Download Cone-beam mammo-computed tomography from data along two

Cone-beam mammo-computed tomography from data along two tilting arcs
Kai Zeng,a兲 Hengyong Yu,b兲 Laurie L. Fajardo,c兲 and Ge Wangd兲
CT/Micro-CT Laboratory, Department of Radiology, University of Iowa, Iowa City, Iowa 52242
共Received 23 December 2005; revised 19 July 2006; accepted for publication 20 July 2006;
published 13 September 2006兲
Over the past several years there has been an increasing interest in cone-beam computed tomography 共CT兲 for breast imaging. In this article, we propose a new scheme for theoretically exact
cone-beam mammo-CT and develop a corresponding Katsevich-type reconstruction algorithm. In
our scheme, cone-beam scans are performed along two tilting arcs to collect a sufficient amount of
information for exact reconstruction. In our algorithm, cone-beam data are filtered in a shiftinvariant fashion and then weighted backprojected into the three-dimensional space for the final
reconstruction. Our approach has several desirable features, including tolerance of axial data truncation, efficiency in sequential/parallel implementation, and accuracy for quantitative analysis. We
also demonstrate the system performance and clinical utility of the proposed technique in numerical
simulations. © 2006 American Association of Physicists in Medicine. 关DOI: 10.1118/1.2336510兴
Key words: cone-beam CT, mammography, exact reconstruction, Katsevich algorithm
I. INTRODUCTION
Breast cancer is ranked as the second leading cause of cancer
death in women in the United States. It has been recognized
that mass screening and early treatment are extremely important to reduce the mortality of breast cancer. Due to its specificity and sensitivity, x-ray mammography has been the
method of choice for screening and diagnosis.1,2 However,
x-ray mammography is far from being perfect because up to
17% of breast cancers are not identified with mammography,
and normal breasts are associated with 70%–90% of mammograms suspicious of cancers.3 A major limitation of x-ray
mammography is its projective nature, while the real
anatomy and pathology is really in three dimensions 共3D兲. To
address this problem, x-ray tomosynthesis and cone-beam
computed tomography 共CT兲 are two compelling solutions.
Tomosynthesis is a three-dimensional 共3D兲 imaging technique to reconstruct a series of images from a limited number of projections.4 Since its introduction in 1972, the area of
tomosynthesis has been significantly advanced largely due to
the development of the area detectors.5 A primary application
of tomosynthesis is for breast imaging.6–8 The tomosynthetic
algorithms are either analytic or iterative. The analytic algorithms are straightforward and efficient, such as
self-masking,9 selective plane removal,10 and matrix inversion tomosynthesis.11 While the iterative algorithms are robust against noisy data and flexible to integrate prior knowledge, as it is done using algebraic reconstruction
techniques,12,13 expectation-maximization,14 etc. none of
these algorithms can avoid the inherent drawback of tomosynthesis due to the data incompleteness.
Technically speaking, a breast volume should be imaged
very well by cone-beam CT. Since more information of the
object is acquired, the image quality of CT is much better
than tomosynthesis, in terms of contrast resolution, geometrical distortion, etc. The concept of breast CT was proposed
two decades ago,15 but little progress had been made initially
because of compromised image quality and involved radia3621
Med. Phys. 33 „10…, October 2006
tion exposure. Again, thanks to the advancement in the digital detector technology, a number of groups investigated the
feasibility and prototypes of cone-beam mammo-CT.16,17
Nevertheless, the algorithms for breast CT are still based on
the traditional Feldkamp-type algorithms,18 and reconstruct
images approximately with various artifacts.
The fundamental classic results on exact cone-beam CT
reconstruction were achieved by Grangeat,19 Smith,20 and
Tuy.21 The recent breakthroughs on exact cone-beam CT algorithms were reviewed by Zhao et al.22 Up to now, there are
a number of accurate and efficient cone-beam CT algorithms
for various scanning trajectories, such as a helix,23–25 an
arc-plus-line,26,27 a circle-plus-arc,28,29 and a saddle
curve.30,31 Also, there are several algorithms which allow
exact image reconstruction in the case of general
trajectories.32–36
To improve image quality with cone-beam mammo-CT,
we are motivated to design a cone-beam scanning mode that
allows theoretically exact image reconstruction. In this article, we propose a novel scheme for cone-beam mammo-CT,
which is theoretically exact, and develop a corresponding
Katsevich-type reconstruction algorithm. In our scheme,
cone-beam scans are performed along two tilting arcs to collect a sufficient amount of information for exact reconstruction. We derive our corresponding algorithm in the framework established by Katsevich,29,33 which may handle axial
truncation of cone beam data. In what follows, we describe
our system setup and derive the algorithm in Sec. II, describe
numerical simulation results in Sec. III, and discuss relevant
issues and conclude the article in Sec. IV.
II. METHODS AND MATERIALS
A. Cone-beam mammo-CT system
In the proposed cone-beam mammo-CT system 共Fig. 1兲, a
patient lays down on a table with one breast hanging through
a hole. The x-ray tube and a flat-panel camera are fixed to a
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© 2006 Am. Assoc. Phys. Med.
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B. Cone-beam reconstruction method
1. Review of general cone-beam reconstruction
formula
First, we need to review Katsevich’s general cone beam
reconstruction framework briefly.33 Let the scanning locus ⌳
be a finite union of smooth curves defined in R3:
I ª 艛 关al,bl兴 → R3,
l
s 苸 I → y共s兲 苸 R3,兩ẏ共s兲兩 ⫽ 0 on I,
共1兲
where −⬁ ⬍ al ⬍ bl ⬍ ⬁ and ẏ共s兲 ª dy / ds. Assume f is
smooth, compactly supported and identically equals zeros in
a neighborhood of the locus ⌳, the cone beam transform of f
along ⌳ is defined as
FIG. 1. Proposed 3D mammography cone-beam CT system. 共a兲 System configuration and 共b兲 the cross section along the dashed line in 共a兲.
rigid frame such as a C-arm to produce cone-beam projections. Then, a beast can be scanned twice along two tilting
arcs respectively 关Figs. 1共b兲 and 2共a兲兴 by rotating the C-arm
in different scanning planes. The other breast can be scanned
in a similar fashion. Compared with the existing breast CT
systems that only perform approximate reconstruction, our
system for the first time aims at theoretically exact conebeam mammo-CT with fewer artifacts and better accuracy.
Unlike other C-arm based algorithms,27,29 our scanning trajectory is more symmetric, which helps improve image quality in general.
D f 共y, ␤兲 ª
冕
⬁
f共y + ␤t兲dt,
y 苸 ⌳, ␤ 苸 S2 ,
共2兲
0
where S2 is the unit sphere in R3. For given x 苸 R3 and ␰
苸 R3 − 0, we also define
␤共s,x兲 ª
x − y共s兲
,
兩x − y共s兲兩
x 苸 R3 \ ⌳,
s 苸 I,
⌸共x, ␰兲 ª 兵z 苸 R3:共z − x兲 · ␰ = 0其.
共3兲
With y共s j兲 denoting points of intersection of ⌸共x , ␰兲 and ⌳
and ␤⬜ denoting the great circle 兵␣ 苸 S2 : ␣ · ␤ = 0其 which
consists of unit vectors perpendicular to ␤, we introduce the
sets
Crit共s,x兲 ª 兵␣ 苸 ␤⬜共s,x兲:⌸共x, ␣兲 is tangent to ⌳ or
contains an end point of ⌳其
Ireg共x兲 ª 兵s 苸 I:Crit共s,x兲 傺 ␤⬜共s,x兲其
Crit共x兲 ª 艛 Crit共s,x兲.
s苸I
FIG. 2. Scanning geometry for exact cone-beam reconstruction. 共a兲 3D view
of the tilting arcs and the breast and 共b兲 the exact reconstruction region U.
Medical Physics, Vol. 33, No. 10, October 2006
共4兲
To reconstruct f at any fixed x 苸 R3, Katsevich suggested
the following main assumptions about the locus ⌳.33
Property C1. (Tuy’s condition Ref. 21). Any plane through
x intersects ⌳ at least at one point.
Property C2. There exists a constant N1 such that the
number of directions in Crit共s , x兲 does not exceed N1 for any
s 苸 Ireg共x兲.
Property C3. There exists a constant N2 such that the
number of points in ⌸共x , ␣兲 艚 ⌳ does not exceed N2 for any
␣ 苸 S2 − Crit共x兲.
If we only use a subset ⌳共x兲 傺 ⌳ for image reconstruction
at x, ⌳ and I are replaced by ⌳共x兲 and I共x兲 for the earlier
mentioned formulas, respectively.
Let us consider a weighting function n共s , x , ␣兲, s
苸 Ireg共x兲, ␣ 苸 共␤⬜共s , x兲 − Crit共s , x兲兲. If n共s , x , ␣兲 is the piecewise constant and 兺 j:y共s j兲苸⌳艚⌸共x,␣兲n共s j , x , ␣兲 = 1 for almost all
␣ 苸 S2, we arrive at Katsevich’s general inversion formula33
as follows:
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f共x兲 = −
⫻
1
4␲2
冏冕
冕
3623
c 共s,x兲
m
兺
I共x兲 m 兩x − y共s兲兩
2␲
0
⳵
D f 共y共q兲,cos ␥␤共s,x兲
⳵q
+ sin ␥␣⬜共s,x, ␪m兲兲
冏
d␥
ds,
sin ␥
q=s
共5兲
␣⬜共s,x, ␪兲 ª ␤共s,x兲 ⫻ ␣共s,x, ␪兲, ␣共s,x, ␪兲 苸 ␤⬜共s,x兲, 共6兲
where ␪ is a polar angle in the plane perpendicular to ␤共s , x兲
and ␪m 苸 关0 , ␲兲 are the points where ␾共s , x , ␪兲 are discontinuous, and cm共s , x兲 are values of the jumps
cm共s,x兲 ª lim 共␾共s,x, ␪m + ␧兲 − ␾共s,x, ␪m − ␧兲兲,
␧→0+
共7兲
␾共s,x, ␪兲 ª sgn共␣ · ẏ共s兲兲n共s,x, ␣兲,
␣ = ␣共s,x, ␪兲 苸 ␤⬜共s,x兲.
共8兲
2. Existence of PI segments
For our two-tilting-arcs scanning locus 关Fig. 2共a兲兴, we describe our geometry first, and then validate every property
required by Katsevich’s general reconstruction framework.
Let the three components of x 苸 R3 are x1 , x2 , x3, respectively. f共x兲 苸 C⬁0 is a smooth function inside the cylinder x21
+ x22 ⬍ r2. Our scanning orbit ⌳ = Ca 艛 Cb defined on the real
interval I = I1 艛 I2 consists two tilting arcs 关see Fig. 2共a兲兴 is
parameterized by
Ca ª 兵y共s兲:y 1 = R cos共s + smx兲cos t,y 2 = R sin共s + smx兲,y 3
= R cos共s + smx兲sin t,s 苸 I1其,
Cb ª 兵y共s兲:y 1 = R cos共s − smx兲cos t,y 2 = R sin共s − smx兲,y 3
= − R cos共s − smx兲sin t,s 苸 I2其
共9兲
where I1 = 关−␲ / 2 − 2smx , ␲ / 2兴, I2 = 共␲ / 2 , 3␲ / 2 + 2smx兴, R ⬎ r
is the radius of the arcs, t is the tilting angle satisfying
R cos t ⬎ r, and smx = arcsin共r / R cos t兲. smx is the fan angle.
The cone beam projection data of f is defined as in 共Eq. 共2兲兲.
As defined in Appendix A, the reconstruction region Ū can
be decomposed into two kinds of reconstruction zones Ū1
and Ū2 关Figs. 2共b兲 and 14兴. There are exactly two PI segments within Ū2 and there is only one PI segment within Ū1.
Here PI means “␲,” and a PI segment of x is the line segment containing x and its two end points on the scanning
orbit ⌳. Our problem is to reconstruct f inside a region U
which is the intersection of Ū and the cylindrical support
关Fig. 2共b兲兴. In practical clinical applications, the region in the
chest should be avoided and only the breast part is concerned.
Based on the definition of Ū, for any fixed x 苸 Ū, there
exist at least one and at most two PI segments. For each PI
segment, one end point must be on arc A while the other
Medical Physics, Vol. 33, No. 10, October 2006
FIG. 3. Illustration of Pl lines for 共a兲 x 苸 Ū2 and 共b兲 x 苸 Ū1.
must be on arc B if we ignore those points on the scanning
arcs’ planes of zero measure. We denote the corresponding
angular variable as sa = sa共x兲 for arc A and sb = sb共x兲 for arc B.
The existence and uniqueness of these PI segments can be
divided in the two cases: 共1兲 R sin共sa + smx兲 ⬎ x2 ⬎ R sin共sb
− smx兲 and 共2兲 R sin共sa + smx兲 ⬍ x2 ⬍ R sin共sb − smx兲. Moreover,
for any x 苸 Ū2 关Fig. 3共a兲兴, there exist exactly two PI segments with our symmetric imaging geometry. While, for any
x 苸 Ū1 关Figs. 3共b兲 and 14共d兲兴, there exists exactly one PI
segment, either in case 1 or case 2. With our scanning geometry, for a pair of given sa共x兲 and sb共x兲 for x 苸 Ū2
关Fig. 3共a兲兴, the PI segment may correspond to two PI interand
vals Ī1共x兲 = 关−␲ / 2 − smx , sa共x兲兴 艛 关sb共x兲 , 3␲ / 2 + smx兴
mx
mx
Ī2共x兲 = 关sa共x兲 , ␲ / 2 − s 兴 艛 共␲ / 2 + s , sb共x兲兴 yielding two PI
arcs ⌳1 = ⌳1共x兲 = ⌳共Ī1共x兲兲 and ⌳2 = ⌳2共x兲 = ⌳共Ī2共x兲兲 关Fig.
3共a兲兴.
3. Reconstruction method
To derive a theoretically exact image reconstruction algorithm in Katsevich’s framework,33 we need to check the
properties of the locus ⌳ and the weighting function in our
TABLE I. Weighting functions n共s , x , ␣兲 used for our Katsevich-type reconstruction.
Case
Value of n共s , x , ␣兲
1 IP, s1 苸 I1
1 IP, s1 苸 I2
3 IPs, s1 , s2 苸 I1 ; s3 苸 I2
1
1
n共s1 , x , ␣兲 = n共s2 , x , ␣兲 = 1,
n共s3 , x , ␣兲 = −1
n共s1 , x , ␣兲 = n共s3 , x , ␣兲 = 1,
n共s2 , x , ␣兲 = −1
3 IPs, s1 苸 I1 ; s2 , s3 苸 I2 , s2 ⬍ s3;
x is above the plane 关in Ū2, Fig. 2共b兲兴
3 IPs, s1 苸 I1 ; s2 , s3 苸 I2 , s2 ⬍ s3;
x is below the plane 关in Ū1, Fig. 2共b兲兴
n共s1 , x , ␣兲 = n共s2 , x , ␣兲 = 1,
n共s3 , x , ␣兲 = −1
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particular geometry. Let us only consider the PI-arcs ⌳1 to
reconstruct x 苸 Ū2 in the following, while ⌳2 can be similarly utilized. As there exists at least one PI segment for any
x 苸 Ū, property C1, Tuy’s condition, is satisfied. Also, for
y共s兲 苸 ⌳1共x兲 there are a limited number of planes ⌸共x , ␣兲,
␣ 苸 ␤⬜共s , x兲, which satisfies ⌸共x , ␣兲 contains an endpoint of
⌳1 or is tangent to ⌳1. Thus property C2 is satisfied. Moreover, apart from a set of zero measure, any plane ⌸共x , ␣兲,
␣ 苸 S2, through x intersects ⌳1共x兲 at an odd number of intersection points 共IP兲. And the number of IP is either one or
three which validates the property C3. Overall, ⌳1 satisfies
the three properties C1 – C3 required in Ref. 33. Following
the work by Katsevich,29 we define a similar weighting function n共s , x , ␣兲 in Table I, where s j, j = 1 , 2 , 3 are the angular
parameters satisfying y共s j兲 苸 共⌳1 艚 ⌸共x , ␣兲兲. Thus, we have
兺 j:y共s j兲苸⌳1艚⌸共x,␣兲n共s j , x , ␣兲 = 1 for almost all ␣ 苸 S2, and
n共s , x , ␣兲 is the piece-wise constant. Therefore, our scanning
arcs and weighting function are qualified to be fitted into
Katsevich’s general reconstruction framework.
Then, we need to find the discontinuities in ␪ of 共2.8兲 for
each s 苸 Ī1共x兲, and determine the filtering directions. Similar
to Katsevich’s treatment,29 we can analyze the discontinuities
by projecting the source trajectory ⌳ = Ca 艛 Cb onto the detector plane DP共s0兲 共Fig. 4兲, which is defined as the plane
through the origin O and perpendicular to the line through
y共s0兲 and O. In this article, all the variables are denoted with
hat signify the objects on a projection plane. The detector
coordinates are defined along the directional vector u and v.
For the sources on the arc A, that is, s0 苸 关−共␲ / 2
+ 2smx兲 , ␲ / 2兴,
u = 共sin共s0 + smx兲cos t,− cos共s0 + smx兲,sin共s0 + smx兲sin t兲,
v = 共− sin t,0,cos t兲.
共10兲
And for the source on the arc B, that is, s0 苸 关␲ / 2 , 共3␲ / 2
+ 2smx兲兴
FIG. 4. Projections of a Pl line on a detector plane for the configuration of
Fig. 3共a兲. 共a兲 The projected trajectory with the source on Ca, 共b兲 the projected trajectory with the source on Cb, and 共c兲 an impossible trajectory
projection.
u = 共sin共s0 + smx兲cos t,− cos共s0 − smx兲,− sin共s0 − smx兲sin t兲,
v = 共sin t,0,cos t兲.
With the source on Ca, we can describe the projection of Cb
on the detector plane as
u共s兲 = R
sin共s0 + smx兲cos共s − smx兲cos 2t − cos共s0 + smx兲sin共s − smx兲
,
cos共s − smx兲cos共2t兲cos共s0 + smx兲 + sin共s − smx兲sin共s0 + smx兲 − 1
v共s兲 = R
cos共s − smx兲sin 2t
,
cos共s − s 兲cos共2t兲cos共s0 + smx兲 + sin共s − smx兲sin共s0 + smx兲 − 1
mx
where s0 苸 关−␲ / 2 − 2smx , ␲ / 2兴 indicates the source position
on the arc A, s 苸 共␲ / 2 , 3␲ / 2 + 2smx兲 and u共s兲 and v共s兲 are the
detector coordinates. With the source on Cb, by geometric
symmetry we can immediately obtain the same formula except for a changed range of s. When the source is on Ca 关Fig.
4共a兲兴, the discontinuities may occur at polar angle ␪1, ␪2, and
␪3 which corresponding to the plane P1, P2, and P3. It is easy
to compute that the magnitudes of the jumps at these three
Medical Physics, Vol. 33, No. 10, October 2006
共11兲
共12兲
positions are 2, 0, and 0, respectively. Thus, we only have a
jump along direction parameterized by ␪1, which is the same
filtering direction used in the Feldkamp algorithm.18 When
the source is on Cb 关Fig. 4共b兲兴, the projection of a PI segment
can only have one IP with the projection of ⌳1共x兲, giving the
tangential direction ␪1 with a jump of magnitude 2. The last
case 关Fig. 4共c兲兴 is simply impossible, as shown in Appendix
B. Thus, the filtering directions should be along horizontal
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TABLE II. Linear attenuation coefficients at 38 keV.
Material
Attenuation coefficient 共cm−1兲
Skin
Breast
Calcifications
Mass
Fibrous
0.2963
0.2465
3.2822
0.2667
0.2667
1
2␲2
f¯I2共x兲 = −
⫻
冕
2␲
0
冕
兺
¯I 共x兲 m=1,2
2
1
兩x − y共s兲兩
冏
⳵
D f 共y共q兲,cos ␥␤共s,x兲
⳵q
+ sin ␥␣⬜共s,x, ␪m兲兲
冏
d␥
ds.
sin ␥
q=s
共14兲
Combining these two formulas, we have
f共x兲 = 共f¯I1共x兲 + f¯I2共x兲兲/2.
共15兲
Consequently, our reconstruction algorithm can be divided into four steps:
FIG. 5. Uncompressed breast phantom which contains masses, calcifications, and fibroses of different sizes and densities.
lines 关Fig. 4共a兲兴 or tangential lines 关Fig. 4共b兲兴, depending on
␪m determined based on our above analysis. This is exact the
basic same result obtained by Katsevich.29 Therefore, the
reconstruction formula can be expressed as
f¯I1共x兲 = −
⫻
1
2␲2
冕
2␲
0
冕
1
兺
¯I 共x兲 m=1,2 兩x − y共s兲兩
1
冏
⳵
D f 共y共q兲,cos ␥␤共s,x兲
⳵q
+ sin ␥␣⬜共s,x, ␪m兲兲
冏
共1兲 differentiate the projection data collected along PI-arc
⌳1共x兲 to obtain ⳵ / ⳵qD f ;
共2兲 filter every projection along the direction defined by ␪m.
Projection data on arc A are horizontally filtered, and
projection data on arc B are tangentially filtered;
共3兲 backproject the filtered data to reconstruct an image; and
共4兲 repeat the above three steps for the projection data collected along the other PI-arc ⌳2共x兲, and average the two
images for the final reconstruction.
For x 苸 Ū1 关Fig. 3共b兲兴, we can derive the formula and
steps in a similar fashion as that for x 苸 Ū2 关Fig. 3共a兲兴. Since
our reconstruction formula is derived in Katsevich’s
framework,29,33, it can be regarded as a variant of the conebeam reconstruction algorithm in the case of circle-and-arc.29
C. Uncompressed breast phantom
d␥
ds.
sin
␥
q=s
共13兲
On the other hand, we can also reconstruct x from ⌳2共x兲.
Similarly, we have
A 3D mathematical mammography phantom 共Fig. 5兲 was
designed in reference to several commercially available
mammography phantoms, such as the American College of
Radiology 共ACR兲 accreditation phantom, Mammography
Imaging Screening Trial phantom and Uniform phantom.37,38
TABLE III. Dimensions and linear attenuation coefficients of the fibroses.
Fibrous No.
1
2
3
4
5
6
7
8
9
10
Diameter 共mm兲
Height 共mm兲
AC 共cm−1兲
1.56
10
0.267
1.12
10
0.267
0.89
10
0.267
0.75
10
0.267
0.54
10
0.267
0.4
10
0.267
1.12
10
0.267
1.12
10
0.253
1.12
10
0.249
1.12
10
0.248
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TABLE IV. Dimensions and linear attenuation coefficients of masses.
Mass No.
1
2
3
4
5
6
7
8
9
10
Diameter 共mm兲
AC 共cm−1兲
8.00
0.267
6.00
0.267
3.00
0.267
2.00
0.267
1.00
0.267
0.5
0.267
3.00
0.267
3.00
0.253
3.00
0.249
3.00
0.248
The breast was modeled as half an ellipsoid. While the existing phantoms contain important structures simulating
mass, fibrous and calcification, they mimic a compressed
breast for x-ray mammography, being suboptimal for our 3D
CT simulation. To address this limitation, our phantom targeted an uncompressed breast with representative anatomical
and pathological features, including structures of different
sizes and contrasts. A detailed description of the phantom is
as follows.
The three ellipsoidal semiaxes were set to 50, 50, and
100 mm. The skin thickness was set to 2.5 mm. While the
fibroses were modeled as cylinders, the calcifications and
mass as balls 共Fig. 5兲. The phantom was positioned in the
nonnegative space, attached to the chest wall defined on z
= 0 mm. Fibroses were placed on the planes of z = 22.5 mm
and z = 62.5 mm. Masses were centered on the planes of z
= 35 mm and z = 75 mm. Calcifications were scattered on the
plane of z = 42.5 mm.
The linear attenuation coefficients 共AC兲 of the breast
structures were specified for 38 keV x-rays16,39 as listed in
Table II. The dimensions of the features in the phantom were
specified in reference to those in the existing mammography
phantoms, especially the ACR phantom. Tables III–V enlist
the sizes and attenuation properties of the fibroses in the
planes z = 22.5 and 62.5 mm, the parameters of the masses in
the planes z = 35.0 and 75.0 mm, and the characteristics of
the calcifications at z = 42.5 mm, respectively.
III. NUMERICAL SIMULATION
To corroborate the correctness of our proposed algorithm
and demonstrate its clinical utility, we implemented it in
C⫹⫹ for numerical tests. These results were also compared
with a circular cone-beam CT scan using the FDK algorithm.
The system setup and geometric parameters were summarized in Tables VI and VII, respectively. In our simulation,
the reconstruction region U was delimited by the chest wall
关Fig. 6共a兲兴. The FDK algorithm can only approximately reconstruct the volume above the scanning plane 关Fig. 6共b兲兴,
while our proposed method can exactly reconstruct a larger
volume. In all the cases, the reconstruction results obtained
using our algorithm were excellent except for the streak artifacts primarily caused by the high-density calcifications
共ten times higher than the background兲 共Figs. 6–8兲, which
were also observed in the FDK results. The other structures,
such as fibrous and mass, were reconstructed very well. Although the detector element size was comparable or even
larger than the fine structures in our phantom, these features
were clearly visible in the reconstructed images. As far as the
calcifications are concerned, despite that they were substantially smaller than one pixel, they were revealed in the reconstruction. The low-contrast structures were well revealed
共Figs. 7 and 8兲 using our algorithm.
Our algorithm generally produced more accurate results
than the FDK algorithm, because there were significant density drops in the FDK results, especially far away from the
central plane 共Figs. 7–10兲. Also, due to the approximate nature of the FDK algorithm, significant streak artifacts were
observed 共Figs. 7 and 8兲 in the FDK results. Besides, our
algorithm preserved the shape of the breast phantom, but
there were substantial shape distortions in the FDK results
especially near the top of the phantom 共Figs. 7 and 8兲.
A number of numerical tests with noisy projection data
were performed to simulate the practical imaging process. As
the image quality is closely related to radiation dose, which
is proportion to the total number of involved x-ray photons
given other conditions being equal, the same total number of
photons was used in each test for image quality comparison.
In our study, the same radiation dose was uniformly distributed to every detector cell in each projection view for simplicity. Then, the Poisson noise was added to the projection
data. The image quality difference between our algorithm
and the FDK algorithm was similar to that in the noise-free
case, except that in the noisy case our algorithm showed a
better noise tolerance than the FDK algorithm.40 Specifically,
the images obtained using our algorithm were smoother than
the FDK results, and had no geometric distortion that was
shown in the FDK counterparts 共Figs. 7–11兲. In Figs. 10 and
11, the image noise associated with the proposed algorithm
appeared to be larger than that with the FDK algorithm. The
reason was that the density drop inherent in the FDK reconstruction, made many pixel values go outside the fixed display window. As a result, a large portion of noise became
invisible in the displayed image.
The aforementioned image artifacts in our simulation
were around the high contrast structures. The profiles across
their boundaries can be modeled as piece-wise constant func-
TABLE V. Dimensions and linear attenuation coefficients of calcifications.
Calcification No.
1
2
3
4
5
6
Diameter 共mm兲
AC 共cm−1兲
0.7
3.282
0.54
3.282
0.4
3.282
0.32
3.282
0.24
3.282
0.16
3.282
Medical Physics, Vol. 33, No. 10, October 2006
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Zeng et al.: Cone-beam mammo-CT
TABLE VI. Simulated imaging parameters for tilting arc based mammo-CT
scan.
Number of projections
Detector array size
Number of detector
elements
Detector element size
Number of photons per
view 共for noisy projections兲
Arc radius
Arc tilting angle
Arc angular range
Reconstruction grid
Voxel size
500⫻ 2
132⫻ 155 mm2
256⫻ 300
0.57⫻ 0.57 mm2
1e7 ⫻ 256⫻ 300
330 mm
20°
226°
256⫻ 256⫻ 320
0.43⫻ 0.43⫻ 0.43 mm3
3627
TABLE VII. Simulated system setup parameters of a circular scan for FDK
algorithm.
Number of projections
Two-dimensional detector array size
Detector elements
Detector element size
Radius of circle
Total photon number per
view 共for noisy projections兲
Reconstruction grid
Voxel size
1000
132⫻ 133 mm2
256⫻ 256
0.57⫻ 0.57 mm2
330 mm
1e7 ⫻ 256⫻ 300
256⫻ 256⫻ 320
0.43 mm3
flawless, precisely as we expected, but the results using the
FDK algorithm contained significant density dropping artifacts 共Figs. 12兲.
tions, which do not satisfy the assumption f 苸 C⬁0 共R3兲 that
was made for the formulation of Katsevich’s general exact
cone-beam reconstruction scheme.33 Therefore, it should not
be surprising that such high-contrast nondifferentiable features had caused intensity fluctuations in the reconstructed
images, as we previously discussed.41 To confirm the correctness of the implementation of our proposed reconstruction
formula, we also applied it to reconstruct the traditional 3D
Shepp-Logan phantom. The reconstruction results looked
IV. DISCUSSIONS AND CONCLUSION
Our reconstruction algorithm was derived from Katsevich’s general scheme and Katsevich’s circle-and-arc algorithm, which can survive the longitude data truncation. As a
result, no x-rays need to be sent into the chest wall. Also, this
algorithm can be efficiently implemented in parallel similar
to what was recently reported of the parallel implementation
of the Katsevich algorithm.42 Additionally, our scanning or-
FIG. 6. Representative phantom slices. 共a兲 The ideal
phantom slice at x2 = 0.0 mm, 共b兲 x1 = 0.0 mm, 共c兲 x3
= 35 mm, and 共d兲 x3 = 75.0 mm. The dotted lines shows
the exact reconstruction region. The display window is
关0.244, 0.255兴.
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Zeng et al.: Cone-beam mammo-CT
FIG. 7. Reconstructed images at x2 = 0.0 mm using the proposed algorithm
and the FDK algorithm. The images in the left column are from the proposed algorithm, and that in the right column are from the FDK algorithm.
The images in the top row are from noise free projection data, and that in the
bottom row from noisy data with the same number of photons. The display
window is 关0.244, 0.255兴.
bit is symmetric, and may be numerically more stable than
other less symmetric scanning trajectories, such as circleplus-line and circle-plus-arc.27,29
Although our algorithm requires a longer scanning trajectory than a circular scan, it does not mean that our algorithm
require more radiation dose. In our imaging protocol, we can
distribute the same dose as that with the circular scan to each
view and reconstruct 3D images from these projections. As
our algorithm can utilize all the information from two arcs,
the final image shall not be inherently noisier than the circular scan reconstruction. Our numeric experiments have indicated that our results were actually less noisy than that from
the circular scan using the FDK algorithm.
To compare the noise levels in the images reconstructed
using our algorithm and the FDK algorithm respectively, the
standard deviations of pixel values within a homogeneous
region-of-interest were computed 共Fig. 11兲. Generally speaking, the variance of a stochastic process cannot be computed
from a single realization of that process. To compare the
noise properties of the reconstructed images, the ensemble
averages should be used, instead of the averages based on a
single image. These averages are equivalent only in the case
of an ergodic stochastic process. However, it is not common
in either the quality-control practice or the reconstruction
literature to do the scan and reconstruction many times for
evaluation of the noise level.40,43–45 Hence, in this project the
noise image has been approximately considered as being ergodic.
In conclusion, we have proposed a novel cone-beam
mammo-CT scheme based on two tilting arcs and formulated
Medical Physics, Vol. 33, No. 10, October 2006
3628
FIG. 8. Reconstructed images at x1 = 0.0 mm using the proposed algorithm
and the FDK algorithm. The images in the left column are from the proposed algorithm, and that in the right column are from the FDK algorithm.
The images in the top row are from noise free projection data, and that in the
bottom row from noisy data with the same number of photons. The display
window is 关0.244, 0.255兴.
a Katsevich-type reconstruction algorithm. The numerical
simulation results are consistent with the ideal 3D image
volume within the numerical error. Compared with other existing mammo-CT algorithms, our work promises better diagnostic performance for breast imaging, and may have a
FIG. 9. Reconstructed images at x3 = 35.0 mm using the proposed algorithm
and the FDK algorithm. The images in the left column are from the proposed algorithm, and that in the right column are from the FDK algorithm.
The images in the top row are from noise free projection data, and that in the
bottom row from noisy data with the same number of photons. The display
window is 关0.244, 0.255兴.
3629
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3629
x1 = ␭共R cos s̄a cos t兲 + 共1 − ␭兲共R cos s̄b cos t兲,
共A1a兲
x2 = ␭共R sin s̄a兲 + 共1 − ␭兲共R sin s̄b兲,
共A1b兲
x3 = ␭共R cos s̄a sin t兲 + 共1 − ␭兲共− R cos s̄b sin t兲.
共A1c兲
We can rewrite 共A1a兲–共A1c兲 as
FIG. 10. Reconstructed images at x3 = 75.0 mm using the proposed algorithm and the FDK algorithm. The images in the left column are from the
proposed algorithm, and that in the right column are from the FDK algorithm. The images in the top row are from noise free projection data, and
that in the bottom row from noisy data with the same number of photons.
The display window is 关0.244, 0.255兴.
commercial potential. Our scheme also can be used to other
similar tomographic imaging applications, such as singlephoton emission computed tomography.
ACKNOWLEDGMENTS
This work is partially supported by the NIH/NIBIB Grant
Nos. EB002667 and EB004287.
APPENDIX A: DEFINITION OF REGION Ū, Ū1,
and Ū2
1. Monotony of x3 with respect to sa and sb
First, we prove that x3 is monotonic for the first case
R sin共sa + smx兲 ⬎ x2 ⬎ R sin共sb − smx兲 with respect to sb as what
Katsevich did.29 Without loss of generality, let us parameterize x by 0 ⬍ ␭ ⬍ 1 with s̄b = sb − smx and s̄a = sa + smx. Hence,
we have
x1
= ␭ cos s̄a + 共1 − ␭兲cos s̄b ,
R cos t
共A2a兲
x2
= ␭ sin s̄a + 共1 − ␭兲sin s̄b ,
R
共A2b兲
x3
= ␭ cos s̄a − 共1 − ␭兲cos s̄b .
R sin t
共A2c兲
Then, let us fix x1 and x2, and let ␭, s̄a be functions of s̄b, i.e.,
␭共s̄b兲 and s̄a共s̄b兲. Differentiating 共A2a兲 and 共A2b兲, we have
共cos s̄a − cos s̄b兲
共sin s̄a − sin s̄b兲
d␭
ds̄b
d␭
ds̄b
− ␭ sin s̄a
+ ␭ cos s̄a
ds̄a
ds̄b
ds̄a
ds̄b
= 共1 − ␭兲sin s̄b ,
共A3a兲
= − 共1 − ␭兲cos s̄b . 共A3b兲
Solving 共A3兲 for d␭ / ds̄b and ds̄a / ds̄b, we obtain
d␭
ds̄b
ds̄a
ds̄b
=
=
共1 − ␭兲sin共s̄b − s̄a兲
1 − cos共s̄a − s̄b兲
共A4a兲
,
共1 − ␭兲
.
␭
共A4b兲
Also, differentiating 共A2c兲 with respect to s̄b, we have
1 dx3
d␭
= 共cos s̄a + cos s̄b兲
R sin t ds̄b
ds̄b
− ␭ sin s̄a
ds̄a
ds̄b
+ 共1 − ␭兲sin s̄b .
共A5兲
Noting
FIG. 11. Reconstructed images at x3 = 46.8 mm from the
proposed algorithm and the FDK algorithm with the
same number of photons. The noise level is measured
as the standard deviation from the homogeneous region
indicated by the dotted rectangle. 共a兲 The noise from the
proposed algorithm is 2.03e-0.3, and 共b兲 the noise from
the FDK algorithm is 2.92e-03. The display window is
关0.244, 0.255兴.
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3630
FIG. 12. Reconstructed Shepp-Logan phantom images
at x2 = −0.25 and profiles along the dash lines. 共a兲 is
reconstructed from the proposed algorithm and 共b兲 from
the FDK algorithm. The images are displayed with the
window 关1.0, 1.05兴.
2. Upper and lower limits of x3 given „x1 , x2…
to admits a PI line
sin共s̄b − s̄a兲共cos s̄a + cos s̄b兲 − 共sin s̄b − sin s̄a兲cos共s̄b − s̄a兲
= 共sin s̄b cos s̄a − cos s̄b sin s̄a兲共cos s̄a + cos s̄b兲 − 共sin s̄b
− sin s̄a兲共cos s̄a cos s̄b + sin s̄a sin s̄b兲
= cos2 s̄a sin s̄b − sin s̄a cos2 s̄b − sin s̄a sin2 s̄b
共A6兲
+ sin2 s̄a sin s̄b = sin s̄b − sin s̄a
and substituting 共A4兲 and 共A5兲, we have
dx3
ds̄b
= 共1 − ␭兲R sin t
冋
共cos s̄a + cos s̄b兲sin共s̄b − s̄a兲
册
1 − cos共s̄b − s̄a兲
+ 共sin s̄b − sin s̄a兲 = 2共1 − ␭兲R sin t
共sin s̄b − sin s̄a兲
1 − cos共s̄b − s̄a兲
.
共A7兲
Because 0 ⬍ ␭ ⬍ 1 and R sins̄a ⬎ x2 ⬎ R sins̄b, dx3 / ds̄b does
not change its sign. Hence, x3 is monotonic decreasing with
s̄b 共or sb兲. As s̄a 共or sa兲 rotates together with s̄b 共or sb兲 in the
same direction, x3 is also monotonic decreasing with s̄a 共or
sa兲.
Similarly, for the second case R sin共sa + smx兲 ⬍ x2
⬍ R sin共sb − smx兲, we can prove that x3 is monotonous increasing with s̄b 共or sb兲. As s̄a 共or sa兲 rotates together with s̄b
共or sb兲 in the same direction, x3 is also monotonic increasing
with s̄a 共or sa兲.
It is easy to see that if we project all the x that admit at
least one PI line onto the plane x3 = 0, the results must be
within an ellipse 兵共x1 , x2兲 兩 x21 / cos2 t + x22 = R2其. Moreover,
given a 共x1 , x2兲 within this ellipse, we can readily get the
limit of x3 with respect to the assumption of cases 1 and 2
based on the aforementioned monotony property. If we do so
for every point 共x1 , x2兲 within the ellipse, we can obtain an
image showing the appearance of the region. Based on assumption for the first case, we have x2共y共sa兲兲 艌 x2
艌 x2共y共sb兲兲 where x2共y共sb兲兲 represents the second component
of y共sb兲. For any fixed 共x1 , x2兲 within the ellipse and all the
points admitting a PI line 共Fig. 13兲, we have the monotony of
x3 with respect to sa or sb. Therefore, we can determine the
upper and lower limits of x3 for any given 共x1 , x2兲 as follows:
Let sa−共x2兲 , sa+共x2兲 be two particular sa± on arc A 关Fig.
13共a兲兴, at which x2共yArcA共sa±兲兲 = x2, x1共yArcA共sa+兲兲 ⬎ 0 and
x1共yArcA共sa−兲兲 ⬍ 0. Note that for 兩x2兩 ⬍ 兩R cos共smx兲兩 there does
not exist sa−. Let sb−共x2兲, sb+共x2兲 be two particular sb± on arc
B 关Fig. 13共b兲兴, at which x2共yArcB共sb±兲兲 = x2, x1共yArcB共sb+兲兲
⬎ 0 and x1共yArcB共sb−兲兲 ⬍ 0. Note that for some 兩x2兩
⬍ 兩R cos共smx兲兩 there dose not exist sb+.
According to the monotony, i.e., x3 is monotonous decreasing with sa, the upper limit of x3 can be reached at the
FIG. 13. Projection of the scanning arcs, the Pl line
共case 1兲 and X onto the traverse plane. 共a兲 The projection of ArcA, and 共b兲 the projection of ArcB. The dotted
curve shows the range of sa and sb.
Medical Physics, Vol. 33, No. 10, October 2006
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Zeng et al.: Cone-beam mammo-CT
3631
minimum sa, i.e., sa = sa+. As far as the lower limit of
x3 case1共x1 , x2兲 is concerned, it is more complicated to calculate the maximum sa for a particular 共x1 , x2兲. We must check
all the possible boundaries from arc A 关the dotted line in Fig.
14共a兲兴, such as sa− and sa max. Besides, as y共sa兲, x, and y共sb兲
are on the same line, the maximum sa is also limited by the
boundaries on arc B 关the dotted line in Fig. 14共b兲兴, such as
sb+ and sb max. Hence, the lower limit of x3 is the maximum
one of those corresponding x3s at those four critical positions. For example, in Fig. 14, the maximum sa is achieved
when the PI line intersects arc B on y共sb max兲.
Finally, the strategy to determine the limits can be described as follows:
Up共x3
case 1共x1,x2兲兲
= x3共sa+兲 = x3共sb−兲
冦
x3共sa−兲, if sa− exists
冉
冊
␲
2
Low共x3 case 1共x1,x2兲兲 = max
x3共sb+兲, if sb+ exists
3␲
x3 sb max =
+ 2sm
2
x3 sa max =
冉
冊
冧
where Up共x3 case1共x1 , x2兲兲 is the upper limit of x3,
Low共x3 case1共x1 , x2兲兲 is the lower limit of x3. And for the second case’s assumption, we can analysis it similarly.
3. Definition of Ū, Ū1, and Ū2
Now, we can define the region Ū as 共A9a兲, which is the
disjunction of the regions determined in cases 1 and 2. Based
on the earlier results, it is easy to see that for any x 苸 Ū, it
admits at least one PI line and at most two PI lines. The
upper and lower surfaces of the region Ū are shown in Fig.
14. Furthermore, the two PI-segments region Ū2 is defined as
in 共A9b兲, which admits two PI segments and has the same
共A8兲
upper surface as region Ū. Ū2’s lower surface is shown in
Fig. 14共c兲. Ū1 is defined as 共A9c兲, which admits only one PI
line. Figure 14共d兲 shows the relative position between Ū2 and
Ū1,
FIG. 14. Geometry of the exact reconstruction region Ū. 共a兲 The upper surface of the region Ū with respect to two tilting arcs, 共b兲 the lower surface of the
region Ū, 共c兲 the lower surface of the region that allows 2 Pl lines, and 共d兲 the 2 Pl lines region vs the 1 Pl line region 共the 1 Pl line region in green, and the
2 Pl line region in red兲.
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3632
冦冨
冦冨
冧
冧
x21
+ x22 ⬍ R2, and given a 共x1,x2兲,
Ū = x cos2 t
,
x3 苸 关min共Lowcase 1,Lowcase 2兲,max共Upcase 1,Upcase 2兲兴
共A9a兲
x21
2
2
2 + x2 ⬍ R , and given a 共x1,x2兲,
cos
t
,
Ū2 = x
x3 苸 关max共Lowcase 1,Lowcase 2兲,min共Upcase 1,Upcase 2兲兴
共A9b兲
Ū1 = Ū − Ū2 .
共A9c兲
APPENDIX B: PROOF OF THE IMPOSSIBLE CASE †FIG. 4„c…‡
Again, let us denote s̄b = sb − smx, s̄ = s − smx, and s̄t = st + smx. If the case exists 关Fig. 4共c兲兴, as pointed out by Katsevich29 for the
circle-and-arc trajectory, there must be a plane tangent to Ca at s̄t. Thus, we must have
冨
− sin s̄t cos t
cos s̄t
− sin s̄t sin t
共cos s̄ − cos s̄t兲cos t
sin s̄ − sin st
− 共cos s̄t + cos s̄兲sin t
共cos s̄b − cos s̄t兲cos t sin s̄b − sin st − 共cos s̄t + cos s̄b兲sin t
where 3␲ / 2 ⬎ s̄ ⬎ s̄b ⬎ ␲, sin s̄b ⬍ sin s̄t. Simplifying 共B1兲, we
have
2 cos t sin t关sin s̄t sin共s̄ − s̄b兲 + 共cos s̄ − cos s̄b兲兴 = 0.
共B2兲
Since 0 ⬍ t ⬍ ␲ / 2, cos t sin t ⫽ 0, we have
sin s̄t sin共s̄ − s̄b兲 + 共cos s̄ − cos s̄b兲 = 0.
共B3兲
On the other hand, we obtain that
cos s̄ − cos s̄b + sin s̄t sin共s̄ − s̄b兲
=cos s̄ − cos s̄b − sin s̄b sin共s̄b − s̄兲
+ 共sin s̄b − sin s̄t兲sin共s̄b − s̄兲
=cos s̄ − cos s̄b − sin2 s̄b cos s̄
+ sin s̄b cos s̄b sin s̄
+ 共sin s̄b − sin s̄t兲sin共s̄b − s̄兲
=− cos s̄b共1 − sin s̄b sin s̄兲 + cos2 s̄b cos s̄
+ 共sin s̄b − sin s̄t兲sin共s̄b − s̄兲
=− cos s̄b共1 − cos共s̄b − s̄兲兲 + 共sin s̄b − sin s̄t兲sin共s̄b − s̄兲
⬎ 0,
共B4兲
where we have used the fact that for 3␲ / 2 ⬎ s̄ ⬎ s̄b ⬎ ␲,
sin s̄b ⬍ sin s̄t. Therefore, we arrive at a contradiction.
a兲
Electronic mail: [email protected]
Electronic mail: [email protected]
c兲
Electronic mail: [email protected]
d兲
Corresponding author. Electronic mail: [email protected]
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