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Local and global 3D noise power spectrum in cone-beam CT system

with FDK reconstruction

Jongduk Baeka?

Department of Radiology, Stanford University, Stanford, California 94305

Norbert J. Pelc

Department of Radiology, Stanford University, Stanford, California 94305; Department of Electrical

Engineering, Stanford University, Stanford, California 94305; and Department of Bioengineering,

Stanford University, Stanford, California 94305

?Received 5 July 2010; revised 24 January 2011; accepted for publication 29 January 2011;

published 23 March 2011?

Purpose: The authors examine the nonstationary noise behavior of a cone-beam CT system with

FDK reconstruction.

Methods: To investigate the nonstationary noise behavior, an analytical expression for the NPS of

local volumes and an entire volume was derived and quantitatively compared to the NPS estimated

from experimental air and water images.

Results: The NPS of local volumes at different locations along the z-axis showed radial symmetry

in the fx-fyplane and different missing cone regions in the fzdirection depending on the tilt angle

of rays through the local volumes. For local volumes away from the z-axis, the NPS of air and

water images showed sharp transitions in the fx-fyand fy-fzplanes and lack of radial symmetry in

the fx-fyplane. These effects are mainly caused by varying magnification and different noise levels

from view to view. In the NPS of the entire volume, the fx-fyplane showed radial symmetry

because the nonstationary noise behaviors of local volumes were averaged out. The nonstationary

sharp transitions were manifested as a high-frequency roll-off.

Conclusions: The results from noise power analysis for local volumes and an entire volume dem-

onstrate the spatially varying noise behavior in the reconstructed cone-beam CT images. © 2011

American Association of Physicists in Medicine. ?DOI: 10.1118/1.3556590?

Key words: cone-beam CT, NPS, FDK, nonstationary noise

I. INTRODUCTION

In any medical imaging system, objective metrics of image

noise are needed to assess the system performance. To de-

scribe the noise in reconstructed CT images, the image vari-

ance has been studied for parallel-beam,1–4fan-beam,5–9and

cone-beam CT systems.10,11While the image variance pro-

vides information on the noise magnitude, it cannot charac-

terize the noise correlation introduced by the reconstruction

algorithm and detector blurring and therefore the perfor-

mance in object or feature detection tasks cannot be fully

described by the variance alone.12The most complete char-

acterization of noise is to describe for each pixel, the vari-

ance and the covariance with all other pixels.13This may be

difficult to accomplish reliably in experiments. When the

noise behavior is the same for pixels in a region, an average

characterization provides a more stable noise description.

The noise power spectrum ?NPS? does this and provides in-

formation on the noise components for different spatial fre-

quencies. It can be used to predict detection performance14,15

and the system detective-quantum-efficiency.16–19In addi-

tion, the noise variance, the detectability ?SNR?, and the cor-

relation in the noise can be determined from the NPS.20,21

Some previous researches used a cascaded system analysis to

derive the NPS and its dependence on the signal detection

and processing chain for a single region at the center of the

field of view.22,23In this paper, we instead focus on the spa-

tially variant behavior starting with the noise in the measured

projection data.

Much work has been done to study the propagation of

noise in the raw data into the NPS of different CT systems.

For parallel-beam CT, the NPS was analytically derived and

shown to be dependent on the type of reconstruction filter.24

The NPS for a discrete reconstruction was also studied25,26

and the effects of the discrete sampling within the projection,

angular sampling, interpolation, and noise aliasing caused by

pixel sampling were considered. Recently, an analytical for-

mula of the NPS for direct fan-beam CT reconstruction was

derived and its spatially varying noise behavior was

investigated.27Location dependent noise was investigated

using 2D NPS and covariance matrix.28

In this paper, we examine the nonstationary noise behav-

ior of cone-beam CT systems with FDK reconstruction.29We

first present an analytical derivation of the propagations of

noise in the projections into the NPS and then investigate the

nonstationary behavior using NPS of local volumes. We

study the NPS of local volumes from reconstructed air and

water images and examine the effects of spatially varying

magnification, cosine weighting, backprojection weighting,

21222122Med. Phys. 38 „4…, April 20110094-2405/2011/38„4…/2122/10/$30.00© 2011 Am. Assoc. Phys. Med.

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noise statistics, and detector noise apodization. In addition,

the NPS of the entire volume is estimated. In all cases, we

quantitatively compare the experimentally estimated NPS to

the analytical prediction.

II. METHOD

II.A. Analytical derivation of the 3D NPS

We assume that noise in the raw data contain only quan-

tum noise that is additive20,21and follows a Poisson

distribution30and we allow for correlation due to cross-talk

across detector cells. The detector signal is normalized by the

incident intensity and the negative logarithm is taken to yield

projection measurements. Extending the derivation in Ref.

27 to FDK reconstruction, the NPS of a cosine weighted

projection is

Sj?fu,fv? =1

kj?

l=1

kjwlj

Nlj

2

?D?fu,fv??2

for

?fu? ? fuc

and

?fv? ? fvc,

wlj= cos??lj?cos??lj?,

?1?

where kjis the number of detector cells contributing to the

volume of interest in the jth view and each of these, indexed

by the letter l; u and v are 2D spatial coordinates in the

detector plane in the transverse and longitudinal direction,

respectively; fuand fvare the corresponding spatial frequen-

cies; wljand Nljare the cosine weighting factor and number

of detected photons of the lth detector cell in the jth view; fuc

and fvcare the Nyquist frequencies of the projection data; ?lj

and ?ljare the view dependent fan angle and tilt angle for the

lth detector cell in the jth view ?corresponding to the recon-

struction point shown in Fig. 1?; and D?fu,fv? is the fre-

quency response of the detector noise apodization ?due to

cross-talk?. Since projection measurements are unitless, the

NPS of a single view has units of distance squared ?i.e.,

cm2?.22

In FDK reconstruction, the reconstruction filter ?e.g.,

apodized ramp filter? is applied only in the transverse ?fu?

direction. Since filtering the projections multiplies the power

spectrum by the squared magnitude of the filter’s frequency

response,31the NPS of a filtered projection ?indicated by the

subscript “fp”) in the jth view is

Sfp,j?fu,fv? =1

kj?

l=1

kjwlj

Nlj

2

?fu?2?D?fu,fv??2?H?fu,fv??2

for

?fu? ? fuc

and

?fv? ? fvc,

?2?

where ?fu?H?fu,fv? is the apodized ramp filter used in the

reconstruction. Note that Eqs. ?1? and ?2? are valid for the

local and global projection NPS ?see the derivation in Ref.

27?, where the set of detector cells “l” and the number of

cells in that set “kl” need to be changed to reflect the volume

being considered.

Since the noise in the FDK reconstruction is nonstationary

across the image, we investigate the noise behavior by sub-

dividing the reconstructed entire volume I into n small sub-

volumes

FIG. 1. Three parameters characterizing the lth ray in the jth view. View

angle ?jwith respect to the x-axis, view dependent fan angle ?ljwith respect

to the central ray, and tilt angle ?ljwith respect to the plane of the x-ray

source.

FIG. 2. NPS of small volumes. ?a? Two small volumes along the z-axis. The

contribution of the 2D Fourier plane and the corresponding 3D NPS for ?b?

an off-centered volume ?zero fan angle and constant tilt angle ? over all

views? and ?c? an isocentered volume ?zero fan angle and tilt angle over all

views?.

2123J. Baek and N. J. Pelc: 3D noise power spectrum with FDK reconstruction2123

Medical Physics, Vol. 38, No. 4, April 2011

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I = i1+ i2+ , ¯ ,+ in,

?3?

where each icis a masked subvolume from I. We assume that

the mask is small enough to ensure the noise within it is

approximately stationary. At the same time, the subvolume

has to be large enough to contain the vast majority of the

correlation for pixels within the subvolume. For regions sub-

stantially larger than the central lobe of the reconstruction

kernel, this condition will be obeyed except for the pixels

near the edges of the region. This is discussed further below.

Also, it should be noted that the frequency resolution of the

NPS is inversely related to the width of the region used to

compute it. We next derive the NPS for a small volume.

For a small volume ic, the cone-beam rays can be approxi-

mated as parallel rays and therefore the central-slice theorem

holds locally.32During the backprojection, each view con-

tributes a 2D Fourier plane to the 3D NPS, but as was shown

in Ref. 27 for fan-beam reconstruction, the bandwidth and

amplitude of the 2D Fourier plane changes according to the

view dependent magnification and backprojection weighting.

Thus, the contribution onto the 2D Fourier plane from the jth

view is

Sfp,j?fu,fv? →

Sfp,j?fu/aic,j,fv/aicj?

aic,j

4uic,j

4

,

?4?

where aic,jand uic,jare the magnification factor and back-

projection weighting for the small volume icin the jth view.

In addition, the 2D Fourier plane is rotated and tilted due to

the view dependent fan angle and tilt angle.

For example, if we consider the two small volumes along

the z-axis in Fig. 2?a?, the contribution from all views is

similar except for the effect of view dependent intensity

variations and tilt angle. For the off-centered volume, a

larger angle with respect to the x-y plane tilts the 2D Fourier

plane which, when rotated for all views, produces a symmet-

ric missing cone region in the 3D NPS ?Fig. 2?b??. For the

isocentered volume, the 3D NPS does not have a missing

cone region because of its very small tilt angle ?Fig. 2?c??.

Note that the tilt angle is constant and fan angle is zero for

small volumes along the z-axis.

The 3D NPS of the small volume in Fig. 3, which is offset

in all three directions, has a different behavior. When the

source is located at the 0° view direction, shown in Fig. 3?a?,

the 2D Fourier plane is stretched ?that is, has a larger band-

width? because of the higher magnification. The Fourier

plane is also rotated by the view dependent fan angle and

tilted by the larger tilt angle. However, when the source is

located at 180°, shown in Fig. 3?b?, the 2D Fourier plane is

squeezed ?that is, has a smaller bandwidth? because of the

lower magnification, is rotated by the view dependent fan

angle, and is tilted by the smaller tilt angle. Considering all

views, the 3D NPS has an asymmetric missing cone region.

In addition, backprojection weighting and cosine weighting

are different from view to view. Since the contribution of 2D

FIG. 3. Description of the view dependent fan angle ? and tilt angle ? and the corresponding 2D Fourier planes at ?a? 0° and ?b? 180° view direction for a

small volume.

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Medical Physics, Vol. 38, No. 4, April 2011

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Fourier planes onto the 3D NPS are additive over all views

and the rotation in image space results in an equal rotation in

Fourier space,333D NPS for a small volume iccan be ex-

pressed as

Sic?fx,fy,fz? =??

j=1

??

aic,j

m?

2?

m

R

Sfp,j?fu/aic,j,fv/aicj?

4uic,j

4

??fx?,?j+ ?ic,j,?ic,j?,

?5?

where R?h,?,?? is a counterclockwise rotation operator act-

ing on h by angle ? about the fzaxis and ? about the axis

perpendicular to the source-isocenter direction; ?ic,jand ?ic,j

are the tilt angle and fan angle in the jth view; ?jis the view

angle shown in Fig. 1; and m is the number of views equally

spaced over 2?. Note that aic,j, and uic,jare constant over all

views for a small volume centered along the z-axis, but vary

from view to view for off-centered small volumes. Since our

reconstruction produced images of the linear attenuation co-

efficient, 3D NPS has units of distance ?i.e., cm?.22

As described above, a cone-beam system has a different

3D NPS at different locations due to the nonstationary noise

behavior and therefore the best way to characterize the image

noise is to use 3D NPS of small volumes. However, one may

wish to compute the 3D NPS of a large volume which char-

acterizes the average noise behavior over the whole volume

and previous investigators have done this.34If we assume the

cross-correlation between neighboring small volumes is neg-

ligible, the NPS of the reconstructed entire volume is the

sum of the NPS over n small volumes

SI=?

c=1

n

Sic,

?6?

although one must be cognizant that due to nonstationarity,

the resulting NPS may not be valid throughout the volume.

Of course, the pixels at the edge of a small region will

have some correlation with pixels in neighboring regions, but

for large enough regions, this correlation should be a small

fraction of the net behavior. If that is not the case, then Eq.

?6? would need to be modified to include the correlations

among small subregions.

II.B. Experimental setup

Experiments were performed on a tabletop CBCT system

?shown in Fig. 4? consisting of a generator ?Indico 100, CPI

Communication & Medical Products Division, Georgetown,

Ontario, Canada?, an x-ray tube ?G-1950SP Varian X-ray

TABLE I. Parameters for the experiment.

X-ray source

Pulse width

Total mA s

120 kVp, 12 mA

15 ms

72

1000 mm

670 mm

Source to detector distance

Source to isocenter distance

Detector size

Magnification

Tilt angle

Number of views

Reconstructed voxel size

Reconstructed volume size

400 mm?300 mm ?512?384 pixels?

1.5

?3° to 14°

400 evenly spaced over 360°

0.26 mm in all directions

26.6 cm?26.6 cm?13 cm centered

at ?0, 0, 6.7 cm? ?1024?1024?512 voxels?

FIG. 4. The tabletop cone-beam CT system with a 260 mm diameter water

phantom.

FIG. 5. Measured projection NPS of air.FIG. 6. Nonuniform noise statistics across detector cells.

2125 J. Baek and N. J. Pelc: 3D noise power spectrum with FDK reconstruction2125

Medical Physics, Vol. 38, No. 4, April 2011

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Products, Salt Lake City, UT? with a nominal 0.6 mm focal

spot and 1.0 mm Al inherent filtration, a rotation stage, and a

400 mm?300 mm flat panel x-ray detector ?PaxScan

4030CB, Varian Medical Systems, Palo Alto, CA? operated

in a 2?2 pixel binning mode ?388 ?m?388 ?m pixel

size? with 1024?768 pixels per frame at 30 frames per sec-

ond with dark field correction applied. Further 2?2 pixel

binning was performed producing 512?384 pixels with

776 ?m?776 ?m pixel size. The detector center was off-

set by 10 cm in the z-direction to be similar to a system for

cone-beam breast CT.34To measure ?D?fu,fv??2?see Eq. ?1??

for our detector, we measured the projection NPS of air

?shown in Fig. 5?. The measured projection NPS was fit to a

two-dimensional Gaussian function G?fu,fv? ?Ref. 35?

G?fu,fv? = e−fu

and a good fit was found with ?2=2.86?cm−2?. This fit was

used for comparison of the analytical prediction to the mea-

2+fv

2/?2

?7?

sured CT NPS ?i.e., ?D?fu,fv??2=G?fu,fv?? in Eq. ?1??.

Scan data of air and a 260 mm diameter water cylinder

placed at the isocenter were reconstructed using a FDK al-

gorithm and an unweighted ramp filter. To avoid unwanted

additional apodization and noise aliasing, the projections

were filtered using 10-fold Fourier interpolation ?i.e., zero

padding in frequency space followed by an inverse Fourier

transform?23,36and then voxel-driven backprojection with

linear interpolation was performed using a small voxel size

?0.26 mm in all directions?. The parameters for imaging and

reconstruction are summarized in Table I.

II.C. NPS estimation

Experimental 3D NPS were estimated from the Fourier

transform ?squared? of noise-only 3D images37

FIG. 7. fx-fyplane and fy-fzplane of the 3D NPS and radial NPS from water phantom. Estimated NPS ?first column?, analytical NPS ?second column?, and

radial NPS ?third column? for ?a? an isocentered volume and ?b? a volume at ?0, 0, 13 cm?. The display range in both cases is ?0 to 3?10−8?.

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S?fx,fy,fz? =bxbybz

LxLyLz

??DFT?i?x,y,z???2?,

?8?

where Lx, Ly, and Lzare the number of elements in three

dimensions and bx, by, and bzare the corresponding voxel

size in each direction. i?x,y,z? is a noise-only image ac-

quired by subtracting the image data from two identical

scans, thereby containing only stochastic noise and dividing

the result by?2. The symbol ? ? indicates the ensemble av-

erage of 32 independent realizations ?i.e., 64 images divided

into 32 pairs?.

The 3D NPS at three local volumes centered at ??11 cm,

0, 0?, ?0, 0, 0?, and ?0, 0, 13 cm?, respectively, were esti-

mated. Each local volume was composed of 128?128

?128 elements corresponding to a cube of ?3.3 cm per

side. To estimate the 3D NPS for an entire volume while

avoiding memory limitations, the 1024?1024?512 matrix

was separated into 32 nonoverlapped 2563subvolumes and

then the NPS of the nonoverlapped subvolumes were aver-

aged. This was done for both the air scan and water phantom

data.

Analytical 3D NPS for the three local volumes and the

entire volume were calculated using Eqs. ?5? and ?6?. To

compare the analytical NPS to the estimated NPS quantita-

tively, we first found a conversion factor ?4.375? between

detector counts and number of detected photons by compar-

ing the measured projection NPS of air at zero frequency to

the analytical projection NPS. Since the detector counts in

the air scan were ?46 000, the analytical 3D NPS for the air

scan was calculated using uniform noise statistics of 201 250

photons per detector cell over all views. For the analytical

calculation of 3D NPS for the water phantom, the counts

across one row of the detector in a scan of the water phantom

were scaled to produce the nonuniform noise statistics across

detector channels shown in Fig. 6 and this was used for each

detector row and view. To calculate the analytical 3D NPS

for the local and entire volume, 3D NPS at 125 ?i.e., 5?5

?5? and 32 000 ?i.e., 40?40?20? locations equally spaced

in the x-, y-, and z-directions were calculated and then aver-

aged.

III. RESULTS

Figure 7 shows the experimentally estimated and analyti-

cal 3D NPS of the water phantom for two local volumes

centered at ?0, 0, 0? and ?0, 0, 13 cm?, respectively, displayed

up to 3/4 Nyquist sampling frequency at isocenter, and plots

of the radial NPS at fz=0, 1/4, and 1/2 Nyquist sampling

frequency. The fx-fyplanes of 3D NPS show the radial sym-

metry because the number of photons through each volume

is the same over all views. Compared to the 3D NPS of the

isocentered volume, the 3D NPS of the volume at ?0, 0, 13

cm? shows the missing cone region near the fzaxis because

of its tilt angle ??11°?, a corollary of the insufficient cover-

age of Fourier space in cone-beam CT, and higher amplitude

because of its longer path length ?or equivalently lower num-

ber of detected photons?. We also estimated the 3D NPS of

air scans for the same volume locations and verified the same

symmetry and missing cone. The effect of the detector noise

apodization is most easily appreciated as a decreasing spec-

tral power in the fzdirection. The experimental 3D NPS and

the analytical 3D NPS both show this effect, which was in-

cluded in the analytical model as the Gaussian fit described

above. In all cases, the estimated NPS shows excellent agree-

ment with the analytical NPS.

Figure 8 shows the estimated and analytical 3D NPS of

the water phantom and air for the local volume centered at

??11 cm, 0, 0?. In contrast with the two volumes along the

z-axis, the 3D NPS at this location shows a different shape

and amplitude distribution. The sharp transitions in the fx-fy

and fy-fzplanes of the 3D NPS demonstrate the effect of the

varying bandwidth of the sampled 2D Fourier plane during

the backprojection in opposed directions ?described in Fig.

3?. The different angular dependent amplitude distributions

shown in the fx-fyplanes of the water phantom 3D NPS ?Fig.

8?b?? are caused by the different noise levels in views in

FIG. 8. fx-fyplane, fy-fzplane, and fx-fzplane of the 3D NPS for the local

volume centered at ??11 cm, 0, 0?. Estimated NPS ?first column?, analytical

NPS ?second column?, and difference NPS image ?third column? for ?a? air

and ?b? water phantom.

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Medical Physics, Vol. 38, No. 4, April 2011

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FIG. 9. fx-fyplane and fy-fzplane of the 3D NPS and radial NPS for an entire volume. Estimated NPS ?first column?, analytical NPS ?second column?, and

radial NPS ?third column? for ?a? air and ?b? water phantom. The display window level of NPS images of air and water are ?0 to 5.8?10−10? and ?0 to

1.9?10−8?.

FIG. 10. fu-ftplane of the 3D NPS and a vertical profile

at fu=0 from ?a? measured air scan data and ?b? simu-

lated air scan data with 10% detector lag. The display

window is ?10−5to 2.5?10−5?

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Medical Physics, Vol. 38, No. 4, April 2011

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different directions. In contrast with the air scan, the number

of photons through the local volume in the water phantom is

smaller in the 0° view direction ?horizontal in Fig. 3?a?? than

at 90° view direction and therefore the fx-fyplane shows

higher amplitude in the fydirection. The 3D NPS of a local

volume centered at ?0, 11 cm, 0? has this pattern rotated by

90°. From Figs. 7 and 8, it can be observed that the noise

behavior varies significantly for different locations.

Figure 9 shows the estimated and analytical 3D NPS and

radial NPS of the water phantom and air for the entire vol-

ume. The fx-fyplane of the 3D NPS does not show the an-

gular dependent amplitude variation since the angular depen-

dence of local volumes in Fig. 8 has been averaged out.

Instead, the different noise behaviors of local volumes are

reflected as a roll-off in the spectrum at high frequencies

?shown in the radial NPS?. The missing cone region is again

seen in the fy-fzplane of the 3D NPS. However, the size of

the missing cone is relatively smaller and less distinct than in

the local volume at ?0, 0, 13 cm? due to the averaging across

the entire volume. This effect is clearly seen in the radial

NPS, where the low frequency of the radial NPS is lower

than that of the local NPS in Fig. 7?b?, especially at high fz.

Since the number of detected photons through the water

phantom is lower than that through air, the amplitude of the

noise power is much higher in the NPS of the water phan-

tom. The radial NPS shows excellent agreement between the

analytical and experimental 3D NPS, demonstrating that the

cross-correlation between neighboring small volumes is

small compared to the noise power within each small vol-

ume.

FIG. 11. NPS at the isocenter for different detector lag. ?a? 2D NPS ?up? and radial NPS ?down?. ?b? The comparison of radial NPS.

2129J. Baek and N. J. Pelc: 3D noise power spectrum with FDK reconstruction2129

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IV. DISCUSSION AND CONCLUSIONS

Cone-beam CT systems with FDK reconstruction have

nonstationary noise across the FOV. While the NPS of a

large volume provides information about the average noise

behavior, the NPS of local volumes show significant differ-

ences in the noise behavior in different locations. In this

work, we estimated the 3D NPS of local volumes and an

entire volume from reconstructed water and air images and

compared them to the analytical 3D NPS quantitatively.

The NPS of different local volumes showed symmetric

and asymmetric shapes and different angular dependent am-

plitude variations. For the local volumes along the z-axis, the

fx-fyplane of the 3D NPS showed a radial symmetry because

of the circular symmetry of our test object, but the fy-fzplane

did not. The 3D NPS of local volumes at larger tilt angles

showed the missing cone region caused by the insufficient

sampling in Fourier space of the cone-beam CT system. The

3D NPS of a local volume centered at ??11 cm, 0, 0?

showed sharp transitions in the fx-fyand fy-fzplanes and

different amplitude distributions for the water and air im-

ages. These location dependent noise behaviors were mainly

caused by the effects of varying magnification, backprojec-

tion weighting, cosine weighting, and noise level from view

to view. In the 3D NPS of an entire volume, the nonstation-

ary noise behaviors of local volumes were averaged out, pro-

ducing high-frequency roll-off. Radial symmetry was ob-

served in circular symmetric test objects and a missing cone

region was observed in the fy-fzplane of the 3D NPS. It is

important keep in mind that while the NPS of the entire

volume may characterize the average behavior, it may not

reflect the noise properties in specific regions.

In the estimated 3D NPS, the effect of the detector noise

apodization was most easily observed as a decreasing spec-

tral density in the fzdirection although some effect is also

present at high fxand fyfrequencies. The detector noise

apodization was modeled using a 2D Gaussian and incorpo-

rated into the analytical NPS. The use of linear or nearest

neighbor interpolation can also introduce decreasing spectral

density at high frequencies. However, as described in Refs.

23 and 38, Fourier interpolation can prevent this effect.

In this work, the effect of detector lag was assumed to be

small since the estimated 3D NPS of raw data through air in

the ?fu,fv,ft? domain showed “white” noise in the ftdirec-

tion. For example, Fig. 10?a? shows the fu-ftplane of the

measured raw data NPS of the projections and the vertical

profile at fu=0. For comparison, 10% detector lag was simu-

lated and demonstrates decreasing spectral density in the ft

direction ?Fig. 10?b??. With this level of detector lag, our

initial CT simulations with parallel-beam geometry showed

noise correlation in the azimuthal direction and decreasing

high-frequency noise with increasing detector lag ?Fig. 11?.

The spatially varying effect of detector lag on the NPS is

under investigation.

In our experiment, the water phantom data contained scat-

tered x-ray photons, which can degrade image quality. Initial

studies with computer simulations showed that if uncor-

rected, scattered photons in the raw data decrease the ampli-

tude of the NPS, but an ideal correction for x-ray scatter

increases the amplitude of the NPS by the scatter-to-primary

ratio. The impact of nonstationary noise due to x-ray scatter

on detectability is a subject of future research.

ACKNOWLEDGMENTS

This work is supported by GE Healthcare, NIH Grant No.

EB006837, and the Lucas Foundation.

a?Electronic mail: bjd1219@stanford.edu

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