A new weighting function to achieve high temporal resolution in circular cone-beam CT with shifted detectors.

Abstract

The size of the field of measurement (FOM) in computed tomography is limited by the size of the x-ray detector. In general, the detector is mounted symmetrically with respect to the rotation axis such that the transaxial FOM diameter approximately equals the lateral dimensions of the detector when being demagnified to the isocenter. To enlarge the FOM one may laterally shift the detector by up to 50% of its size. Well-known weighting functions must then be applied to the raw data prior to convolution and backprojection. In this case, a full scan or a scan with more than 360 degrees angular coverage is required to obtain complete data. However, there is a small region, the inner FOM, that is covered redundantly and where a partial scan reconstruction may be sufficient. A new weighting function is proposed that allows one to reconstruct partial scans in that inner FOM while it reconstructs full scan or overscan data for the outer FOM, which is the part that contains no redundancies. The presented shifted detector partial scan algorithm achieves a high temporal resolution in the inner FOM while maintaining truncation-free images for the outer part. The partial scan window can be arbitrarily shifted in the angular direction, what corresponds to shifting the temporal window of the data shown in the inner FOM. This feature allows for the reconstruction of dynamic CT data with high temporal resolution. The approach presented here is evaluated using simulated and measured data for a dual source micro-CT scanner with rotating gantry.

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A new weighting function to achieve high temporal resolution in circular
cone-beam CT with shifted detectors
Clemens Maaßa?and Michael Knaup
Institute of Medical Physics, University of Erlangen-Nürnberg, Erlangen, Germany
Robert Lapp
VAMP Verfahren und Apparate der Medizinischen Physik GmbH, Erlangen, Germany
Marek Karolczak, Willi A. Kalender, and Marc Kachelrieß
Institute of Medical Physics, University of Erlangen-Nürnberg, Erlangen, Germany
?Received 30 June 2008; revised 16 October 2008; accepted for publication 16 October 2008;
published 24 November 2008?
The size of the field of measurement ?FOM? in computed tomography is limited by the size of the
x-ray detector. In general, the detector is mounted symmetrically with respect to the rotation axis
such that the transaxial FOM diameter approximately equals the lateral dimensions of the detector
when being demagnified to the isocenter. To enlarge the FOM one may laterally shift the detector
by up to 50% of its size. Well-known weighting functions must then be applied to the raw data prior
to convolution and backprojection. In this case, a full scan or a scan with more than 360° angular
coverage is required to obtain complete data. However, there is a small region, the inner FOM, that
is covered redundantly and where a partial scan reconstruction may be sufficient. A new weighting
function is proposed that allows one to reconstruct partial scans in that inner FOM while it recon-
structs full scan or overscan data for the outer FOM, which is the part that contains no redundan-
cies. The presented shifted detector partial scan algorithm achieves a high temporal resolution in the
inner FOM while maintaining truncation-free images for the outer part. The partial scan window
can be arbitrarily shifted in the angular direction, what corresponds to shifting the temporal window
of the data shown in the inner FOM. This feature allows for the reconstruction of dynamic CT data
with high temporal resolution. The approach presented here is evaluated using simulated and
measured data for a dual source micro-CT scanner with rotating gantry. © 2008 American Asso-
ciation of Physicists in Medicine. ?DOI: 10.1118/1.3013700?
Key words: cone-beam CT ?CBCT?, dynamic CT, micro-CT, image quality
I. INTRODUCTION
The size of the x-ray detector in CT determines the size of
the transaxial field of measurement ?FOM?. Longitudinally,
detectors that are smaller than the z extent of the object are
typically unproblematic. To cover a long object one may
simply use multiple circular scans, spiral scans, or other tra-
jectories. In the lateral dimension, however, the detector size
has a significant impact on the maximum object size. Basi-
cally, the lateral size of the detector, when being demagnified
to the center of rotation, corresponds to the diameter of the
FOM and objects exceeding the FOM will result in signifi-
cant truncation artifacts.
Methods to perform approximate high quality reconstruc-
tions in the region where no truncation occurs are well
known.1–6However, these methods are approximative, they
require extrapolation, and the reconstructed CT values are
not quantitative. The images often still suffer from truncation
artifacts.
To avoid approximations, one uses the possibility to shift
the detector laterally away from the isocenter.Ashift of up to
50% of the detector size combined with an appropriate
weighting function that must be applied prior to convolution
and backprojectionthen
reconstruction.7,8This shifted detector weight correctly ac-
allows for exactimage
counts for all data redundancies and no extrapolation is re-
quired. However, partial scan reconstruction was not possible
with shifted detectors and the scan must cover 360° or more.
This paper proposes a solution and shows how partial scan
reconstruction can be easily combined with a shifted
detector.
In the scanner used here, the dual source cone-beam
micro-CT scanner TomoScope Duo ?VAMP GmbH, Erlan-
gen, Germany?, the detector is not shifted to the extreme. It
is shifted by approximately 25% of its size, thereby resulting
in an inner FOM of radius rM=12 mm and an outer FOM of
radius RM=25 mm ?Fig. 1?. Hence, it allows one to scan
objects with a diameter of up to 50 mm. Objects that are
smaller than 24 mm in diameter are covered redundantly,
because they lie in the detector overlap region of direct and
complementary rays. For those objects, one may do with
scans of less than 360° angular range. Given that at least
180° plus fan angle is available, a Parker-weighting type
weight function is applied to the raw data9to obtain images
from just a short scan. The short scan feature has the advan-
tage of having a temporal resolution that is twice as high as
the temporal resolution of the full scan.
Since the scanner used has two source-detector systems,
also called threads, a 90° plus fan angle rotation is sufficient
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to obtain partial scan data.10With a 4 s rotation time in total
this allows for a temporal resolution of 1 s provided that
partial scan weighting in the inner FOM can be used.
In order to do so, a new weighting function has been
developed that performs shifted detector weighting com-
bined with partial scan weighting, called the shifted detector
partial scan ?SDPS? algorithm. The reconstruction performed
consists of weighting followed by convolution and back-
projection. The algorithm is a Feldkamp-type cone-beam al-
gorithm for flat panel detectors.11Since the new weighting
strategy acts on the lateral coordinates only and since the
length correction that is applied in the longitudinal direction
is the same as for other approximate cone-beam algorithms,
one may safely ignore the z coordinates during the following
considerations. It should further be noted that the weighting
mechanism applies to all possible detector geometries and is
not restricted to flat panel detectors.
II. MATERIALS AND METHODS
II.A. Geometry
Reconstruction of cone-beam CT data of the form follow-
ing is discussed:
q?t,u,v? =?
0
?
d?f?s + ?Θ?,
?1?
with s=s?t? being the source position at time t and
Θ=Θ?t,u,v? being the direction vector of a ray emerging at
s?t? toward the detector element ?u,v? ?cf. Fig. 2?. The raw
data are denoted as q?t,u,v? and the object to be recon-
structed is f?r?. The detector coordinates u and v parametrize
the two-dimensional detector surface that may be planar,
curved, or of any other shape. It is assumed that the source
trajectory is an exact or an approximate circle trajectory
where any image reconstruction algorithm for circular cone-
beam CT data applies. The micro-CT scanner used follows
an exact circle trajectory and the trajectory parameter t is
proportional to the rotation angle of the gantry. The approxi-
mate circle trajectory becomes relevant when reconstructing
data from C-arm CT scanners where slight mechanical de-
viations from the ideal circle may occur. For those devices,
the weighting functions apply, too, and precise knowledge of
the trajectory plays a role mainly during backprojection. It is
assumed that the rotation center is the center of the global
coordinate system, as depicted in Fig. 3.
To perform data weighting, the parameterization of the
source vector and of the ray direction vector needs to be
changed as follows:
s =?
RFsin ?
− RFcos ?
0?
?2?
and
Θ =?
Note that the parameters ?=??t?, RF=RF?t?, ?=??t,u,v?,
and ?=??t,u,v?—the angle and distance of the source posi-
tion with respect to the global coordinate system and the
angle in the fan and the angle within the cone, respectively—
can be computed from s?t? and Θ?t,u,v? at any time.
Further, the angle ?=?+? is introduced, which is the
ray’s angle with respect to the coordinate system. It is im-
portant to realize that one of the parameters ?, ?, and ? is
redundant due to the relation ?=?+?. For convenience, the
independent pair of parameters will be selected depending on
the type of weighting function.
− sin?? + ??cos ?
cos?? + ??cos ?
sin ? ?.
?3?
II.B. Data weighting
Data weighting compensates for data redundancies before
the data are passed to the Feldkamp-type reconstructor. Ba-
sically, it requires to multiply the projection data by a weight
function, w??,?? or w??,?? prior to filtering and back-
projection. The weight function must correctly count contri-
FIG. 1. The TomoScope Duo micro-CT scanner consists of two threads with
laterally shifted detectors to achieve an inner field of measurement of
24 mm and an outer FOM of 50 mm. The white circle delineates the inner
FOM.
FIG. 2. Illustration of the geometry conventions used.
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butions from direct rays ??,?? and from complementary rays
??+?,−??. To obtain a correct normalization of all direct
and complementary rays, the weight function must have the
property
?
k
w?? + k?,?− ?k?? = 2.
?4?
Note that fan-beam filtered backprojection requires further
weighting in the channel direction prior to convolution and
cone-beam backprojection. This kind of weighting depends
on the detector shape and on the image reconstruction algo-
rithm and it is different for flat and for curved ideal detectors.
Since this paper is about redundancy weighting and not
about filtered backprojection, this type of weighting is not
handled here. The length correction factor cos? is applied
prior to image reconstruction.
In the present case, the authors are interested in combin-
ing partial scan weighting with shifted detector weighting to
achieve both a high temporal resolution and a large diameter
of the FOM. The situation is illustrated in Fig. 3. Here, one is
dealing with two fields of measurement: the inner field of
measurement with radius rM—given by the overlap region of
the detector where both direct and complementary rays are
available—and the outer field of measurement with radius
RM, where only direct rays are measured. Assuming a full
scan acquisition, pixels with a distance r?rMare covered
with a twofold redundancy, while those with rM?r?RMdo
not contain redundant data.
First, let us define the relevant parameters. It is assumed
that the range of the full scan is ??1−??,?2+??? with
?2−?1=2?K, K?N, and 0????? being the overscan
angle. Since there is a shifted detector design, at least 360° of
gantry rotation needs to be covered until one can reconstruct
voxels in the outer FOM. The integer K denotes the number
of full rotations covered and may be rather large for dynamic
studies where the contrast uptake is observed during multiple
gantry rotations.
Let the range of the partial scan be ?? ˆ1−?? ˆ ,? ˆ2+?? ˆ?
with ? ˆ2−? ˆ1=? and ?? ˆ ??F, where ?Fis the half fan angle
of the inner FOM and will be precisely defined later. This
partial scan range may be a subset of the full scan range.
Although this appears to be the typical situation, one can also
think of scans where both ranges do not necessarily overlap.
In dynamic CT, where data from many rotations are acquired
to study the dynamics of contrast uptake, one extracts partial
scans from the data prior to reconstruction. If one is inter-
ested in the dynamics of object parts in the inner FOM there
is no reason why the outer FOM shall be irradiated by x rays
by more than 360°.
To understand the definition of ?Fconsider Fig. 4. Figure
4 illustrates the weighting functions in the ? domain, where
FIG. 3. A typical shifted detector geometry. In the inner FOM, one can
achieve a high temporal resolution. The positions ?Aand ?Hare symmetric
with respect to the central ray, i.e., ?A=−?H, and denote the region of double
redundant coverage. In dynamic scans, which consist of multiple full rota-
tions, a collimator can be moved to the position ?Hafter the first rotation to
significantly reduce the dose to the object. The center of rotation is the
center of both FOM circles.
FIG. 4. Illustration of the shifted detector weight and the transition weight. The angles ?Athrough ?Hmust be symmetric with respect to ?=0. The transition
regions ??A,?B? and ??C,?D? of the shifted detector weight and the transition weight may overlap. However, their size must be chosen large enough to avoid
data inconsistency artifacts due to data discretization. For example, the case ?A=?Bor ?C=?D, which would correspond to step functions, should be avoided.
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? is the angle of a ray within the fan. The angles ?Aand ?H
delimit the region where direct and complementary rays are
available. The authors demand ?A=−?H, ?B=−?G, ?C
=−?F, ?D=−?E, 0??B??A, 0??C??A, and 0??D??C.
The shifted detector covers the channels ??A,?max?; the re-
gion where there is redundant data is ??A,?H?.
Due to the discretization, discontinuities in raw data
weights always need to be avoided to prevent artifacts.
Therefore, a smooth function s?x? is defined that needs to be
continuousfor
x??−1,1?
s?−x?=−s?x? and s?1?=1. This paper always uses s?x?
=sin?1
and has theproperties
2?x?.
II.B.1. Overscan weight
If a scan covers more than 360° in rotation angle, an
appropriate overscan weighting needs to account for this
fact. For K rotations, this weight needs to average by multi-
plying the raw data with 1/K. The remaining angular range
of size ?? can be accounted for by using a smooth transition
from 0 to 1. The overscan weight is defined as
2K?
Due to its property ?kwOS??+2?k?=1, proper normalization
is guaranteed.
wOS??? =
1
0
1 + s?? − ?1
2
1 − s?? − ?2
0
if ? ? ?1− ??
??? else if ? ? ?1+ ??
else if ? ? ?2− ??
??? else if ? ? ?2+ ??
else.
?
?5?
II.B.2. Shifted detector weight
To compensate for the missing raw data in the case of the
shifted detector, the weight function needs to weight areas
where contributions from direct and complementary rays ex-
ist with the value 1, while it needs to weight detector areas
where only direct rays contribute with the value 2. A smooth
transition in between these zones is required to avoid arti-
facts. The shifted detector weight is defined as
2?
which fulfills wSD?−??+wSD???=2 and therefore achieves
the correct normalization ?cf. Fig. 4?. Multiplying the full
scan raw data with this weight function would yield full scan
images in the inner and in the outer FOM. The function
proposed here differs from the function of Refs. 7 and 8 by
having a large plateau region. Apart from this difference,
both methods are identical.
wSD??? =1
0
1 + s?2? − ?A
2
3 + s?2? − ?G
?H− ?G
4
if ? ? ?A
?B− ?A
− 1?
else if ? ? ?B
else if ? ? ?G
− 1? else if ? ? ?H
else,?
?6?
II.B.3. Partial scan weight
Here, our weight function needs to extract a range signifi-
cantly less than 360° data. Direct rays ??,?? and comple-
mentary rays ??+?,−?? need to add up to 2 to obtain exact
images. At least 180° plus fan angle of data is required to do
so. The authors follow Parker9and use the weighting
function
wPS??,?? =?
0
1 + s?? − ? ˆ1
2
1 − s?? − ? ˆ2
0
if ? ? ? ˆ1− ?? ˆ + ?
?? ˆ − ?? else if ? ? ? ˆ1+ ?? ˆ − ?
else if ? ? ? ˆ2− ?? ˆ − ?
?? ˆ + ?? else if ? ? ? ˆ2+ ?? ˆ + ?
else,
?
?7?
where one can easily see the correct normalization
wPS??,??+wPS??+?,−??=2.
II.B.4. Transition weight
Since the inner FOM is covered redundantly, one may
combine the shifted detector weight with the partial scan
weight by applying the partial scan weight only to the inner
region of the detector. A smooth transition between the full
scan shifted detector type weighting of the outer FOM and
the partial scan behavior in the inner FOM is achieved by
introducing a third weight function with the property
wT?−??=wT???,
2?
which is zero in the outer region and smoothly increases to
one in the inner region of the detector ?cf. Fig. 4?.
wT??? =1
0
1 + s?2? − ?C
?D− ?C
2
1 − s?2? − ?E
?F− ?E
0
if ? ? ?C
− 1? else if ? ? ?D
else if ? ? ?E
− 1?
else if ? ? ?F
else,?
?8?
II.B.5. Shifted detector partial scan weight
The combination of the shifted detector and overscan
weight in the outer region and the partial scan weight in the
inner region yields the final shifted detector partial scan
weight:
wSDPS??,?? = wT???wPS??,?? + ?1 − wT????wSD???wOS???.
?9?
Note that wSDPS??,??+wSDPS??+?,−??=2, which implies
correct normalization.
Both terms of wSDPScan be reconstructed separately ?al-
though each reconstruction will show truncation artifacts?.
The function wTthereby serves as a transition function be-
tween partial scan reconstruction ?left term? and full scan
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shifted detector reconstruction ?right term?. Performing two
separate reconstructions may be computationally inefficient
whenever only a single reconstruction is required. However,
in the case of dynamic CT studies where a scan covers sev-
eral rotations ?e.g., K=10? and where the inner FOM shall be
reconstructed for a variety of partial scan ranges, it will be
worthwhile to separate the reconstructions of the inner FOM
and the outer FOM.
II.C. Dual source weighting
The dual source micro-CT scanner available to the au-
thors has both threads mounted under an interthread angle of
about ??12=90°. Dual source weighting is rather simple.
Either the data are averaged to get the maximum dose usage
and the minimum image noise or one uses weighting func-
tions w1??? and w2??? that fade out the contribution of
thread 1 and fade in the contribution of thread 2. In the inner
FOM, the fading approach is used, which allows the authors
to perform reconstructions from a 90° plus fan angle rotation
only. The temporal resolution achievable equals trot/4, which
is a quarter of the rotation time. The TomoScope scanner’s
fast scan mode has trot=4 s, which means that the authors are
aiming at a temporal resolution of about 1 s for the inner
FOM.
II.C.1. Dual source averaging weight
Starting with the averaging approach, let ??1−??,?2
+??? with ?2−?1=2?K be the range of scan angles covered
by both threads, just as it was defined earlier. The first thread
covers the range ??1−??,?2+??−??12?, while the second
thread covers ??1−??+??12,?2+???. The intersection of
both intervals, the range of scan angles that are covered
twice, is given as ??A,?D? with ?A=?1−??+?12and ?D
=?2+??−??12. To allow for a smooth transition between
the ? values covered redundantly and those that are covered
only once, the authors define two additional angles ?Band
?Cwith ?A??B??C??Dsuch that the regions ??A,?B?
and ??C,?D? are the transition regions. For the present stud-
ies, the authors use ?B−?A=?D−?C=10° for transition. The
averaging weights for the first and for the second segment
are given as follows:
4?
4?
wA1??? =1
4
3 − s?2? − ?A
2
1 − s?2? − ?C
?D− ?C
0
if ? ? ?A
?B− ?A
− 1?
else if ? ? ?B
else if ? ? ?C
− 1? else if ? ? ?D
else,?
else.?
?10?
wA2??? =1
0
1 + s?2? − ?A
2
3 + s?2? − ?C
?D− ?C
4
if ? ? ?A
?B− ?A
− 1?
else if ? ? ?B
else if ? ? ?C
− 1? else if ? ? ?D
?11?
One has wA1???+wA2???=1, which means that the data are
normalized correctly after summing up the weighted data of
both threads. Figure 5 illustrates the weights and the defini-
tion of the various ? angles.
II.C.2. Dual source fading weight
Now, let one consider the fading weight. It is supposed to
maximize temporal resolution and shall fade between thread
1 and thread 2 right in the middle of the partial scan range.
With ?? ˆ1−?? ˆ ,? ˆ2+?? ˆ? denoting the partial scan range ?just
as in the previous section?, the transition region is defined as
?? ˆA,? ˆB? and it is centered with respect to the partial scan
window by ensuring ? ˆA+? ˆB=? ˆ1+? ˆ2. For the present stud-
ies, a 10° transition range is used, which implies ? ˆB−? ˆA
=10°. The fading weights for the first and for the second
thread are given as follows:
FIG. 6. Fading weight for a dual source CT scanner.
FIG. 5. Averaging weight for a dual source CT scanner ?not drawn to scale?.
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