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Geometric calibration for a dual tube/detector micro-CT system
Samuel M. Johnston, G. Allan Johnson, and Cristian T. Badeaa兲
Center for In Vivo Microscopy Box 3302, Duke University Medical Center, Durham,
North Carolina 27710
共Received 6 November 2007; revised 29 January 2008; accepted for publication 22 February 2008;
published 16 April 2008兲
The authors describe a dual tube/detector micro-computed tomography 共micro-CT兲system that has
the potential to improve temporal resolution and material contrast in small animal imaging studies.
To realize this potential, it is necessary to precisely calibrate the geometry of a dual micro-CT
system to allow the combination of projection data acquired with each individual tube/detector in a
single reconstructed image. The authors present a geometric calibration technique that uses multiple
projection images acquired with the two imaging chains while rotating a phantom containing a
vertical array of regularly spaced metallic beads. The individual geometries of the imaging chains
are estimated from the phantom projection images using analytical methods followed by a refine-
ment procedure based on nonlinear optimization. The geometric parameters are used to create the
cone beam projection matrices required by the reconstruction process for each imaging chain. Next,
a transformation between the two projection matrices is found that allows the combination of
projection data in a single reconstructed image. The authors describe this technique, test it with a
series of computer simulations, and then apply it to data collected from their dual tube/detector
micro-CT system. The results demonstrate that the proposed technique is accurate, robust, and
produces images free of misalignment artifacts. © 2008 American Association of Physicists in
Medicine.关DOI: 10.1118/1.2900000兴
Key words: x-ray, micro-CT, calibration, small animal
I. INTRODUCTION
Micro-computed tomography 共micro-CT兲is a noninvasive
imaging modality used to assess morphology and function in
small animals.1,2Dual tube/detector micro-CT, in which two
imaging chains consisting of an x-ray tube and a detector
image the same object in parallel, offers several advantages
over more conventional micro-CT. Dual simultaneous imag-
ing can acquire the same amount of images as a conventional
micro-CT in half the time, which facilitates functional imag-
ing studies such as in cardiac,3,4pulmonary,5,6or perfusion
investigations in small animals.7Dual tube/detector
micro-CT also facilitates dual energy imaging, in which two
different absorption coefficients at two different energies are
recorded for the same location simultaneously, enhancing
sensitivity and material differentiation.8
To realize these advantages of dual tube/detector
micro-CT imaging, it is necessary to reconstruct the images
from the individual systems in one shared geometry. To ac-
complish this, geometric calibration must be performed, in
which a set of parameters describing the geometry of the
system is calculated from a set of images acquired of some
ideal object. The calibration parameters can then be provided
to the CT reconstruction algorithm to map data from the
pixels in the detector to voxels in the object.
Several techniques have been proposed over the years for
finding these calibration parameters in individual micro-CT
systems for various geometries. These techniques have gen-
erally fallen into two categories: those based on iterative
nonlinear optimization9–13 and those based on the direct so-
lution of geometric equations.14–18 The latter category has
become favored in recent years because of superior perfor-
mance and ease of implementation. Both categories entail the
imaging of phantoms containing point-like structures, such
as various arrangements of small metal beads, with varying
requirements about the a priori knowledge of the phantom
and the geometry.
Although similar work has been done with multiple de-
tectors in SPECT and PET,12 we have not found any prior
work on the calibration of dual tube/detector x-ray micro-CT.
Therefore, in the construction of our dual tube/detector sys-
tem, it was necessary to develop a new geometric calibration
method. In this method we first employ an analytic algorithm
to find estimates of the most important calibration param-
eters and then use nonlinear optimization to find additional
parameters and refine the results. Once this is done for both
tube/detector chains, we use the parameters to create the
cone beam projection matrices required by the reconstruction
process for each imaging chain. We then find a transforma-
tion between the two projection matrices to combine the two
sets of projection data in a single reconstructed image. In this
work we describe this method and test it in computer simu-
lations and then apply it to data collected from our dual
tube/detector system.
II. MATERIALS AND METHODS
II.A. Projection matrices
Our approach is based on the use of projection
matrices.19–22 A cone beam projection matrix Ahasa4⫻3
dimension and relates the mapping of a point in three-
1820 1820Med. Phys. 35 „5…, May 2008 0094-2405/2008/35„5…/1820/10/$23.00 © 2008 Am. Assoc. Phys. Med.
dimensions 共3D兲共x,y,z兲to its projection 共u,v兲on a two-
dimensional detector in homogeneous coordinates
冤
uw
vw
w
冥
=A
冤
x
y
z
1
冥
,
where wis an arbitrary scale factor. Each projection matrix is
constructed from seven geometric parameters that define the
cone beam geometry of a single imaging chain made of one
x-ray source and one detector 共see Fig. 1兲. These are dso,ddo,
u0,v0,
,
, and
, where dso and ddo are the source-to-origin
and detector-to-origin distances 共for simplicity, we will often
use the source-to-detector distance dsd =dso +ddo instead of
ddo兲;共u0,v0兲is the pixel location where the central ray inter-
sects the detector plane 关see Fig. 1共a兲兴; and
,
, and
are
the detector rotation angles around each of the x,y, and z
axes, respectively 关see Figs. 1共b兲–1共d兲兴. Additionally, the
horizontal pixel distance ⌬uand vertical pixel distance ⌬vof
the detector must be known in advance.
The rotation of the detector around the three axes is de-
scribed by the three rotation matrices R
,R
, and R
,
R
=
冤
10 0 0
0cos
− sin
0
0sin
cos
0
00 0 1
冥
,
R
=
冤
cos
0 sin
0
0 100
− sin
0 cos
0
0 001
冥
,
R
=
冤
cos
− sin
00
sin
cos
00
0010
0001
冥
.
With these matrices we construct the vectors uand vwhich
describe the displacement of a pixel from the detector origin,
and the detector normal vector n,
u=⌬uR
−1R
−1R
−1
冤
0
1
0
1
冥
,
v=⌬vR
−1R
−1R
−1
冤
0
0
1
1
冥
,
n=v⫻u.
We can now construct the projection matrix A,
A=
冤
01/⌬u00
001/⌬v0
00 01
冥
R
R
R
冤
100u0u1+v0v1−dsd
010 u0u2+v0v2
001 u0u3+v0v3
000 1
冥冤
n1dsd 00
n1dsddso
0n1dsd 00
00
n1dsd 0
n1n2n3n1dso
冥
.共1兲
To describe the planar circular trajectory of the source and
detector by angle
around the rotation axis, in this case z
axis, we multiply A, given by Eq. 共1兲, by a rotation matrix to
obtain the projection matrix for each sampling angle A
,
A
=A
冤
cos
− sin
00
sin
cos
00
0010
0001
冥
.
The tomographic reconstruction algorithm uses the A
ma-
trices to perform the 3D backprojection process. We note that
with our dual tube/detector micro-CT system, the scanned
object is rotating and the tubes/detectors are stationary. Nev-
ertheless, the two geometries with rotating tube/detector or
rotating object are equivalent. Therefore, to calibrate a dual
tube/detector system, it is necessary first to find the param-
eters describing the cone beam geometry for each individual
tube/detector and to construct the projection matrices 关see
Eq. 共1兲兴. Next, we find a transformation that relates these two
projection matrices in one shared geometry.
II.B. Geometric calibration for a single imaging chain
To find dso,ddo,u0,v0,
,
, and
for a single imaging
chain, we have taken a two-step approach. First, we obtain
initial estimates following the method described by Yang et
al.,18 and then we refine the values with a nonlinear optimi-
zation procedure. We use a phantom consisting of 20 metal-
lic beads witha2mmdiameter inserted in an acrylic rod and
arranged along a vertical axis with distance l=5 mm be-
tween each adjacent pair of beads. The phantom is scanned
through one complete rotation 关see Fig. 2共a兲兴of 360° using
1821 Johnston, Johnson, and Badea: Dual tube/detector micro-CT system geometric calibration 1821
Medical Physics, Vol. 35, No. 5, May 2008
both imaging chains and projection images acquired every
1°. Following scanning, the phantom’s projection images are
processed in MATLAB 共The MathWorks, Inc., Natick, MA兲to
compute the trajectories of the center of mass of each imaged
bead. Finding the beads’ centers of mass is a semiautomatic
process that requires the user to select rectangular regions of
interest 共ROIs兲around each bead only in the first projection
image. In each ROI, the bead is segmented using adaptive
local thresholding. An initial threshold is chosen halfway be-
tween the maximum and minimum brightness values, and
this threshold is iteratively improved. At each iteration, the
threshold divides the pixels inside each ROI into two sets
corresponding to the bead and background. The bead set is
assigned the connected component containing the central
pixel. The background set is assigned the corner pixels, and
the threshold is recalculated to be halfway between the av-
erages of these two sets. The iterations stop when the thresh-
old no longer changes. After segmentation, the center of
mass 共uc,vc兲for each bead is found using the bead pixel
values of the segmented bead image I共u,v兲,
uc=⌺beadI共u,v兲·u
⌺beadI共u,v兲,
vc=⌺beadI共u,v兲·v
⌺beadI共u,v兲.
The radius of the bead projection is found by calculating
the average distance from the perimeter to the center
r=⌺perimeter冑共u−uc兲2+共v−vc兲2
⌺perimeter1.共2兲
Unlike in the first projection image, the ROIs in subsequent
projections are constructed automatically by predicting the
location of the bead center using a linear fit of the most
recent locations and setting a square around the predicted
location with a side length equal to twice the radius given by
Eq. 共2兲.
For each bead i, an ellipse is fitted to the set of all centers
of mass over all projection images. Point pin ellipse iis
FIG. 1. The geometry of an x-ray CT
system, defined by 共a兲dso,ddo,u0,v0,
共b兲
,共c兲
, and 共d兲
.
FIG.2.共a兲An x-ray image of the calibration phantom,
with the centers of the projections of the beads overlaid
in white, and 共b兲the ellipses that are fit to the centers.
1822 Johnston, Johnson, and Badea: Dual tube/detector micro-CT system geometric calibration 1822
Medical Physics, Vol. 35, No. 5, May 2008
denoted aip=共uip,vip兲, and the distance between points
aip and ajp is denoted aipajp. The center, ai0=共ui0,vi0兲, and
four principal corners, ai1,ai2,ai3, and ai4, are recorded for
each ellipse 关see Fig. 2共b兲兴. The centers of the ellipses, ai0,
are computed as the mean of all points, aip, on the ellipse.
The major axis line is obtained by linear regression on all the
points, and the minor axis line is the line perpendicular to the
major axis that passes through the center. The angle of the
major axis is obtained as the inverse tangent of the slope. All
points aip are then projected onto the ellipse axes, and the
corners are the extreme points of the projections. With these
ellipse parameters we now calculate the calibration param-
eters, using geometric relations derived by Yang et al.18
II.B.1. Calculation of dsd and v0
For each bead i, we construct
xi=
vi1−vi2
ai3ai4
,
yi=
vi1+vi2
2.
The xiand yiare related by the linear function
yi=v0+dsd⌬vxi.
We find v0and dsd by linear regression over all i.
II.B.2. Calculation of
and u0
For each bead i,ui0, and vi0are related by the linear
function
ui0=a+bvi0.
We find aand bby linear regression over all i, and then
calculate
u0=a+bv0,
= arctan b.
II.B.3. Calculation of dso
For each pair of beads iand j,
dso =l
ai0aj0⌬v
dsd.
We average dso for each adjacent pair iand j. This formula is
simpler than the one described by Yang et al.,18 since we
assume that our phantom is placed parallel to the zaxis.
While this assumption may not always be valid, we note that
the geometric parameters obtained by this analytic approach
are just initial estimates that are further refined through an
optimization procedure.
II.B.4. Nonlinear optimization
The method described by Yang et al.18 does not determine
the out-of-plane detector rotations
and
, since it is
claimed that careful mechanical placement can reduce these
parameters to less than 5°, below which these values have
little impact on reconstruction. However, in an imaging sys-
tem with components that are frequently moved, as in our
case, we cannot guarantee that the detectors will always be
so well aligned. Furthermore, in a dual tube/detector system,
the small changes caused by erroneous values for
and
will be amplified, since overlapping systems with slight in-
dependent misalignments will produce pronounced double
contours in the reconstruction. These rotations may be
present in our system and should be addressed, so we use a
nonlinear optimization program to move from an initial esti-
mate of 关dsd dso u0v0
00兴to 关dsd dso u0v0
兴.
For the objective function, we must first construct the
projection lines from the x-ray source to the beads’ projec-
tion centers of mass using the initial estimates of the geomet-
ric parameters. These projection lines are constructed for all
beads in all projections. For each bead the minimum pair-
wise distance between all its associated projection lines is
computed. The objective function returns a vector with an
entry for each bead representing the sum of the minimum
pairwise distances between projection lines.
We pass this objective function to a nonlinear least-
squares minimization function which successively recom-
putes selected parameters in order to minimize the distances
between projection lines. We run the program in two steps.
First, we allow
and
to vary while holding the other
parameters fixed since these parameters have no values and
would distort the other parameters. Next, we allow all pa-
rameters dsd,dso,u0,v0,
,
, and
to vary together. This
technique estimates
and
accurately and refines the esti-
mates of the other parameters.
II.C. Geometric calibration for a dual tube/detector
system
After computing the geometric parameters for each indi-
vidual imaging chain, we can construct their respective pro-
jection matrices A1and A2using Eq. 共1兲. However, these
matrices describe projection in two different systems of ref-
erence. We need a transformation matrix Tthat will allow us
to transform projection matrix A2into A2
⬘=A2Tin the coor-
dinate system of A1.
Since both systems image the same rotating object, they
must share the same axis of rotation, i.e., the zaxis. Conse-
quently, the transform between the two systems of coordi-
nates can consist only of rotation
␣
around the zaxis and
translation ⌬zalong the zaxis and thus the transformation
matrix Tis given by
T=
冤
cos
␣
− sin
␣
00
sin
␣
cos
␣
00
001
⌬z
0001
冥
.
Therefore, once we calibrate the individual systems, we need
an additional step to find ⌬zand
␣
to calibrate the entire dual
system 共see Fig. 3兲.
1823 Johnston, Johnson, and Badea: Dual tube/detector micro-CT system geometric calibration 1823
Medical Physics, Vol. 35, No. 5, May 2008
We first compute ⌬z, using the sets of projections of the
same calibration phantom described previously which were
acquired with both imaging chains simultaneously. The
movement of each bead of the phantom is described by ro-
tation around the zaxis, therefore we expect the zcoordinate
to remain constant at each rotation step, and a single zvalue
for each bead can be found in each system of geometry 关see
Fig. 4共a兲兴. The same beads should be matched in the two sets
of projection images from the two imaging chains. Knowing
the system parameters allows us to write the expressions of
vectors sfor the x-ray source and pfor a detector point 共u,v兲
in the object system of reference
s=
冤
−dso
0
0
1
冥
,
p=
冤
p1
p2
p3
1
冥
=
冤
ddo
0
0
1
冥
+⌬u共u−u0兲R
−1R
−1R
−1
冤
0
1
0
1
冥
+⌬v共v−v0兲R
−1R
−1R
−1
冤
0
0
1
1
冥
.共3兲
In a rotating object geometry as in our dual micro-CT sys-
tem, the point at the center of the rotation of a bead should
project to the center of the ellipse trajectory that we found in
the individual system calibration, and therefore we set 共u,v兲
in Eq. 共3兲to be equal to 共ui0,vi0兲for ellipse i关see Fig. 4共b兲兴.
The line from the x-ray source sto the pixel pis constructed
and its intersection with the zaxis is given by
冤
−dso
0
0
1
冥
+k
冢
冤
p1
p2
p3
1
冥
−
冤
−dso
0
0
1
冥
冣
=
冤
0
0
z
1
冥
,
FIG. 3. The geometry of a dual micro-CT system, defined by ⌬zand
␣
in
addition to the parameters of the individual systems.
FIG.4.共a兲The projection of beads
onto detectors in the dual system cali-
bration technique, 共b兲the method for
finding ⌬zusing the line segments
from the x-ray sources through the
axis of rotation to the centers of the
ellipses, 共c兲the lines from the x-ray
sources through a bead to the ellipses,
and 共d兲the method for finding
␣
using
the different xand yvalues found from
the lines through the same bead.
1824 Johnston, Johnson, and Badea: Dual tube/detector micro-CT system geometric calibration 1824
Medical Physics, Vol. 35, No. 5, May 2008
k=dso
p1+dso
,
z=p3k,
where kis the parameter describing the location of a point
along the line. This gives us the zvalue for each bead in each
geometry. The vertical displacement ⌬zbetween the two sys-
tems is the difference between the zcoordinates for the same
bead in the two different systems. We compute this value for
each bead iand report the average value as ⌬z关see Fig.
4共b兲兴.
Following the computation of the zcoordinate for each
bead, we proceed to determine the xand ycoordinates at
each rotation step by constructing the line from sto pas
described before, where the pixel location 共u,v兲is now the
center of the projection of the bead 关see Fig. 4共c兲兴. The xand
ycoordinates for each bead at each rotation step are com-
puted
k=z
p3
,
x=k共p1+dso兲−dso,
y=kp2.
We can now compute the angle
␣
around the zaxis between
the two imaging chains. For this purpose, we first find the
angle of rotation
of each 共x,y兲from some arbitrary starting
point. Since tan
=共y/x兲,wefind
=arctan共y/x兲. There are
two values of
, i.e.,
1and
2, corresponding to each imag-
FIG. 5. A flowchart of the complete
calibration process.
1825 Johnston, Johnson, and Badea: Dual tube/detector micro-CT system geometric calibration 1825
Medical Physics, Vol. 35, No. 5, May 2008
ing chain 关see Fig. 4共d兲兴. The rotation angle
␣
is computed as
␣
=
2−
1. We perform this calculation for each bead in each
simultaneous projection and report the average value as
␣
关see Fig. 4共d兲兴.
Finally, we refine these estimates with the same nonlinear
optimization program used for the individual calibrations, by
constructing the transformed projection lines from the differ-
ent detectors through the same beads and minimizing the
pair-wise distances between the lines.
A flowchart of the complete calibration process is shown
in Fig. 5.
II.C.1. Implementation
The above equations are implemented in MATLAB 共Ve r -
sion 7.0兲. The optimization function used is lsqnonlin, part of
the MATLAB Optimization Toolbox, which uses precondi-
tioned conjugate gradients constrained by a subspace trust
region. The ray-tracing programs called in the optimization
process were written in Cto reduce computation time.
For our in-house developed dual tube/detector system 共see
Fig. 6兲, we use two Varian A197 x-ray tubes with dual focal
spots fs=0.6/1.0 mm. The tubes are designed for angio-
graphic studies with high instantaneous flux and total heat
capacity. Two high frequency x-ray generators 共EPS 45-80,
EMD Technologies, Quebec, Canada兲are used to control the
x-ray tubes. The system has two identical detectors with a
Gd2O2S phosphor 共XDI-VHR 2 115 mm, Photonic Science,
East Sussex, UK兲with pixels of 22
m, 115 mm input taper
size, and 4008⫻2672 image matrix. Both detectors allow
on-chip binning of up to 8⫻8 pixels, and subarea readout to
allow high speed readout of more than 10 frames/s, i.e., a
time resolution of 100 ms. Both tubes and detectors are
mounted on a table together with the rotation stage. The
vertically positioned animal is placed in a cradle that is ro-
tated via an Oriel model 13049 digital stepping motor. The
x-ray generators, tubes, detectors and the rotation are con-
trolled by a sequencer application written in LABVIEW 共Na-
tional Instruments, Austin, TX兲that also allows for in vivo
studies, the flexible integration of cardiac and respiratory
physiology with the imaging sequence.23 Images of the ro-
tating object are acquired with a step-and-shoot acquisition
scheme.
For geometric calibration, the phantom described previ-
ously, containing an array of steel beads placed in an acrylic
rod, is attached to the imaging cradle and scanned prior to
the animal experiments. The projections and the computed
projection matrices are used with the COBRA EXXIM software
package 共EXXIM Computing Corp, Livermore, CA兲that
implements Feldkamp’s algorithm24 to reconstruct tomogra-
phic data as 3D image arrays 共5123兲. The projection matrices
computed with our method are written to a geometry file
containing one line for each angle, which is read by COBRA.
II.C.2. Experiments
To test our geometric calibration method we used both
simulated and experimental data. Using Eq. 共1兲we simulated
the projection operation on the calibration phantom in MAT-
LAB and performed the calibration method. Since our method
first involves finding the geometric parameters for each im-
aging chain, it made sense to compare our results with those
from previous articles for single chain micro-CT systems.
We used the same parameters as in Yang et al.,18 with the
same number of projections and beads, and the same noise
conditions: 48
m⫻48
m pixel pitch, 500 projection im-
ages over 360°, eight beads, distance between beads l
=2 mm, dsd = 400 mm, dso = 150 mm, u0=1005, v0=480,
=−1°,
=1.2°, and
=1.5°. We ran ten simulations with
Gaussian noise with standard deviation 0.4 pixels added to
the simulated bead projection centers.
We then simulated the projection operations with two or-
thogonal imaging chains as in our dual tube/detector system
in which the parameters for the first chain were the same as
before, and the parameters for the second chain were dsd
=420 mm, dso = 160 mm, u0= 900, v0= 500,
=2°,
=2°,
and
=−2°. The dual parameters were ⌬z=5 mm for the
z-axis displacement and
␣
=90° between the central rays of
the two systems. We again performed ten simulations with
Gaussian noise with standard deviation 0.4 pixels added to
the simulated bead projection centers.
For the validation of our calibration method, experiments
involving our dual tube/detector system were also per-
formed. We performed three sets of scans with the system,
with the parameters of 80 kVp, 100 mA, and 10 ms per
exposure. The small focal spot of 0.6 mm was used, and we
set the sampling distances to approximately dsd =750 mm
TABLE I. Calibration results for the single system parameters, compared with
results in similar conditions by Yang et al. 共Ref. 18兲, from ten computer
simulations, with Gaussian noise of standard deviation 0.4 pixels added to
simulated bead projection centers, 500 images, eight beads.
True values Our method Yang’s method
dsd 共mm兲400.00 399.99⫾0.06 401 ⫾1
dso 共mm兲150.00 149.62⫾0.06 150.2 ⫾0.5
u0共pixel兲1005.0 1005.0⫾0.0 1005.9 ⫾0.3
v0共pixel兲480.00 479.90⫾0.15 480 ⫾1
共°兲−1.0000 −1.0001 ⫾0.0002 −0.99⫾0.03
共°兲1.2000 1.1961⫾0.0116
共°兲1.5000 1.5018⫾0.0046
FIG. 6. Our in-house implemented dual tube/detector micro-CT system.
1826 Johnston, Johnson, and Badea: Dual tube/detector micro-CT system geometric calibration 1826
Medical Physics, Vol. 35, No. 5, May 2008
and dso =650 mm to ensure that the penumbral blurring
caused by focal spot is less than the detector pixel size of
0.88 mm. In the first scan, we acquired 360 images with each
imaging chain over a 360° rotation of the calibration phan-
tom. Next we acquired 372 projections with each imaging
chain over a 186° 共180° +fan angle of 6°兲scan angle of a
cylindrical phantom containing water. Finally, we acquired
372 projections over a 186° scan angle of a dead C57BL/6
mouse.
The images from the first scan were provided as input to
the calibration program, and the system parameters were ob-
tained. These parameters were then used to reconstruct the
objects in the next two scans. Single detector images were
reconstructed using 372 projections acquired with the same
imaging chain. For dual tube/detector reconstructions we
used 186 projections acquired with the first imaging chain
over 93° and 186 projections over the other 93° acquired
with the second imaging chain. All projection images are
corrected for distortions by the acquisition software of the
detectors. The reconstructed data of the cylinder phantom
was used to calculate the modulation transfer function of the
individual and dual systems according to the method de-
scribed in the ASTM.25
III. RESULTS
The values estimated in the simulation by our calibration
program for the geometric parameters of an individual sys-
tem are shown alongside the values found by Yang et al.18 in
Table I. Overall the two sets of results compare well and they
show similar performance in the noise-affected situation. Un-
like Yang’s method, note that our method also gives esti-
mates of two detector rotation angles
and
. This is pos-
sible due to the refinement part based on optimization.
Next, the values estimated for the dual tube/detector sys-
tem parameters are shown in Table II. Again, the results pro-
vided by the calibration procedure match the known values
quite well.
The geometric parameters estimated for our dual tube/
detector system are shown in Table III. While the real values
of these variables are unknown, we can judge the perfor-
mance of the calibration results by the image quality of the
reconstructions. We know that imperfect calibration would
cause double contours, and blur the reconstructed images.
Therefore, we used the modulation transfer functions
共MTFs兲as a more quantitative figure of merit to assess the
performance of the calibration. Figure 7presents the MTFs
plots for the following: single detector reconstruction using
refined parameters; single detector reconstruction using un-
refined parameters as in Yang’s method, i.e., without the two
angles
and
; dual tube/detector reconstruction using re-
fined parameters; and dual tube/detector reconstruction using
unrefined parameters. The MTF at 10% appears to be about
3.4 lp/mm for refined single detector reconstructions and 3.3
lp/mm for unrefined single detector reconstructions, 2.3
lp/mm for refined dual tube/detector reconstructions, and 1.9
lp/mm for unrefined dual tube/detector reconstructions.
Finally, Fig. 8displays micro-CT images of slices in
axial, coronal and sagittal orientations of the mouse head
TABLE II. Calibration results for the dual system parameters from ten computer simulations, with Gaussian
noise of standard deviation 0.4 pixels added to simulated bead projection centers, 500 images, eight beads.
System 1 True values Estimates System 2 True values Estimates
dsd 共mm兲dsd 共mm兲400.00 399.99⫾0.06 dsd 共mm兲420.00 419.84 ⫾0.14
dso 共mm兲150.00 149.62⫾0.06 dso 共mm兲160.00 159.96 ⫾0.10
u0共pixel兲1005.0 1005.0⫾0.0 U0共pixel兲900.00 899.99 ⫾0.01
v0共pixel兲480.00 479.90⫾0.15 V0共pixel兲500.00 500.19 ⫾0.18
共°兲−1.0000 −1.0001 ⫾0.0002
共°兲2.0000 2.0000⫾0.0002
共°兲1.2000 1.1961⫾0.0116
共°兲2.0000 1.9997⫾0.0065
共°兲1.5000 1.5018⫾0.0046
共°兲−2.0000 −1.9985 ⫾0.0029
Dual True values Estimates
⌬z共mm兲5.0000 4.9947⫾0.0062
␣
共°兲90.000 4.9947⫾0.0062
TABLE III. Calibration results for our dual micro-CT system.
System 1 Estimates System 2 Estimates Dual Estimates
dsd 共mm兲808.80 dsd 共mm兲753.22 ⌬z共mm兲1.2357
dso 共mm兲706.41 dso 共mm兲653.32
␣
共°兲−90.846
U0共pixel兲543.95 u0共pixel兲459.43
v0共pixel兲326.80 v0共pixel兲356.20
共°兲1.9983
共°兲1.4183
共°兲−4.4962
共°兲2.1522
共°兲1.1031
共°兲2.7240
1827 Johnston, Johnson, and Badea: Dual tube/detector micro-CT system geometric calibration 1827
Medical Physics, Vol. 35, No. 5, May 2008
using the single and dual tube/detector reconstructions. Al-
though projections from both imaging chains were used in
the reconstruction of the dual tube/detector micro-CT im-
ages, they show no misalignment artifacts and the image
quality of single and dual chain micro-CT are comparable,
visual proof that the calibration method performed well.
IV. DISCUSSION
We have presented here a geometric calibration method
suitable for a dual tube/detector micro-CT system. We are
not aware of other published prior work on this subject.
Since our method involves the geometric calibration of each
individual chain, we could compare part of our results with
the previous work.18 Table Idemonstrates that our method
finds the calibration parameters with good accuracy and pre-
cision, and improves upon the results of previous work for
single chain imaging systems. We find values for dsd,u0,v0,
and
that are both closer to the correct values and have less
variance in the presence of noise than the method of Yang et
al.18 The values for dso are within the margin of error of
previous methods, but are not improved by the optimization,
so we typically exclude dso from the optimization process.
The values for angles
and
that were not estimated with
other methods are now found accurately. Although we par-
tially use the same formulas as Yang et al.18 to initially esti-
mate five of the geometric parameters, our method for single
chain calibration shows advantages due to added refinement
based on optimization. Table II demonstrates that we also
find accurate and precise values for the dual tube/detector
system parameters including ⌬zand
␣
. The discrepancy be-
tween the two chains in the accuracy of estimation of dso
further indicates the fragility of this parameter estimation.
The optimization-based refinements improve the quality
of reconstruction as shown by the MTF plots 共see Fig. 7兲.
The dual tube/detector system does not match the quality of
the single chain, but is reasonably close. We suspect that this
loss in quality is due to the compounding of slight errors in
the separate single chain calibrations, since the impact of
optimization is much stronger on the dual tube/detector MTF
than the single tube/detector MTF. Further reductions in im-
FIG. 7. A plot of the modulation trans-
fer functions for the reconstructions
with different calibration parameters
for the single and dual micro-CT
systems.
FIG. 8. Axial 关共a兲,共b兲兴, sagittal 关共c兲,共d兲兴, and coronal 关共e兲,共f兲兴 slices from a
reconstruction of a mouse from a single chain micro-CT system 关共a兲,共c兲,共e兲兴
and a dual micro-CT system 关共b兲,共d兲,共f兲兴.
1828 Johnston, Johnson, and Badea: Dual tube/detector micro-CT system geometric calibration 1828
Medical Physics, Vol. 35, No. 5, May 2008
age quality are caused by detector distortion, which is not
completely corrected by the imaging software. We will ad-
dress this issue in future work.
Figure 8demonstrates that reconstructions from the dual
micro-CT system look very much like the reconstructions
from a single micro-CT system, and both reconstructions
show few misalignment artifacts.
In our step-and-shoot acquisition scheme, with each tube/
detector acquiring one quadrant, the radiation dose should be
the same as in the single tube/detector system.
Although the method shown here was tested for our dual
tube/detector micro-CT system that is built with a rotating
object geometry, we believe that the method could be
adapted for rotating gantry geometry and could be used with
other cone beam CT systems. We hypothesize that our
method could be extended to correct for other sources of
misalignment artifacts, such as reproducible wobbling gantry
motion in C-arm-based systems.26 This could be accom-
plished by adding an angle-dependent perturbation parameter
that could be found in the optimization step in the same
manner as
and
. Further work would be required for the
validation of this hypothesis.
V. CONCLUSIONS
We have developed a method that accurately finds the
parameters necessary to perform reconstruction with dual
micro-CT imaging systems. The method is currently used
with a newly developed dual micro-CT system and is robust.
The accuracy of the parameters estimated for the individual
sources and detectors is equal to or higher than the accuracy
of previous single micro-CT methods, and the parameters
found for the combined system enable accurate reconstruc-
tions free of misalignment artifacts.
ACKNOWLEDGMENTS
All work was performed at the Duke Center for In Vivo
Microscopy, NCRR National Biomedical Technology Re-
source Center 共P41 RR005959兲, with additional support from
NCI 共R21 CA124584-01, U24 CA092656兲.
a兲Author to whom correspondence should be addressed. Telephone: 919
684-7509. Electronic mail: chris@orion.duhs.duke.edu
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