Home
Search
Collections
Journals
About
Contact us
My IOPscience
A cone-beam tomography algorithm for orthogonal circle-and-line orbit
This content has been downloaded from IOPscience. Please scroll down to see the full text.
1992 Phys. Med. Biol. 37 563
(http://iopscience.iop.org/0031-9155/37/3/005)
View the table of contents for this issue, or go to the journal homepage for more
Download details:
IP Address: 155.98.164.38
This content was downloaded on 26/05/2015 at 19:53
Please note that terms and conditions apply.
Phys. Med. B i d , 1992, Vol. 37, N o 3 , 563-571. Printed in the UK
A cone-beam tomography algorithm for orthogonal
circle-and-line orbit
Gengsheng L Zeng and Grant T Gullberg
Department of Radiology, Medical Imaging Research Laboratory, University of Utah,
Salt Lake City, UT 84132, USA
Received 24 October 1991
Abstract. A cone-beam algorithm which provides a practical implementation of B D Smith's
cane-beam inversion formula is presented. For a cone-beam vertex orbit consisting of a
circle and an orthogonal line. This geometry is easy to implement in a s p ~ c ~ s y s t e m
and
,
it satisfies the cone-beam data sufficiency condition. The proposed algorithm is i n the form
of a convolution-back projection, and requires a pre-filtering procedure. Computer rimulalions show a reduction of the artifacts that are found with the Feldkamp algorithm where
the cone-beam vertex orbit is a circle.
1. Introduction
Single-photon-emission computed tomography (SPECT) with a cone-beam collimator
has been under investigation for several years (Jaszczak et a / 1986, 1988, Floyd el al
1986). The main advantage of using a cone-beam collimator is a gain in sensitivity
and resolution over images formed using standard parallel-hole or fan-beam collimators. Comparisons of cone-beam, parallel-hole and fan-beam collimators for brain
and heart imaging have been reported by Jaszczak et U / (1988), Hahn et a / (1988),
Gullberg et a/ (1990, 1991) and Terry et a / (1991). Cone-beam tomography also finds
applications in x-ray imaging. In cone-beam x-ray medical imaging and non-destructive
testing, a two-dimensional detector is used to speed up the data acquisition and improve
the sensitivity. The x-ray source defines the cone-beam focal point, and an image
intensifier can be used as a two-dimensional defector (Ning er Q/ 1988).
Currently, circular cone-beam vertex geometry is used and a popular algorithm to
process cone-beam projections is that of Feldkamp et a/ (1984). This algorithm was
originally derived from the fan-beam reconstruction formula by an approximation,
and has since been extensively analysed mathematically (Yan and Leahy 1991). The
Feldkamp algorithm is a convolution and back projection method and can be implemented efficiently. It is known that if the focal point of the collimator follows a single
planar orbit, then the data obtained d o not satisfy this cone-beam data sufficiency
condition (Smith 1985) for exact three-dimensional reconstruction. Reconstructions
by the Feldkamp algorithm suffer distortions and artifacts, and only the central plane
is reconstructed free of artifacts.
Non-planar orbits must be used in order to fulfil the data sufficiency condition.
The cone-beam data sufficiency condition requires that every plane which passes
through the imaging field of view must also cut through the orbit at least once. As
mentioned above, the conventional single planar orbit does not satisfy this condition,
whereas two orbits arranged as illustrated in figure 1 d o satisfy this condition.
0031-9155/92/030563+15$04.50
@ 1992 IOP Publishing Ltd
563
564
G L Zeng and G T Gullberg
I
T
axis of rotation
y
orbit
Figure 1. Cone-beam venex orbits: one-circle and a perpendicular line. For the orthogonal
circle-and-line orbit. the vertex of the cone-beam geometry traverses one complete circle
of radius D and a line of length 2n.
There are basically two approaches to exactly reconstructing cone-beam images
from sufficient data. The first is to convert the cone-beam projections into parallel
radon transform data, and to use the radon inversion formula to reconstruct the image.
Grangeat (1987) established a formula between the cone-beam data and the first
derivative of the three-dimensional radon transform, which provided an inversion
formulie suitable for a genera! orbit. However, Grangea.!’s algorithm involver; !he
sorting (or rebinning) of the cone-beam data. Clack et ol(1991) proposed an algorithm
for two orthogonal orbits. The algorithm was not an exact inversion formula, but
demonstrated that there exist practical, effective methods to reconstruct cone-beam data.
The second approach is based on Kirillov’s pioneering work (1961). Based on
Kirillov’s formulation, Tuy (1983) proposed the first inversion formula, but his
algorithm did not lead directly to an efficientnumerical implementation. Smith (1985,
1987, 1990) derived a more practical formula.to invert the cone-beam data sets. Smith’s
algorithm has a simple implementation with the assumption that for any line that
contains a vertex point and a reconstruction point, there is an integer M (which remains
constant for this line) such that almost every plane that contains this line intersects
the orbit exactly M times. However, this condition is too restrictive since the only
known cone-beam vertex geometry meeting this assumption is an infinite straight line,
which is not practical.
In this paper, we modify Smith’s inversion formula for the cone-beam vertex
geometry containing a circle and a finite, orthogonal line. Our modified algorithm
consists of two steps. In the first step, we filter the cone-beam projection data twodimensionally, and in the second step, we apply a Feldkamp-type algorithm to the
filtered data. The two-dimensional filter used in the first step depends on the orbit
geometry. This is a practical algorithm for reconstructing a complete cone-beam data
set.
2. Theory
2.1. Cone-beom projections
The cone-beam geometry is shown in figure 2. The focal point trajectory is referred to
as the orbit. The focal length D is the distance between the focal point and the axis
+
Circle-and-line orbit tomography algorithm
565
axis of rotation
Figure 2. Cone-beam geometry. The detector plane P, is at the axis of rotation. Thc orbit
is the trajectory of the cone venex, and is described by a global vector @(A), where A is
a real parameter. The cone-beam projection p(P, @) is determined by a line integral along
the unit vector B.
of rotation. The detector plane is assumed to be at the axis of rotation. This assumption
is valid, because one can always scale the projection images if the axis of rotation is
moved onto the detector plane when reconstructing the image. The object density
function i s f ( x ) , where x is a vector in W'. The cone-beam data are expressed as
m
g ( P , a) = (_-f(a+tP) d t
where CP is the orbit vector for the location of the focal point and P is a unit vector
along the ray of the line integral. Our aim is to reconstruct the object f (x) from the
cone-beam projections g ( p , a).
2.2. Smith's algorithm
The major steps in Smith's reconstruction algorithm (1990) are quoted here. Let us
566
G L Zeng and G T Gullberg
first define a one-dimensional filter kernel
The filter kernel defined in equation (2) is a ramp filter commonly used in tomography.
An intermediate function G (y. @) is obtained from the two-dimensional cone-beam
projection data g:
where S denotes (any) half of the unit sphere and d p is the surface element on the
unit sphere.
Modified projection data g, are obtained from G via
where B and q are the azimuth and longitude of vector yL, respectively, in the local
coordinate system as shown in figure 4 ( a ) . In equation (4), the function M ( y L , @ )
denotes the number of times the orbit intersects the plane which is represented by the
normal vector y L and passes through the point @. Thc local coordinatc system (SCC
figure 4 ( a ) ) is spanned by W(A),D and E, and is determined by the current focal
y"
-n:nt
-,,., nnrl r.>.- h n
.er-..rtn.rt:-..
+..:.,..
*. I,--+-- ?! is orthogor.z! to "(A!
and is in the plane which contains vector @'(A) and reconstruction point x. Vector E
is defined by @'(A) x D. Here, A E A is the real parameter of the orbit function a,and
@'(A) is the derivative of @ ( A ) with respect to the parameter A. Finally, the inversion
formula is (Smith 1990)
....
Y.."
XICOS
rY
*
. .Ir
.C
re
I ..+
p"L1.L
.~C"IIY..Ylll"ll
-
Y C I I " .
'PI dqll@'(A)ll dA.
(5)
It can be shown that equation (5) is identical with the Feldkamp algorithm if the orbit
is a circle (ll@'(A)ll = D ) and g, is cone-beam projection data.
In order to derive an easier way to evaluate g,, we first study equations (3) and
(4). Figure 3 is a pictorial description of equation (3). At a fixed focal point location
@ associated with a detector plane Pe, let us consider a direction y. The unit vector
y has an azimuth B and a longtitude q, where q is in the orbit plane which is spanned
by @ ( A ) and W ( A ) .The set { a p l p . y = constant, IpI = I y l = 1, V a E 3 defines a plane
or a cone of projection rays, which intersect the detector plane Pe at a straight line
or a quadratic curve. In fact, equation (3) can be rewritten as
G(y,@)=21im
E-0
I:l I
H,(O
g ( P , @ ) S ( t - ( P . y ) ) d P dl.
(6)
S
Therefore, the intermediate function G (y, @) can be formed in three steps: (1) treat
the detector plane P* as a two-dimensional image plane and perform projections on
the plane Pe along the curves which are the intersections of the cone-beam projection
rays with 8.7 =constant and the plane Pe;(2) weight the one-dimensional projected
data by H , ( p . y ) ; and (3) integrate the weighted one-dimensional data to obtain
G( y, a). Hence we may refer to equation (3) as projection and weighted summation
on the detector plane.
Circle-and-line orbit tomography algorithm
561
Figure 3. Geometric illustration of equation (3). For any fixed Q and unit vector y. the
cone-beam projection data g(/3, Q ) are first projected along p. y = eonslam The projected
values are then multiplied by the filter kernel H,(/3.y ) . The intermediate function G ( y , a)
is the integral o f the product. The vector W A ) is the tangent vector to orbit @(A).
Figure 4 depicts the procedure in equation ( 4 ) . I f we once again treat the detector
plane Pm as a two-dimensional image plane, this is a weighted back-projection. In
equation (4) we use a local coordinate system, where the longitude ‘p of vector yL is
fixed, and the azimuth 0 is varying. In fact, g, depends on neither the local co-ordinate
system nor point x. In figure 4 ( b ) , we show a n equivalent way to evaluate g,:
where we use p and @ as variables of g, similar to g(/3,@) in equation ( 1 ) . I n equation
(4) the first variable, q, of g, is a scalar, whereas in equation ( 7 ) the first variable, p ,
568
G L Zeng and G T Gullberg
/
E = @'(h)xD
(b)
of the local coordinate system, i n which vector LJ i s orthogonal
to *(A)
and the reconstruction point x is in the plane spanned by @'(A) and D. In equation
(4). angle 'p is fired and angle 0 i s varying. ( h ) An equivalent way to obtain g,(B,'b) by
back-projecting G ( y , a)/M , ( y , * ) for ply.
Figure 4.
(0)
Illustration
of g , is a vector. We have
where p is in the plane spanned by @'(A) and D, p l y , ( O , 9 )and y = yL.I f M , ( y ,@) =
I , g , is the back-projection of G ( y ,a)and g , ( p , @) = g ( p , @), which was shown by
Smith (1987). Usually, M , ( y , a)is not constant, the back-projection is ,weighted by
l/M,(y,@D)and hence gLP, @I # g ( P , @I.
Circle-and-line orbit tomography algorithm
569
2.3. Frequency domain implementation of g,
In this section, a frequency domain approach to evaluate g, is investigated in order
to obtain gs without computing the three-dimensional function. Let us first assume
that M , ( y ,8 ) 1, thus gXp, 8 )= g(p, 8 ) .I n this section, 8 is fixed and therefore can
be dropped to simplify our notations. We project the cone-beam ray-sum g ( p , 8 )on
the two-dimensional detector plane Po to obtain a two-dimensional function gl(r) as
shown in figure 2, where r is a two-dimensional vector. Similarly we define g , , ( r ) on
P*. The two-dimensional Fourier transform of g , ( r ) (or g , , ( r ) ) is denoted by i , ( w )
(or & , ( U ) ) , where w is a two-dimensional vector in the frequency domain. We have
i l ( w ) = i t i ( wsince
)
gl(r)=stl(r).
We mentioned in section 2.2 that equation (4) rewritten as equation (7) is merely
a parallel-beam backprojection, hence a slice through the origin of the two-dimensional
Fourier transform i t , ( w ) taken at a n angle is the one-dimensional Fourier transform
of the projection of g , , ( r ) at the same angle or is the one-dimensional Fourier transform
of G divided by IwI. Since f , ( w )= & , ( w ) , the one-dimensional Fourier transform of
G divided by I w Jis one slice of f , ( w ) .
Now we can remove the restriction M , ( y , a)=1, but assume that M , ( y , 8 ) is a
constant for any family of y's lying in a plane. This assumption is not true in general.
However, we will provide a means to handle the projection data so that this assumption
is true for our particular cone beam vertex orbit in the following section.
In the situation of M , ( y ,8 )as a constant for any family of y's lying in a plane, g,
can be evaluated as follows.
( 1 ) Compute f , ( w ) , the two-dimensional Fourier transform of g , ( r ) .
( 2 ) To obtain & ( w ) , scale i , ( o )by F ( o ) .
(3) Compute the two-dimensional inverse Fourier transform of & ( w ) to obtain
g t , ( r ) . Finally,
sd(~.@)
=gt,(r)
(10)
with
tanv=r.5/II@II
(11)
where 6 is a two-dimensional unit vector orthogonal to the axis of rotation as shown
in figure 2. Note that F ( w ) is a two-dimensional filter kernel in the frequency domain
defined to be 1/ M , ( y , O ) , where the orientation of w is the same as the orientation of
the projection of the three-dimensional vector y on the two-dimensional plane Po.
The filter kernel F ( o ) is constant along lines through the origin.
2.4. Circle-and-line orbit algorithm
In section 2.3, we derived a two-dimensional filter F ( w )= 11M , ( y ,8 )in the frequency
domain in order to compute gs. However, the assumption that M, is not a constant
for any family of y's lying in a plane is not valid in general. In this section, we consider
a special case and tailor it to meet this assumption.
In this paper, we only consider the orbits consisting of a circle (in the x-y plane)
and an orthogonal line (parallel to the z-axis) as shown in figure 1 . Suppose that our
object has a support of a sphere [ [ x ( l < R as shown in figure I , then the required
half-length a of the line-orbit is 2 D R ( D ' - R 2 ) - " 2 by the data sufficiency condition.
Consequently, any plane cutting through the object will intersect the orbit, and may
intersect the orbit 1, 2, 3 or an infinite number of times.
570
G L Zeng and G T Gullberg
Let us define a plane that cuts through the object as a cutting plane, and define the
plane angle as the angle between the plane normal and the z-axis. It can be verified
that a cutting plane with a plane angle less than sin-'(R/D) intersects the linear orbit
and a cutting plane with a plane angle greater than sin-'(R/D) intersects the circular
orbit. Some cutting planes may intersect both orbits. We can divide the cutting planes
into two groups: (1) with plane angle >sin-'(R/D) and (2) with plane angle
<sin-'(R/D) as shown in figure 5. The planes in the first group always cut the circular
orbit, and may cut the linear orbit also. Similarly, the planes in the second group
always cut the linear orbit, and may cut the circular orbit also.
lb)
Figure5 T h e regiansaftheplanenormal:
( U ) the first group and ( b ) the second
group.
V
Figure 6. For the linear orbit, the filter is spatially varying,
different filters for different slices.
For the circular orbit data, we use the planes restricted to the first group as shown
in figure 5 ( a ) . For the linear orbit data, we use the planes restricted to the second
group as shown in figure 5 ( b ) . Consequently, we have M , ( y , 9 ) - 2 for CP on the
circular orbit and M,(y, a)= 1 for 9 on the linear orbit. We thus separate the data
into two groups, and make M,( y, e)such as to satisfy our assumption that M,(y, 9)
is a constant for any family of y's lying in a plane. Further, we even make M , be a
constant for each orbit.
The filter F ( w ) is also used to select the planes. For the circular orbit, F ( w ) = f if
the plane belongs to the first group and F ( w ) = O otherwise. For the linear orbit,
F ( w ) = 1 if the plane belongs to the second group and F ( w )= 0 otherwise.
The filter F ( w ) for the circular orbit is defined by
Circle-and-line orbit tomography algorithm
571
At each projection angle, g, is obtained by equation (10) and
(13)
I F R ( F c i r c d w ) X FFT(gi))
gti
where FFT is the two-dimensional fast Fourier transform operator and IFFT is the
two-dimensional inverse fast Fourier transform operator. In order to avoid aliasing,
we zero-pad g, before taking the FFT.
For the linear orbit, due to the cone shape in figure 5 ( b ) , the selecting angle for
each row of the projection data is different. The selecting angle depends on the distance,
d, between the particular row and central cone plane. Therefore, the filter F ( w ) is a
spatially varying two-dimensional filter, one filter kernel for each row. The filter F ( w )
for the linear orbit is defined by
L A w ,4=
( b ( d ) l w , l a D J o , l ) and
( b ( d ) l w , l < D l w h l ) or
1
IO
(d<O.Sa)
(d>O.Sa)
(14)
where oh and wv are the horizontal and vertical components of w, respectively, and
b ( d ) is defined as
b
(
d
)
=
m
(15)
as illustrated in figure 6. Thus, g,, of a certain row d is obtained by equation (10) and
= IFm(F~ine(w,d ) X Fm(gi)).
Different filters are used to obtain g,, for different rows
3. Computer simulation
3.1. Phantom and orbit
A mathematical three-dimensional phantom (see figure 7) was used to verify our
algorithm. The phantom was a uniform sphere containing four hot and three cold
t
I
i
Figure 7. A malhematical phantom: a uniform sphere containing four hot discs and three
cold discs.
512
G L Zeng and G T Gullberg
discs. The sphere was of radius 25 and density 1. The four hot discs had identical
densities of 2. The densities for the three cold discs were identically equal to 0. All
discs had a radius 16 and thickness 2. The distance between the adjacent discs was 5.
The central disc was at the circular orbit plane ( x - y plane). The radius of the circular
orbit was 112 and the half-length of the linear orbit was 51.294. The units are in
sampling bin widths.
3.2. Methods
We generated the cone-beam projections in 128 64 x 64 arrays. Both the circular and
the linear orbits had 64 views with uniform spacing between two adjacent views. For
each view, we did the following:
(1) Zero-padded the 64 x 64 array into a 128 x 128 array, in order to reduce aliasing.
(2) Computed the FFT for the 128x 128 zero-padded array.
(3.C) (For the circular orbit only)
(3.1.C) Multiplied the 128 x 128 Fourier transformed array by Fc;rc,e(w).
(3.2.C) Computed the inverse FFT of the array, obtaining gs.
(3.L) (For the linear orbit only)
(3.1.L) Made 26 copies of the 128 x 128 Fourier transformed array obtained
from step (2). [Here we considered the case that a=51.294; thus 26
was the smallest integer greater than 0.5a (see equation (IZ)).]
n
. ... ..
(3.2.i) iviuiiipiied i i x x 26 copies by F;j,,z(e,
4 ) .?OK d = 1 , 2 , , , , ,L",
!cL.:cL;tively.
(3.3.L) Computed the inverse m-r of these 26 arrays.
(3.4.L) Constructed a new 128 x 128 array g s from these 26 arrays. The central
52 rows of g s were numbered from -26 to 26 (without 0). Rows *k of
g, were taken from rows z t k in the kth array in (3.3.L). Rows + k ( k > 26)
were filled with zeros.
(4) Applied equation ( 5 ) to g , obtained in step (3).
(4.1) Pre-weighted the array with lcos PI.
(4.2) Filtered the array one-dimensionally row-by-row with a ramp filter.
(4.3) Performed weighted back-projection of the array into a 64x 64 x 64 image
volume.
I
3.3. Results
The cone-beam reconstruction algorithm was coded on a SUN4 computer in Fortran-77.
The image was reconstructed separately for the circular and linear orbits. The final
image was the sum of these two reconstructions. The reconstruction time for the circular
orbit was 1120 s and for the linear orbit 3200 s. The projection data from the circular
orbit alone were also used to reconstruct an image by the Feldkamp algorithm, with
reconstruction time of 1000 s. The final images are compared in figure 8, where the
central sagittal views are shown at the top and the transaxial views of the top disc are
shown at the bottom. It was observed that the artifacts which are very severe in
Feldkamp's reconstruction are greatly reduced with the orthogonal circle-and-line orbit
geometry, yet some artifacts can still be observed which require further investigation.
In order to reconstruct an object defined in llxll< R, the half-cone-angle should be
at least sin-'(R/D), that is, the detector plane should contain a disc of radius
D R / ( D 2 - R 2 ) p 2 . Intuitively, the detector plane should be large enough so that the
573
Circle-and-line orbit tomography algorithm
--w
2.:
- 3 :
52 $
=.
3 P.. sx
,?
p
,a
t2 sZ -z
:
'G
a.:
A*
'$
E
L
c
.
3"
-e
M
f s *
0
.E
2m su.cU
E Z U
C
L
yI
a'?2
sj>
:n;.
:2 :
LL
..
0
&.C$
5 - 0
z.5 s
-2 -
? E 8
V
U
C
.L
Y
~
Z V S
.ev E
Z.5 2
E E
$ 2 5
.E-.c
.= z
o sz
M
S
E$
-.c
ii .E 2
-$22
.-E 2 25 owg
yI
- ; ; o x .
nn
VI
P O .
g o a o
" 5 s ;
m
A;'=
-,a o w
g g x ;c
G V "
- - *&- z
'j 2
2 2,s:
-
.L
2I v ) x $ .bz
;:2
L U -
g .E
&""
.8:
VI
.-E
I
- 0
$
514
G L Zeng and G 7 Gullberg
projection data from the circular orhit does not have any truncations. The same detector
is used for the linear orbit. As the detector moves along the z-axis, the object may be
truncated (i.e outside the field of view). However, the truncated data are usually
unwanted. If they had been measured by a large field-of-view camera, we would have
used the spatially varying filter to discard them as in step (3.4.L), where all values
above row 26 or below row -26 are set to zero. In figure 9, one cone-beam projection
( g ) from the linear orbit is shown on the left, and its pre-processed version ( g , ) is
shown on the right. These two images were scaled separately. The minimum value
corresponds to the darkest point (0) and the maximum value corresponds to the
brightest point (255). The background in each image is zero. There are no negative
values in g, whereas there are negative values in g,. One also observes that only the
central part of gs is non-zero.
4. Conclusions
In this paper, we have modified Smith's cone-beam reconstruction algorithm and
applied it to an orthogonal circle-and-line orbit. The basic idea of our modification is
to discard unwanted data such that a cutting plane either intersects the circular orbit
twice or intersects the linear orbit once. In order to achieve this consistency, some
projection data are rejected by the pre-filtering procedure. For the circular orbit, this
filter is spatially invariant. However, for the linear orbit, it is spatially varying. The
filters were implemented in the frequency domain.
When one uses this algorithm, the whole object should he within the field of view
when the camera rotates in the circular orbit. When the camera moves in the linear
orbit, parts of the object may be outside the field of view. This situation has been
handled properly by the algorithm and will not cause any problems.
This one-circle and perpendicular Fine orbit is easy to realize in a SPECT system by
rotating the gamma camera 360" and translating the camera or patient bed. For a
~
Circle-and-line orbit tomography algorithm
Figure 10. Orbit geometries for two-head and three-head
575
SPECT systems.
two-head or three-head SPECT system, an orbit of one-circle and two (or three)
perpendicular lines is formed as shown in figure 10. The reconstruction algorithm is
identical with that presented in this paper, except that the reconstructed image for
each linear orbit should be divided by two (or three) before summing up to obtain
the final image.
Acknowledgments
The research work presented in this paper was partially supported by N I H Grant R 0 1
H L 39792 and Picker International. We thank D r R Clack of Albert Ludwig University,
Germany, for useful discussions, and Dr B D Smith of the University of Cincinnati,
Cincinnati, OH, for providing us with his dissertation. We also thank Biodynamics
Research Unit, Mayo Foundation, for the use of the Analyze software package.
516
G L Zeng and G T Gullberg
Rhm6
Un algorithme tomographie adapt6 aux faisceaux
lineairen orthagonales.
P geometrie coniques pour des orbites cicculaires et
Ce travail pr6sente un algorithme adapt6 aux faisceaux P geometric conique qui rend possible l a mise en
oeuvre pratique d e la formule d'inversion d e B D Smith pour les faisceaux P geometric conique. D a m cet
algorilhme, lesommet du faisccau conique decril une orbite resultant de la combinaison d'un cercle et d'une
ligne orthagonale. Cette geomCtrie est facile P implanter sur un systeme SPECT et repand P l a condition d e
suffiiance des donnees en faisceaux coniques. L'algorithme propose est du type rttroprolection-convolulion
et necessite une procedure de prefiltrage. Des simulations par ordinateur montrent m e reduction des artefacts
risultant de l'algorithme d e Feldkamp pour lequel le sommet du faisceau conique est un cercle.
Zusammenfassung
Ein Kegelstrahltomographieverfahrenalgorifhm"~fkr onhogonden Kreis- und Linienkurven
In der vorliegenden Arbeit wird ein Kegelstrahlalgorithmus vorgertellt, der tine praktische Einfiihrung der
Kegelstrahlinersionsformel YO" B D Smith datstellt. In diesem Algorithmus besteht die Kurve der Kegelslrahlspitze aus einem Kreis und einer orthogonalen Linie. Diese Geometric ist leicht zu implementiercn in
einem SECT System und erfkllt die Hinl~nglichkeitibedingungyon Kegelstrahldaten. Der vorgeschlagene
und erfarden ein Vorfilterverfahren. Computersimulationen
Algorithmus ist eine Entfaltungs-Riickpra~kli~"
zeigen eine Verringerung der Anefakte, die sich a u dem Feldkamp-Algorithmus ergeben, wen" die Kurve
der Kegeistrahispitze ein Kreis ist.
References
Clack R, Zeng G L, Weng Y, Christian P E and Gullberg G T 1991 Cone beam single photon emission
computed tomography using two orbits Informorion Processing in Medical Imaging, XIIlh IPMI Internolionol Conference ( K e n t , 1991) e d A Colchester and D Hawker (New York: Springer) pp45-54
Feldkamp L A, Davis L C and Kress J W 1964 Practical cone-beam algorithm 3. Opr Soc. Am. A I 612-9
Floyd C D Jr, Jaszczak R J, Greer K L a n d Coleman R E 1986 Cone beam collimation for SPECT: Simulation
and reconstruction IEEE Tronr. Nucl. Sci. NS-33 511-4
Grangeat P 1987 Analysis d'un s y s t h e d'lmagerie 3D par reconstruciton P panir d e X en g6ometrie conique
PhD T h i s I'Ecole Nationale Superieure des Telecommunications
Gullberg G T. Zeng G L, Christian P E, Tsui B M W and Morgan H T 1991 Single photon emission
computed tomography of the heart using cone beam geometry and noncircular detector rotation
Informorion Proeersing in Medical Imaging, Xlrh IPMl lnrernorionol Conference (Berkeley, CA 198'2)
ed D A Ortendahl and J Llacer ( N e w York: Wiley-Liss) p p 123-38
Gullberg G T, Christian P E, Zeng G L, Datz F L and Morgan H T 1991 Cone beam tomography of the
heart using single-photon emission-computed tomography Inuerr. Radiol. 26 681-8
Hahn L J, Jaazczak R J, Gullberg G T, Floyd C E, Manglos S H, Greer K L a n d Coleman R E 1988 Noise
characteristics for cone beam collimators: a comparison with parallel hole collimator Phys. Med. Bid.
33 541-55
Jaszczak R J, Floyd C E, Manglos S H , Greer K L and Coleman R E 1986 Cone beam collimation for
single photon emission computed tomography: analysis, simulation, and image reconstruction using
filtered backprojection Med. Phy.7. 13 484-9
Jaszczak R J. Greer K L and Coleman R E 1988 SPECT using a specially desiened cone beam Collimator 3.
Nucl. Med. 29 1398-1405
Kirillov A A 1961 On a Droblem of I M Gel'fand Sou. Math. Dokl. 2 268-9
Ning R. Hu H and Kruger R A 1988 Image intensifier-based CT volume imager for angiography SPlE
Medical Imaging II Conf. (Newport Beach, CA, 1988) SPlE 914 282-90
Circle-and-line orbit tomography algorithm
511
Smith B D 1985 Image reconstruction from cone-beam projections: necessary and sufficient conditions and
reconstruction methods IEEE Trans. Med. Imag. M I 4 14-25
-I987 Computer-aided tomographic imaging from cone-beam data PhDDisserrorion University of Rhode
Island
-1990 Cone-beam tomography: recent advances and a tutorial review Opt. Eng 29 524-34
Terry J A, Tsui B M W, Perry J R, Gullberg G T and Zeng G L 1991 A comparison of myocardial defect
detection in TI-201 S E C T using parallel-hole, fan beam and cone beam collimators J. Nucl. Med. 32 946
Tuy H K 1983 An inversion formula far cone-beam remnstwtion SIAM J. Appl. Math. 43 546-52
Yan X H and Leahy R M 1991 Derivation and analysis of a filtered backprojection algorithm for cone beam
projection data IEEE Tmne Med. Imog. MI-10 462.12