL SPOT (FFS) IN CONE-BEAM CT 1243 Fig. 5. Point spread functions and slice sensitivity profiles obtained from the reconstructions of the simulated delta peak. Without zffs one obtains the expected broadening of the SSP due to linear interpolation between adjacent detector rows. The zffs improves sampling and the effective slice width (FWHM) equals the nominal slice width. The same behavior is observed in-plane where the radial point spread function is plotted. (a) Slice sensitivity profiles SSP(z) with and without zffs. (b) Point spread functions PSF(r) with and without FFS. Thus, the patient dose is proportional to which is a factor of compared to an ideal scanner. For scanners without FFS there is only one trapezoidal dose profile and the offset would be zero which means a dose of. Evidently, the dose ratio of scans with FFS and those without is given by. Assuming a focal spot size mm and a collimator distance mm and our scanner s mm collimation ( mm) this evaluates to about 1.06. This implies a dose increase of 6% with the FFS. VI. SIMULATIONS For our simulation we use the Sensation 64 geometry with a slice thickness of mm. The number of readings per rotation is set to, the number of detector channels to and the number of detector rows to. Further, mm and mm. The resulting fan angle and cone angle yield and, respectively. A spiral pitch of, that corresponds to a table increment of 25 mm per rotation, was simulated. To mimic a finite detector area, focal spot area and integration time we used a 3 3 subsampling for the detector and the focal spot and a three-fold subsampling for the view angle. Altogether 243 needle beams were computed and averaged to obtain one measured value. A monochromatic source was assumed to avoid beam hardening effects. The simulation software used was ImpactSim (www.vamp-gmbh.de).

1244 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 3, JUNE 2006 Fig. 6. With double z-sampling only half of the quanta contribute to each detector reading when a constant total dose is assumed. Consequently, image noise (standard deviation) will increase by p 2. TABLE I FFS COMBINATIONS To assess spatial resolution a delta object was simulated and its response was evaluated by computing radial and longitudinal profiles. To study the artifact behavior the FORBILD head phantom (www.imp.uni-erlangen.de/phantoms.htm) was simulated. The FFS combinations, among which only the last two are available at our scanner were simulated, are shown in shown in Table1. Here is the number of projections per rotation, which is the number of independent azimuthal samples per rotation. Each projection is made up of 1, 2, or 4 of the readings depending on whether no FFS, either FFS, or both FFS are used. denotes the number of independent radial samples per projection and it is equal to the number of channels when the FFS is inactive or it equals with the FFS switched on. Similarly, holds the number of independent longitudinal samples and is also known as the number of simultaneously acquired slices. equals the number of active detector rows with the FFS being switched off and whenever the FFS is used. Image reconstruction of simulated and measured data was performed with the Feldkamp-type extended parallel backprojection (EPBP) algorithm [22]. VII. RESULTS Fig. 5 shows the axial and longitudinal profiles through the delta object. Apparently, the double - and the double -sampling improve spatial resolution by about 20 to 30% each. It should be noted that these improvements have been achieved without modifying the kind of interpolation used: in all cases standard linear interpolation was applied. However, implicitly the FWHM of the triangle function that represents linear interpolation shrunk from 0.6 mm (standard sampling) to 0.3 mm (double sampling) when switching from standard to FFS reconstruction. We also evaluated image noise and found a two-fold increase of noise variance with increasing spatial resolution. Quantitatively, the increased noise corresponds to the theoretically expected cost for the increased resolution. This is illustrated in Fig. 6. It compares a standard scanner with single -sampling to a FFS scanner with double -sampling. At constant patient dose (constant number of quanta) each detector receives only half of the number of quanta when using double sampling compared to the standard scanner. Since the longitudinal interpolation algorithm is a linear interpolation between adjacent samples in both cases one finds the said two-fold increase in noise variance or a increase in the noise standard deviation. The figure further shows a similar consideration for a hypothetical high-resolution scanner where the same 0.3 mm sampling has been achieved by halving the detector aperture and spacing. This scanner could yield a higher spatial resolution than the FFS scanner while keeping the same image noise level. On the other hand one could use this scanner together with some smoother -interpolation algorithm to obtain the same SSP as with the FFS scanner. In that case image noise would be significantly lower which implies a better dose usage with the high-resolution scanner. Note that the same point spread function, which implies the same spatial resolution, and the same image noise as with FFS

KACHELRIEß et al.: FLYING FOCAL SPOT (FFS) IN CONE-BEAM CT 1245 can be achieved from a standard scan: just replace the linear in by a conterpolation which is a convolution with, where denotes a volution with a slim triangle triangle function of maximum and full width at half maximum 1. What is the FFS for, if we can obtain the same PSF or SSP from a standard scan with a slim interpolation function? It is the sampling which makes the difference. A slim -filter applied to a standard scan would amplify the windmill artifacts whereas using double -sampling significantly attenuates these artifacts. The main reason why the manufacturers introduced double sampling techniques was the expected reduction of aliasing artifacts. This reduction is clearly visible in the reconstructions of our simulated head phantom that are shown in Fig. 7. Streaks emerging from high contrast objects are strongly suppressed by the Nyquist-conform double sampling. The figure also shows that in-plane aliasing is not influenced by the FFS. Windmill artifacts are present in real patient data, too. They become especially annoying when slicing through a complete volume since the windmill-type streaks that emerge from sharp edges seem to rotate about their origin when switching from one image to the next (hence the name windmill). However, the windmill artifacts without FFS and their suppression with FFS becomes less impressive in static images such as the slice presented in Fig. 8. Nevertheless, we can clearly see low amplitude and low frequency streaks emerging from the high contrast bone. Sliding through adjacent images would show these streaks rotating around their origin with increasing or decreasing -position. An electronic version of this comparison with the possibility to zoom, pan, slide and window is provided at www.imp. uni-erlangen.de/zffs. The Sensation 64 scanner does not allow to switch off the double -sampling feature. Therefore our reconstruction software was modified to simply leave out every other reading to mimic a standard scan and we did the full reconstruction including all FFS position to obtain FFS reconstructions. The alert reader might thus have observed that, due to our manipulation, image noise is the same with and without FFS in Fig. 8. instead of showing the expected factor VIII. SUMMARY AND DISCUSSION We have given a theoretical treatment of the focal spot deflection technique of our cone-beam spiral CT scanner. The flying focal spot allows to double the sampling in the axial and in the longitudinal direction. The corresponding equations of optimized focal spot placement have been derived. For reconstruction based on rebinning to parallel data this is the case for the ASSR and for the EPBP cone beam algorithms that we prefer to use the exact equations can be conveniently used. If fan-beam filtered backprojection techniques shall be used for image reconstruction subfan approximation is the method of choice. It has been shown that dose due to penumbra effects may insignificantly increase when using the double -sampling. Since our scanner provides only the 2 32 0.6 mm collimation where mm scan mode without the FFS is switched on but no FFS we cannot provide dose measurements that support this result. Our results mainly concentrate upon the new FFS feature since the FFS is available in Somatom scanners for more than Fig. 7. Reconstructions of the FORBILD head phantom. Aliasing artifacts apparent on the left side are greatly reduced with the z FFS. An exception is the second row of images that also shows significant in-plane aliasing due to the periodic high-contrast bubbles in the phantom s inner ear. These in-plane aliasing artifacts are not influenced by the double z -sampling. The increase in noise by a factor of 2 in the right column is due to the higher z -sharpness of the z FFS-reconstruction (50=90). p one decade. Basically, the results show that both, the SSP and the in-plane PSF benefit from the improved sampling assuming that the interpolation between adjacent samples, linear interpolation in our case, is scaled with the sample spacing. Then, when switching from single noise increases by a factor of

1246 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 3, JUNE 2006 The most significant improvement obtained with the FFS is the removal of windmill artifacts that have been present for years in multi-slice spiral CT. These streaks that emerge from high-density objects and appear to be rotating when scrolling through the volume (see www.imp.uni-erlangen.de/zffs) are due to insufficient longitudinal sampling. The reduction of the windmill artifacts might have strong clinical impact since it now allows to go down to the finest effective slice thicknesses without impairing image quality. REFERENCES Fig. 8. Head patient, spiral acquisition with p = 1 and 2 1 32 2 0:6 mm collimation. A significant reduction of windmill artifacts present in the standard scanner (left) is achieved when utilizing a zffs (right). The fact that noise is equivalent in both images is due to skipping every other reading for the standard reconstruction. 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