full article….

Artifacts associated with implementation of the Grangeat formula
Seung Wook Leea)
CT/Micro-CT Laboratory, Department of Radiology, University of Iowa, Iowa City, Iowa 52242 and
Department of Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology,
Daejeon, 305-701, South Korea
Gyuseong Cho
Department of Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology,
Daejeon, 305-701, South Korea
Ge Wangb)
CT/Micro-CT Laboratory, Department of Radiology, University of Iowa, Iowa City, Iowa 52242
共Received 29 April 2002; accepted for publication 30 September 2002; published 27 November 2002兲
To compensate for image artifacts introduced in approximate cone-beam reconstruction, exact
cone-beam reconstruction algorithms are being developed for medical x-ray CT. Although the exact
cone-beam approach is theoretically error-free, it is subject to image artifacts due to the discrete
nature of numerical implementation. We report a study on image artifacts associated with the
Grangeat algorithm as applied to a circular scanning locus. Three types of artifacts are found, which
are thorn, wrinkle, and V-shaped artifacts. The thorn pattern is created by inappropriate extrapolation into the shadow zone in the radon domain. If the shadow zone is filled in with continuous data,
the thorn artifacts along the boundary of the shadow zone can be removed. The wrinkle appearance
arises if interpolated first derivatives of the radon data are not smooth between adjacent detector
planes. In particular, the nearest-neighbor interpolation method should not be used. If the number of
projections is not small, the bilinear interpolation method is effective to suppress the wrinkle
artifacts. The V-shaped artifacts on the meridian plane come from the line integrations through the
transition zones where derivative data change abruptly. Two remedies are to increase the sampling
rate and suppress data noise. © 2002 American Association of Physicists in Medicine.
关DOI: 10.1118/1.1522748兴
Key words: artifact, exact reconstruction, computed tomography 共CT兲, cone-beam CT
I. INTRODUCTION
The CT technology is rapidly evolving from fan-beam to
cone-beam geometry. The two key components are area detectors and reconstruction algorithms. While various types of
area detectors are under development,1 cone-beam reconstruction algorithms are improved in several ways. Wang
et al. categorized the cone-beam reconstruction algorithms
into three categories: exact, approximate, and iterative algorithms, with general comments on their advantages and
disadvantages.2 To compensate for image artifacts introduced
in approximate reconstruction for large cone angles, exact
algorithms are investigated for medical x-ray CT. Although
the exact approach is theoretically error-free, it is also subject to image artifacts due to the discrete nature of numerical
implementation. However, results on artifacts of the exact
reconstruction approach are rare, although we need the artifact characteristics of exact reconstruction for practical use.
The importance of artifact studies was demonstrated in many
situations, and substantial efforts were made to overcome
them in fan-beam reconstruction and approximate conebeam reconstruction.3–11 Hence, we are motivated to study
the artifact behaviors in exact cone-beam reconstruction.
In this paper, we focus on the Grangeat formula as applied
to a circular trajectory, and analyze the artifacts. The
2871
Med. Phys. 29 „12…, December 2002
Grangeat formula is selected because it is widely implemented and geometrically clear.12 In Sec. II, we review the
3D radon transform and the Grangeat formula. In Sec. III, we
summarize each stage of the exact algorithm and present
several implementation conditions. In Sec. IV, we demonstrate three types of image artifacts, explain their causes, and
suggest corrective means. In Sec. V, we discuss relevant issues, and conclude the paper.
II. OVERVIEW OF THE GRANGEAT FORMULA
A. 3D radon transform
As shown in Fig. 1, the 3D radon transform of a 3Dfunction f (x̄) is defined by
R f 共 ␳ n̄ 兲 ⫽
冕 冕 冕
⬁
⬁
⬁
⫺⬁
⫺⬁
⫺⬁
f 共 x̄ 兲 ␦ 共 x̄•n̄⫺ ␳ 兲 dx̄,
共1兲
where n̄ is the unit vector which passes through the characteristic point C described by the spherical coordinate
( ␳ , ␪ , ␸ ), x̄ the Cartesian coordinates (x,y,z). Equation 共1兲
means that the radon value at C is the integral of the object
function f (x̄) on the plane normal to the vector ␳ n̄. It is well
known that the 3D function f (x̄) can be reconstructed from
0094-2405Õ2002Õ29„12…Õ2871Õ10Õ$19.00
© 2002 Am. Assoc. Phys. Med.
2871
2872
Lee, Cho, and Wang: Implementation of the Grangeat formula
2872
B. Cone-beam data to derivative data in the radon
domain
For cone-beam tomography, it is critically important to
link cone-beam projection to 3D radon data. Smith, Tuy, and
Grangeat found such links separately.12,15,16 Grangeat’s formulation is geometrically attractive and becomes popular.
Mathematically, as shown in Fig. 2, the link can be expressed
as follows:
⳵
R f 共 ␳ n̄ 兲 ⬅R ⬘ f 共 ␳ n̄ 兲
⳵␳
FIG. 1. 3D radon transform parameters (x,y,z) represent the Cartesian coordinates, and ( ␳ , ␪ , ␸ ) the spherical coordinate. The 3D radon value at C is
equal to the plane integration through object f (x̄) which is orthogonally
through the vector ␳ n̄.
R f ( ␳ n̄) provided that R f ( ␳ n̄) is available for all planes
through a neighborhood of point x̄ 13,14 as follows:
f 共 x̄ 兲 ⫽
⫺1
8␲2
冕
␲ /2
␪ ⫽⫺ ␲ /2
冕
2␲
␸ ⫽0
⳵2
⳵␳ 2
R f 共共 x̄•n̄ 兲 n̄ 兲 兩 sin ␪ 兩 d ␸ d ␪ .
共2兲
⫽
⳵
2
⳵
cos ␤ s
冕
⫽
⳵
cos ␤ ⳵ s
冕
1
1
2
⬁
⫺⬁
⬁
⫺⬁
SO
X f 关 s 共 ␳ n̄ 兲 ,t 兴 dt
SA
X w f 关共 s 共 ␳ n̄ 兲 ,t 兴 dt,
共3兲
where X f 关 s( ␳ n̄),t 兴 is the detector value a distance of s away
from the center of detector O along the line t perpendicular
to OC D , SO denotes the distance between the source and the
origin, SA the distance between the source and an arbitrary
point A along t, ␤ the angle between the line SO and SC, and
FIG. 2. Meridian and detector planes. 共a兲 Relationship
between meridian and detector planes, 共b兲 a meridian
plane, and 共c兲 a detector plane. (x,y,z): reconstruction
system; (r,z): Cartesian coordinates on the meridian
plane; (p,q): Cartesian coordinates on the detector
plane; C: characteristic point on the meridian plane;
C D : line integration point. The source and detector system is described by the vectors ū, v̄ ,w̄. The first derivative at C is acquired by the line integration along t
through C D . C is described by the spherical coordinate
共␳,␪,␸兲, and C D by the spherical coordinate by (s, ␣ , ␺ ).
Medical Physics, Vol. 29, No. 12, December 2002
2873
Lee, Cho, and Wang: Implementation of the Grangeat formula
2873
FIG. 3. Schematic flowchart of the
Grangeat algorithm.
X w f 关 s( ␳ n̄),t 兴 ⫽(SO/SA)X f 关 s( ␳ n̄),t 兴 . Given a characteristic point C in the radon domain, the plane orthogonal to the
vector ␳ n̄ is determined. Then, the intersection point共s兲 of
the plane with the source trajectory S(␭) can be found, and
the detector plane D ␺ specified, on which the line integration
is performed. Let C D denote the intersection of the detector
plane D ␺ with the ray that comes from S(␭) and goes
through C. The position C D can be described by a vector
sn̄ D . To compute the derivative of radon value at C, the line
integration is performed along t, which is orthogonal to the
vector sn̄ D .
For a digital implementation of Eq. 共3兲, the derivative in
the s direction is reformulated as the sum of its two horizontal and vertical components,
TABLE I. Parameters of the phantoms used in our numerical simulation.
Phantom
a
b
c
x
y
z
␪
␸
Density
3D Shepp and Logan
0.69
0.6624
0.41
0.31
0.21
0.046
0.046
0.046
0.056
0.056
0.046
0.023
0.9
0.88
0.21
0.22
0.35
0.046
0.02
0.02
0.1
0.1
0.046
0.023
0.92
0.874
0.16
0.11
0.25
0.046
0.023
0.023
0.04
0.056
0.046
0.023
0.0
0.0
⫺0.22
0.22
0.0
0.0
⫺0.08
0.06
0.06
0.0
0.0
0.0
0.0
0.0
⫺0.25
⫺0.25
⫺0.25
⫺0.25
⫺0.25
⫺0.25
0.625
0.0625
⫺0.25
⫺0.25
0.0
⫺0.0184
0.0
0.0
0.35
0.1
⫺0.605
⫺0.605
⫺0.105
0.1
⫺0.1
⫺0.605
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
72.0
⫺72.0
0.0
0.0
0.0
90.0
0.0
0.0
0.0
0.0
2.0
⫺0.98
⫺0.02
⫺0.02
0.01
0.01
0.01
0.01
0.02
⫺0.02
0.01
0.01
One sphere
0.9
0.9
0.9
0.0
0.0
0.0
0.0
0.0
2.0
Two sphere
0.9
0.45
0.9
0.45
0.9
0.45
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
2.0
⫺2.0
Three sphere
0.9
0.6
0.3
0.9
0.6
0.3
0.9
0.6
0.3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
2.0
⫺2.0
2.0
Medical Physics, Vol. 29, No. 12, December 2002
2874
Lee, Cho, and Wang: Implementation of the Grangeat formula
2874
III. IMPLEMENTATION IN THE CIRCULAR
SCANNING CASE
A. Two phases
The implementation of the Grangeat formula is divided
into two phases. The first derivatives of radon data are generated from cone-beam projections in phase 1. Images are
reconstructed in phase 2. The recipe is depicted in Fig. 3.
1. Phase 1
FIG. 4. Shadow zone in the radon space associated with a circular scanning
trajectory.
⳵
⳵
X f 关 s 共 ␳ n̄ 兲 ,t 兴 ⫽cos ␣
X f 关 s 共 ␳ n̄ 兲 ,t 兴
⳵s w
⳵p w
⫹sin ␣
⳵
X f 关 s 共 ␳ n̄ 兲 ,t 兴 ,
⳵q w
共4兲
where p and q are the Cartesian axes, s and ␣ define a polar
system on the detector plane 关see Fig. 2共c兲兴. Substituting Eq.
共4兲 into Eq. 共3兲 yields
R ⬘ f 共 ␳ n̄ 兲 ⫽
1
cos ␤
2
冉 冕
cos ␣
冕
⳵
X w f 共 s 共 ␳ n̄ 兲 ,t 兲 dt
⳵
⫺⬁ p
⬁
⳵
X w f 共 s 共 ␳ n̄ 兲 dt 兲 ,
⫹sin ␣
⳵
⫺⬁ q
⬁
where cos ␤ ⫽SO/SC D .
According to Eq. 共5兲, projection data are weighted by the
factor SO/SA 共P1.1 in Fig. 3兲. This cosine weighting is common in analytic cone-beam reconstruction. The weighted
projection data are then differentiated in p and q directions
共P1.2 in Fig. 3兲. The purpose of plane 1 is to generate the
first derivative data in the radon domain for phase 2. Sampling in spherical coordinates 共␳,␪,␸兲 is most convenient for
subsequent parallel backprojection, therefore the characteristics points are specified in polar coordinates 共␳,␪兲 on a meridian plane.
To obtain the first derivative data at a desired sampling
location, we need a geometric relationship between the radon
characteristic point C and the line integration characteristic
point C D . For a radon characteristic point C at 共␳,␪,␸兲, we
should find a corresponding line integration characteristic
point C D at (s, ␣ , ␺ ), where (s, ␣ ) is its polar coordinates on
the detector plane D ␺ with an angle ␺ measured from y axis.
Mathematically, the relationship for the rebinning process is
expressed as 共P1.3 in Fig. 3兲:12,17
s共 ␳ 兲⫽
共5兲
␳ SO
冑SO 2 ⫺ ␳ 2
␪
␣ 共 ␳ , ␪ 兲 ⫽tan⫺1
兩␪兩
共6兲
,
1
冑共 SO
2
⫺ ␳ 兲 / 共 SO 2 cos2 ␪ 兲 ⫺1
2
,
共7兲
FIG. 5. Illustration of the cause of the thorn pattern artifacts. 共a兲 First derivative of radon data on a meridian plane with zero padding in the shadow zone 共b兲,
the profile of the line marked on 共a兲, 共c兲 the derivative filtered data of 共b兲.
Medical Physics, Vol. 29, No. 12, December 2002
2875
Lee, Cho, and Wang: Implementation of the Grangeat formula
2875
FIG. 6. Thorn artifacts with different data filling strategies. 共a兲 Zero padding, 共b兲 horizontal 共␪ direction兲 constant padding, 共c兲 horizontal linear interpolation,
and 共d兲 horizontal quadratic interpolation. 共e兲 Vertical 共␳ direction兲 constant padding, 共f兲 vertical linear extrapolation, and 共g兲 vertical quadratic extrapolation.
First row: first derivative data on the meridian plane; second row: second derivative data on the meridian plane; third row: backprojection image on the
meridian plane; fourth row: vertical slice at y⫽0.26. 共Contrast range: 1.005–1.04.兲
FIG. 7. 共a兲 The profiles of the reconstructed images using interpolations in the ␪ direction, and 共b兲 in the ␳ direction.
Medical Physics, Vol. 29, No. 12, December 2002
2876
Lee, Cho, and Wang: Implementation of the Grangeat formula
2876
FIG. 8. Illustration of the cause of the wrinkle artifacts. 共a兲 Nearestneighborhood, and 共b兲 linear interpolation.
冋
␺ 共 ␳ , ␪ , ␸ 兲 ⫽ ␸ ⫹sin⫺1
册
␳
.
SO sin ␪
共8兲
Once we identify the line integration characteristic point,
the line integration is performed along the line t on the detector plane. The integration line is sampled at the intersection points with either the columns or the rows of the projection grid, depending on the value of ␣共␳,␪兲. If 兩tan ␣兩⬎1,
the intersection points with the columns of the grid are utilized; otherwise, the intersection points with the rows of the
grid are chosen.12,18 The projection value at the sampling
point is 1D-interpolated along the column or row direction
共P1.4 in Fig. 3兲. In addition to the data interpolation on the
detector plane, we need to interpolate data between detector
planes since the detector plane at ␺ ( ␳ , ␪ , ␸ ) is generally unavailable. The data are postweighted by 1/cos2 ␤ in the last
step in phase 1 共P1.5 in Fig. 3兲.
FIG. 9. Wrinkle artifacts with different interpolation methods. 共a兲 Nearestneighborhood interpolation, and 共b兲 linear interpolation. First row: Second
derivative of radon data on the meridian plane; second row: backprojection
image on the meridian plane; third row: Central transverse slice; fourth row:
vertical slice at y⫽0.26. 共Contrast range: 1.015–1.025.兲
2. Phase 2
In phase 2, the first derivatives in the radon domain from
phase 1 are used as input data. The data are derivativefiltered along ␳ direction 共P2.1 in Fig. 3兲. Then, the parallel
backprojection of the filtered data is performed on each meridian phase 共P2.2 in Fig. 3兲. Finally, the subsequent backprojection is performed on each horizontal plane 共P2.3 in
Fig. 3兲.
Medical Physics, Vol. 29, No. 12, December 2002
B. IDL simulator
We developed a software simulator in the IDL Language
共Research Systems, Inc., Boulder, CO兲 to study the Grangeat
formula. In the implementation of the Grangeat formula, the
numerical differentiation is performed with a differentiation
function based on three-point Lagrangian interpolation, em-
2877
Lee, Cho, and Wang: Implementation of the Grangeat formula
2877
FIG. 10. Illustration of the cause of the V-shaped artifacts. 共a兲 V shape artifacts on the meridian plane; and V
shape artifacts propagated to the reconstructed image at
共b兲 y⫽0, 共c兲 y⫽0.26. 共Contrast range: 1.015–1.025.兲
bedded in the IDL. The 3D Shepp and Logan phantom and
other mathematical phantoms were used in the simulations as
shown in Table I.
The numbers of samples were studied for 2D parallelbeam CT.19 It was reported that the relationship between the
number of rays in each projection and the number of projections should be M proj⬇( ␲ /2)N ray . We can extend this to 3D
parallel beam, which is equivalent to the cone-beam geometry with an infinite source to origin distance. In addition to
samples in the spatial domain, samples in the radon domain
are required in the Grangeat framework. The radon domain
consists of meridian planes. The derivative data are required
to be calculated in a polar grid on each plane. Since the
derivative radon data are acquired through line integration on
detector planes, we can apply the above-mentioned sampling
relation to sample the radon space. Then, we have ␶
⫽1/2W, N ray⫽D/ ␶ , M proj⬇( ␲ /2)N ray , N R ⫽N, L R
⫽M proj , and M R ⫽M proj , where W is the bandwidth, D the
diameter of an object, ␶ the sampling interval, N ray the number of rays, M proj the number of projections, N R the number
of samples in the ␳ direction, L R the number of samples in
the ␪ direction, and M R the number of meridian planes. This
consideration only provides a guideline for selecting the
numbers of samples in the Grangeat reconstruction. Based on
the above-mentioned consideration and with some modifications for our purpose, we have chosen the following imaging
parameters for the simulations: the source-to-origin distance
was 3.92, the number of rays per projection 128 by 128, the
size of flat rectangular detector 2.1 by 2.1, the number of
meridian planes 180, and the numbers of radial and angular
samples were 192 and 256, respectively. The maximum radial distance was 2.1. The reconstructed image volume had
dimensions of 2.1 by 2.1, covered by 192 by 192 by 192
voxels.
IV. THREE TYPES OF ARTIFACTS
In our numerical simulation, we found three types of artifacts associated with the Grangeat formula, which are re-
FIG. 11. V shapes from the multiple sphere phantoms:
共a兲 one sphere, 共b兲 two spheres, 共c兲 three spheres. First
row: V-shaped artifacts on the meridian plane; second
row: reconstructed vertical slice at y⫽0. 共Contrast
range: 1.9850–2.0685.兲
Medical Physics, Vol. 29, No. 12, December 2002
2878
Lee, Cho, and Wang: Implementation of the Grangeat formula
2878
FIG. 13. Plot of the V-shaped strength in the first derivative radon domain as
a function of the number of samples per projection.
FIG. 12. Illustrative view of the cause of V-shaped artifacts. 共a兲 Line integration with the projection data filtered in the p direction, 共b兲 line integration
with the projection data filtered in the q direction, 共c兲 trajectory of the data
point related to the transition zones in 共a兲, 共d兲 trajectory of the data point
related to the transition zones in 共b兲, and 共e兲 combination of 共c兲 and 共d兲.
spectively called thorn, wrinkle, and V-shaped artifacts. In
the following, we report the appearance, cause, and remedy
for each of them.
A. Thorn artifacts
In this study, we adopted a circular source trajectory,
which is very common in practice. However, this trajectory
does not satisfy the sufficient condition for exact
reconstruction.15,16 As a result, there exists a shadow zone in
the radon domain, as shown in Fig. 4. This zone is defined by
兩 ␳ 兩 ⬎SO 兩 sin ␪ 兩 .
共9兲
With only a single circular orbit, we must fill in this
shadow zone in some way, e.g., padding zeroes into the region or by extrapolating based on known radon data. Due to
the need for derivative filtering in the ␳ direction, on inadequate extrapolation across the boundary of the shadow zone
may cause some artifacts in reconstructed images, which are
similar to those occurring in local tomography where truncated data are inappropriately handled.20 This type of artifact
Medical Physics, Vol. 29, No. 12, December 2002
is more severe when the cone angle becomes larger or data
are less dense such as with the half-scan Grangeat
algorithm.21 Figure 5 illustrates the cause for the thorn pattern artifacts.
When the first derivative data change abruptly across the
boundary of the shadow zone, the second derivative of the
radon data would have higher values. This phenomenon
would produce stripes in backprojected images on the meridian plane, and consequently in reconstructed images, as
shown in Fig. 6. We tested seven shadow zone filling methods to observe their effects on the thorn pattern artifacts.
Figure 6共a兲 presents the image quality at each stage of the
algorithm when we applied zero padding in the shadow zone.
Figures 6共b兲, 6共c兲, and 6共d兲 show constant 共or nearestneighborhood兲 padding, linear interpolation, and quadratic
interpolation in the ␪ direction, respectively. Figures 6共e兲,
6共f兲, and 6共g兲 are for constant 共or boundary value兲 padding,
linear extrapolation, quadratic extrapolation in the ␳ direction, respectively. The first row of Fig. 6 shows derivative
data
共⳵/⳵␳兲Rf共␳,␪兲.
The
second
row
presents
( ⳵ 2 / ⳵␳ 2 )R f ( ␳ , ␪ ) 兩 sin ␪ 兩 , which is the integrant of Eq. 共2兲. It
can be seen that the thorn artifacts from the zero padding
method significantly degraded the image, while the other six
data-filling methods performed substantially better. However,
there are image quality differences among these six datafilling methods. The representative profiles of each reconstructed image are overlapped together in Fig. 7, which are
extracted from the identical position as marked on the bottom range of Fig. 6共a兲. There is little difference among vertical interpolations but there are noticeable differences
among horizontal extrapolations. Especially, the intensity biases are noticeable with the constant 共or boundary value兲
2879
Lee, Cho, and Wang: Implementation of the Grangeat formula
padding. Further investigation is needed to refine the datafilling scheme for optimal image quality.
B. Wrinkle artifacts
When 共␳,␪兲 are given on a meridian plane M ␸ , a detector
plane D ␺ can be found according to Eq. 共8兲 for the needed
line integration. However, in the digital data acquisition the
exact detector plane D ␺ is generally unavailable. The interpolation between adjacent detector planes is indispensable.
We studied the effects of the two common interpolation
methods on the image quality. These methods are the
nearest-neighbor interpolation and the linear interpolation, as
shown in Fig. 8. In the nearest-neighbor interpolation, the
line integration is performed only on the nearest detector
plane. In the linear interpolation, the line integration is done
by linearly combining data of the two adjacent detector
planes. The former interpolation method is computationally
desirable because a most time-consuming part of the
Grangeat algorithm is the line integration. Nevertheless, we
found that the nearest-neighbor interpolation produced
wrinkle artifacts in the second derivatives of radon data. Figure 9 contains some representative images. In Fig. 9, the first
row presents two images of the first derivative data obtained
from the two interpolation methods, respectively; the second
row shows evident wrinkle artifacts from the nearestneighbor interpolation; the third and fourth rows display reconstructed vertical and horizontal images. Clearly, the linear
interpolation method effectively suppressed these wrinkle artifacts. More sophisticated interpolation methods may further
suppress this type of artifact, but the linear interpolation
should be adequate if the number of projections is not small.
This research is mainly focused on a circular trajectory, but
we think similar artifacts should be found in the Grangeat
reconstruction with discrete vertex sets.22 The interpolation
methods should be optimized to suppress the artifacts in each
of these cases.
C. V-shaped artifacts
In our simulation study, we noticed with great interest that
there exist V-shaped artifacts in the radon domain, which are
also referred to as V-shaped artifacts. They propagate stage
by stage until the final reconstruction. The artifacts are
marked in Fig. 10. The image in Fig. 10共a兲 is
( ⳵ 2 / ⳵␳ 2 )R f ( ␳ , ␪ ) 兩 sin ␪ 兩 , which is again the integrant of the
double integral Eq. 共2兲. Two reconstructed slices are given in
Figs. 10共b兲 and 10共c兲.
To reveal the cause for the V-shaped artifacts, we simulated with mathematical phantoms consisting of single and
multiple spheres, as defined in Table II. As compared to the
3D Shepp–Logan phantom, the sphere phantoms are advantageous in several ways. Because the sphere is symmetric,
the radon data on all the meridian planes are identical. This
property helps save the simulation time greatly, since we
only need to do line integrations on one meridian plane.
Also, interpolation between detector planes is no longer
needed. Therefore, the wrinkle artifacts cannot be introduced.
Medical Physics, Vol. 29, No. 12, December 2002
2879
The simulation with these phantoms graphically revealed
the cause of the V-shaped artifacts, as shown in Fig. 11. We
found that the number of paired V shapes in the radon domain is equal to the number of spheres. The propagation of V
shapes to reconstructed images is also an interesting phenomenon. In addition to the structured interfere, dc shifts
were also observed in the reconstructed images of the phantoms of multiple spheres.
1. Cause for the V-shaped artifacts
V-shaped artifacts came from the line integrals through
the transition zones where derivative data change abruptly,
which are numerically unstable. To simplify the situation, we
let SO be infinite. Then, we have
pSO
s 共 ␳ 兲 ⫽ lim
SO→⬁
冑SO 2 ⫺ ␳ 2
⫽␳,
␪
␣ 共 ␳ , ␪ 兲 ⫽ lim tan⫺1
兩
␪
兩
SO→⬁
1
冑共 SO
␪
1
⫽tan⫺1
,
兩 ␪ 兩 兩 tan ␪ 兩
冉
共10兲
⫺ ␳ 兲 / 共 SO 2 cos2 ␪ 兲 ⫺1
2
共11兲
冋
␺ 共 ␳ , ␪ , ␸ 兲 ⫽ lim ␸ ⫹sin⫺1
SO→⬁
2
␳
SO sin ␪
册冊
⫽␸.
共12兲
Therefore, the characteristic radon point C and the line integration point C D coincide as the detector plane and the meridian plane do. A schematic explanation for the V-shaped
artifacts is given in Fig. 12.
The shadowed disks in Figs. 12共a兲 and 12共b兲, respectively,
denote horizontally and vertically derivative-filtered projection data of a single sphere phantom of diameter D. ␳ 1 , ␳ 2 ,
␳ 3 , ␳ 4 , and ␳ 5 are the radial axes in the meridian plane of
Fig. 2共b兲, with the azimuth angle ␪ of ⫺90°, ⫺45°, 0°, 45°,
and 90° measured clockwise from the vertical axis. The dots
on the radial axes are the line integration points whose corresponding paths go through the transition zones where derivative data change dramatically. Each transition zone is
marked with a small circle, along with the derivative filtering
direction. In the orthogonal system constructed by the radial
and azimuth angle axes, we have Figs. 12共c兲 and 12共d兲,
which are then combined into Fig. 12共e兲 by the weighted
summation according to Eq. 共5兲. Clearly, the patterns of the
dots in Fig. 12共e兲 are close to the V shapes we perceived in
Figs. 10 and 11. We emphasize that although we have explained the cause of the V-shaped artifacts in the case of an
infinite SO for simplicity, the same mechanism is responsible
for the V-shaped artifacts in the case of a finite SO, except
for some degree of shape distortion. Since spherical structures are quite common in practice, we expect to encounter
this type of artifact fairly often.
2. Quantification of the V-shaped artifacts
The V-shaped artifacts are considered directly related to
the aliasing phenomenon. Therefore, we were motivated to
2880
Lee, Cho, and Wang: Implementation of the Grangeat formula
quantify this type of artifact as a function of the number of
samples on the detector plane. We targeted at a single sphere
phantom, and set the number of samples on the detector
plane to N⫻N⫽32⫻32, 64⫻64, 96⫻96, 128⫻128, 256
⫻256, 512⫻512, and 1024⫻1024, respectively. Other sampling numbers were selected to minimize other possible artifacts on the meridian plane. Specifically, we set the sampling
numbers in the ␳ and ␪ directions to 1.5⫻1024 and 2⫻1024,
respectively. We found that the V-shaped artifacts were
weakened as the number of samples on the detector plane
increased. Taking the image reconstructed with the sampling
under 1024⫻1024 as the standard, the mean squared error of
the V-shaped artifacts was measured by
␴ ⫽
2
1
1.5⫻2⫻10242
1.5⫻1024 2⫻1024
兺i
兺j
共 R N⬘ 共 ␳ i , ␪ j 兲
⬘ 共 ␳ i , ␪ j 兲兲 2 .
⫺R 1024
共13兲
The simulated curve is plotted in Fig. 13.
V. DISCUSSIONS AND CONCLUSIONS
Similar to previous artifact studies in other contexts,3–11
our studies on artifacts associated with the Grangeat algorithm should be valuable to guide clinical and other conebeam CT applications. Artifacts studies on exact cone-beam
algorithms are relatively sparse, because the exact conebeam approach has a much shorter track record in applications than that of fan-beam methods and approximate conebeam methods. However, efforts along this direction must be
taken to gain a full understanding of the performance of the
exact algorithms and optimize their performance for healthcare benefits.
Although the Grangeat formula was studied only in the
circular scanning case, the experimental design can be
adapted for other scanning loci and even other exact conebeam image reconstruction algorithms. When the Grangeat
algorithm is applied to a circular locus, the reconstruction is
exact, but we believe that the artifact mechanisms we have
discussed should be the same, hence the artifacts associated
with a complete locus should be similar. It would be particularly valuable to generalize the results into the spiral/helical
cone-beam geometry. By doing so, both common and unique
features of image artifacts may be discovered and compensated for.
In conclusion, we have discovered three types of image
artifacts associated with the Grangeat algorithm in the circular scanning case. Three types of artifacts are thorn, wrinkle,
and V-shaped artifacts. The characteristics, causes, and remedies of these artifacts have been revealed in the numerical
simulation. Further work is under way to extend our results
in general and practical situations.
ACKNOWLEDGMENTS
We would like to thank Ming Jiang and Christopher W.
Piker for setting up the computing environment and John F.
Medical Physics, Vol. 29, No. 12, December 2002
2880
Meinel for helpful discussions and comments on this paper.
This work is supported in part by an NIH grant 共No. R01
DC03590兲 and a development grant from the University of
Iowa.
a兲
Electronic mail: [email protected]
Electronic mail: [email protected]
1
M. J. Yaffe and J. A. Rowlands, ‘‘X-ray detectors for digital radiography,’’ Phys. Med. Biol. 42, 1–39 共1997兲.
2
G. Wang, C. R. Crawford, and W. A. Kalender, ‘‘Multirow detector and
cone-beam spiral/helical CT,’’ IEEE Trans. Med. Imaging 19, 817– 821
共2000兲.
3
M. Gado and M. Phelps, ‘‘The peripheral zone of increase density in
cranial computed tomography,’’ Radiology 117, 71–74 共1975兲.
4
P. J. Joseph and R. D. Spital, ‘‘Method for correcting bone induced artifacts in computed tomography scanners,’’ J. Comput. Assist. Tomogr. 3,
52–57 共1978兲.
5
G. Wang, T. H. Lin, P. C. Cheng, and T. M. Shinozaki, ‘‘A general
cone-beam reconstruction algorithm,’’ IEEE Trans. Med. Imaging 12,
483– 496 共1993兲.
6
G. Wang and M. W. Vannier, ‘‘Stair-step artifacts in three-dimensional
helical CT: An experimental study,’’ Radiology 191, 79– 83 共1994兲.
7
J. Hsieh, in Medical CT and Ultrasound: Current Technology and Applications, edited by L. W. Goldman and J. B. Fowlkes 共American Association of Physicists in Medicine, 1995兲.
8
J. H. Siewerdsen and D. A. Jaffray, ‘‘Cone-beam computed tomography
with a flat-panel imager: Effects of image lag,’’ Med. Phys. 26, 2635–
2647 共1999兲.
9
J. Hsieh, R. C. Molthen, C. A. Dawson, and R. H. Johnson, ‘‘An iterative
approach to the beam hardening correction in cone beam CT,’’ Med.
Phys. 27, 23–29 共2000兲.
10
X. Tang, R. Ning, R. Yu, and D. Conover, ‘‘Cone beam voltage CT image
artifacts caused by defective cells in x-ray flat panel imagers and the
artifact removal using a wavelet-analysis-based algorithm,’’ Med. Phys.
28, 812– 825 共2001兲.
11
T. Kohler, R. Proksa, C. Bontus, M. Grass, and J. Timmer, ‘‘Artifact
analysis of approximate helical cone-beam CT reconstruction algorithm,’’
Med. Phys. 29, 51– 64 共2002兲.
12
P. Grangeat, in Mathematical Framework of Cone Beam 3D Reconstruction via the First Derivative of the Radon Transform, Oberwolfach, 1990
共Lecture notes in Mathematics, Springer, Verlag, 1991兲, pp. 66 –97.
13
F. Natterer, The Mathematics of Computerized Tomography 共SIAM,
Philadelphia, 1986兲.
14
S. R. Deans, The Radon Transform and Some of its Applications 共WileyInterscience, New York, 1983兲.
15
B. D. Smith, ‘‘Image reconstruction from cone-beam projections: Necessary and sufficient conditions and reconstruction methods,’’ IEEE Trans.
Med. Imaging MI-4, 14 –25 共1985兲.
16
H. K. Tuy, ‘‘An inversion formula for cone-beam reconstruction,’’ SIAM
共Soc. Ind. Appl. Math.兲 J. Appl. Math. 45, 546 –552 共1983兲.
17
C. Jacobson, Ph.D. dissertation thesis, Linkoeping University, 1996.
18
P. M. Joseph, ‘‘An improved algorithm for reprojecting rays through
pixel images,’’ IEEE Trans. Med. Imaging MI-1, 192–196 共1982兲.
19
A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, 2nd ed. 共SIAM, Philadelphia, 2001兲.
20
B. Ohnesorge, T. Flohr, K. Schwarz, J. P. Heiken, and K. T. Bae, ‘‘Efficient correction for CT image artifacts caused by objects extending outside the scan field of view,’’ Med. Phys. 27, 39– 46 共2000兲.
21
S. W. Lee and G. Wang, ‘‘A Grangeat-type half-scan algorithm for conebeam CT,’’ Med. Phys. 共submitted兲.
22
F. Noo, R. Clack, and M. Defrise, ‘‘Cone-beam reconstruction from general discrete vertex sets using Radon rebinning algorithms,’’ IEEE Trans.
Nucl. Sci. 44, 1309–1316 共1997兲.
b兲