Cone Beam Fan Beam

Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular
harmonics
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INSTITUTE OF PHYSICS PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 49 (2004) 2239–2256 PII: S0031-9155(04)70992-6
Cone-beam and fan-beam image reconstruction
algorithms based on spherical and circular harmonics
Gengsheng L Zeng
1
and Grant T Gullberg
2
1
Utah Center for Advanced Imaging Research, University of Utah, 729 Arapeen Drive,
Salt Lake City, Utah 84108, USA
2
E O Lawrence Berkeley National Laboratory, One Cyclotron Road, Mail Stop 55R0121,
Berkeley, CA 94720, USA
E-mail: [email protected] and [email protected]
Received 27 October 2003
Published 19 May 2004
Online at stacks.iop.org/PMB/49/2239
DOI: 10.1088/0031-9155/49/11/009
Abstract
A cone-beam image reconstruction algorithm using spherical harmonic
expansions is proposed. The reconstruction algorithm is in the form of a
summation of inner products of two discrete arrays of spherical harmonic
expansion coefficients at each cone-beam point of acquisition. This form is
different from the common filtered backprojection algorithm and the direct
Fourier reconstruction algorithm. There is no re-sampling of the data, and
spherical harmonic expansions are used instead of Fourier expansions. As a
special case, a new fan-beam image reconstruction algorithm is also derived
in terms of a circular harmonic expansion. Computer simulation results for
both cone-beam and fan-beam algorithms are presented for circular planar
orbit acquisitions. The algorithms give accurate reconstructions; however, the
implementation of the cone-beam reconstruction algorithm is computationally
intensive. A relatively efficient algorithm is proposed for reconstructing the
central slice of the image when a circular scanning orbit is used.
1. Introduction
Over the last ten years there have been remarkable advancements in cone-beam image
reconstruction algorithms. Concurrent to the algorithm developments there have been
significant advancements in x-ray CThelical detector hardware. This technology has been, and
remains, a fundamental component of the medical industry. Continuing advancements in cone-
beam reconstruction theory are important for the development of better and faster algorithms.
The goal of this paper is to develop a cone-beam data-interpolation-free algorithm, which
may have a potential to outperform the current cone-beam image reconstruction algorithms in
terms of image resolution.
0031-9155/04/112239+18$30.00 © 2004 IOP Publishing Ltd Printed in the UK 2239
2240 G L Zeng and G T Gullberg
This paper presents a cone-beam image reconstruction algorithm that utilizes spherical
harmonic expansions. Cormack and others first applied a circular harmonic expansion to
the reconstruction of 2D images from parallel projections (Cormack 1963, 1964, Marr 1974,
Hansen 1981). Hawkins et al (1988) successfully used circular harmonic expansions to
develop an algorithm for the reconstruction of exponential Radon projection data and applied
it to single photon emission computed tomography (SPECT). Later, 2D image reconstruction
of data acquired using converging geometries was performed using harmonic expansions
(You et al 1998). Our paper is the first to propose the use of spherical harmonic expansions to
reconstruct 3D images from cone-beam projection data.
In previous work (Cormack 1963, 1964, Hawkins et al 1988, You et al 1998) the
expansion of the projection data were first found; then a relationship between the expansion
coefficients of the data and the expansion coefficients of the image was derived. Using this
relationship, the expansion coefficients of the image are determined and the final image is
reconstructed by synthesizing these expansion coefficients. Only for the parallel geometry
(Cormack 1963, 1964, Marr 1974, Hansen 1981) does an orthogonal polynomial expansion in
the image space correspond to an orthogonal polynomial expansion in the projection space.
In our work we take a different approach from those mentioned previously. We first
followed a similar approach wherein we developed a relationship between the expansion
coefficients of the data and the expansion coefficients of the image. However, we found
implementation of this approach to be numerically unstable. Instead we developed a method
that does not require a spherical expansion of the final image. We simply keep the spherical
expansion coefficients of the data until the final step of the image reconstruction procedure.
Our final image is not the synthesis of the image expansion but rather it is an inner product
of the expansion coefficients of the cone-beam data with the expansion coefficients of the
filter, which is summed over all sampled cone-beam vertex points. In our earlier publications
(Basko et al 1999, Taguchi et al 2001), we used the spherical harmonic expansion to convert
the cone-beam line integrals into the first derivative of the Radon planar integral; second half
of Grangeat’s algorithm (Grangeat 1991) was used to reconstruct the image. In this paper,
we use the spherical expansion to convert the data, and to transform the tomographic filter
kernel as well, in order to reconstruct the 3D image. The approach is shown to lead to a new
formulation of the fan-beam reconstruction problem.
For the case of fan beam, a closely related method is the direct Fourier reconstruction
method. In that method the Fourier transform of the projection data is first calculated via
rebinned parallel beam data. Then, by using the well-known central slice theorem the Fourier
transformof the image is determined. The final image is obtained by taking the inverse Fourier
transform of the result (Bracewell 1956). Our method is similar except that the fan beam data
is not rebinned into the parallel geometry, the central slice theorem is not used, and there is no
re-sampling of the data.
This paper first presents the development of the algorithmbased on spherical harmonics for
3Dcone-beamreconstruction. It is then shown howsimilar results based on circular harmonics
can be derived for 2D fan-beam reconstruction. Subsequent to that, computer simulations are
presented showing results of both the reconstruction of cone-beam data acquired from a
circular orbit and the reconstruction of fan-beam data. Our cone-beam result is then compared
with that obtained from Grangeat’s algorithm. Finally a discussion is presented.
2. Cone-beam data reconstruction
In our algorithm, we first find the spherical harmonic expansion of the cone-beamprojections at
each cone-beam focal point location. We then transform the expansion coefficients into those
Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics 2241
ψ
n
Cone-beam detector
Vertex orbit
Cone-beam ray
g
ψ
n ( )
z
Ray with θ = π/2 and ϕ = 0
θ = 0
Figure 1. In a cone-beam imaging geometry g


(

n ) is a line integral of an object, and

n is defined
in a local coordinate system with latitude and longitude angles θ and ϕ.
of the first-order derivative of Radon integrals. We finally reconstruct the image based on the
Radon inversion formula. This method is different fromthe existing cone-beamreconstruction
algorithms, e.g., Grangeat’s and Katsevich’s cone-beam reconstruction algorithms (Grangeat
1991, Katsevich 2003) in the sense that the proposed algorithm does not require forming
line-integrals on the projection plane, rebinning data or evaluating derivatives at some certain
directions.
2.1. Step 1: the data acquisition
Cone-beam projection g


(

n) are acquired at vertex location

, where

n is a unit vector
indicating the direction of each cone-beam projection ray (see figure 1). Here it is assumed
that the detector is large enough so that the projection data are not truncated.
2.2. Step 2: the harmonic expansion of the data
It is known that any periodic function (or any function defined on a circle) can be expanded as
a Fourier series (circular harmonics). Similarly, any function defined on the unit sphere can
be expanded in spherical harmonics. At each fixed vertex location

, a spherical harmonic
expansion of g


(

n) is
g


(

n) =
N
l

l=0
l

m=−l
g


lm
Y
lm
(

n) (1)
where N
l
is determined by the bandwidth of the projection data and Y
lm
(

n) can be any
orthonormal basis on the surface of the sphere. Here we use
Y
lm
(

n) = Y
lm
(θ, ϕ) =
1
2π

lm
(cos θ) e
imϕ
(2)
with 0 θ < π, −π ϕ < π, and

lm
(t ) =
_
¸
¸
¸
¸
_
¸
¸
¸
¸
_
_
2l + 1
2
(l − m)!
(l + m)!
P
m
l
(t ) for m 0
(−1)
m
_
2l + 1
2
(l − |m|)!
(l + |m|)!
P
|m|
l
(t ) for m < 0
(3)
2242 G L Zeng and G T Gullberg
where P
m
l
(t ) is the associated Legendre function and
lm
(t ) is the normalized associated
Legendre polynomial. Both P
m
l
(t ) and
lm
(t ) can be readily evaluated with commercially
available software such as Matlab (The MathWorks 2002).
Taking the advantage of orthogonality of Y
lm
(

n), i.e.
_ _
S
Y
lm
(

n)Y
∗
l

m
(

n) d

n = δ
ll
δ
mm
(4)
the expansion coefficients can be calculated as
g


lm
=
_ _
S
g


(

n)Y
∗
lm
(

n) d

n (5)
or
g


lm
=
1
2π
_
π
−π
_
π
0
g


(θ, ϕ)
lm
(cos θ) e
−imϕ
sin θ dθ dϕ (6)
where S represents a unit sphere. If the half-cone-angle of the cone-beam detector is α, (6)
can be evaluated as
g


lm
=
1
2π
_
α
−α
_
π/2+α
π/2−α
g


(θ, ϕ)
lm
(cos θ) e
−imϕ
sin θ dθ dϕ. (7)
2.3. Step 3: the derivative of the Radon planar integral
In order to reconstruct (see equation (11)), we need to obtain the derivatives of Radon planar
integrals for all planes intersecting the point

. Let us denote the derivatives of Radon planar
integrals for all planes intersecting the point

as p


(

n), with

n being the normal vector
of the associated plane (see figure 3). The derivatives are taken in the direction of

n. The
function p


(

n) is defined on the unit sphere and has a spherical harmonic expansion:
p


(

n) =
N
l

l=0
l

m=−l
p


lm
Y
lm
(

n) (8)
where

n is the normal vector of the plane. Previously, we have shown a relationship between
coefficients for the expansions (1) and (8) (Basko et al 1999, Taguchi et al 2001)
p


lm
= −2πlP
l−1
(0)g


lm
(9)
where P
l−1
(.) is the Legendre function and P
l−1
(0) = 0 if l is even. Therefore,
p


(

n) =

l odd

m
_
−2πlP
l−1
(0)g


lm
_
Y
lm
(

n). (10)
In (1), the

n is the direction of a cone-beam ray, while the same

n in (10) is the normal
direction of a plane. This plane is orthogonal to the ray with the direction

n.
2.4. Step 4: the image reconstruction
According to Radon inversion formula, a 3D image f (

x ) can be reconstructed from its Radon
projection by
f (

x ) = −
1
8π
2
_ _
S
∂
∂t
p(t,

k )
¸
¸
¸
¸
t =

x ·

k
d

k (11)
Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics 2243
x
y
ψ
n
k
ϕ
λ λ π ϕ – +
Orbit
Radon plane with
as its normal vector
n
t
Cone-beam ray
Focal point
R
0
Figure 2. Top view of the xyz coordinate system with a special case where

n is in the xy-plane.
where p(t,

k ) is the first-order derivative of the Radon projections with respect to t (t 0), and
S represents a unit sphere. The derivative operation in the Fourier domain is a multiplication
by iν where ν is the frequency, and can be implemented as a convolution with a kernel h
h(t ) =
_
∞
−∞
2πiν e
2πit ν
dν. (12)
For band-limited signals we can use
h(t ) =
_

−
2πiν e
2π it ν
dν
= −
_

−
2πν sin(2πt ν) dν
=
_
_
_
0 if t = 0
2cos(2πt )
t
−
sin(2πt )
πt
2
if t = 0.
(13)
Here is the signal bandwidth and = 0.5 corresponds to the highest frequency that is
determined by the sampling interval. Thus (11) can be rewritten as
f (

x ) = −
1
8π
2
_ _
S
_
∞
−∞
p(t,

k )h(

x ·

k − t ) dt d

k . (14)
In the above equations, p(t,

k) can be related to p


(

n). Here

n is defined in the local
coordinate system as shown in figure 1, and

n =

0 points to the centre of the cone-beam
detector. On the other hand,

k is defined in the global coordinate system where the vertex-
orbit vector

is defined.
If we consider a circular orbit (see figure 2)

= R
0
(cos λ, sin λ, 0) (15)
2244 G L Zeng and G T Gullberg
ψ
k
Vertex orbit
t
p
ψ
n ( )
n
z
x
y
ϕ
g
θ
g
Figure 3. A Radon plane that contains a cone-beam vertex

. Both

n and

k are normal to the
plane.
and let the unit vectors

k be

k = (sin θ
g
cos ϕ
g
, sin θ
g
sin ϕ
g
, cos θ
g
) (16)
then the unit vector

n points at the same direction as

k (see figure 3) except that

n is defined
in a local coordinate system as shown in figure 1. The unit vector

n is parametrized by its
latitude and longitude angles θ and ϕ. When θ = 0,

n points to the z-axis direction, and when
ϕ = 0,

n points to the centre of the detector. We have (see figure 2)
p(t,

k ) = p


(

n) (17)
where
θ
g
= θ (18)
ϕ
g
= λ + π − ϕ (19)
t =

·

k = −R
0
sin θ cos ϕ. (20)
If we change the variables (t, θ
g
, ϕ
g
) to variables (λ, θ, ϕ), the transformation Jacobian is
given by
J = R
0
sin θ|sin ϕ|. (21)
By changing the variables, (14) becomes
f (

x ) = −
1
4π
2
_ _
S
_
2π
0
p


(

n)h(g)J dλ d

n (22)
where g is defined as
g =

x ·

k − t =
_
_
x
y
z
_
_
·
_
_
−sin θ cos(ϕ − λ)
sin θ sin (ϕ − λ)
cos θ
_
_
+ R
0
sin θ cos ϕ. (23)
Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics 2245
Changing the order of integrations, we have
f (

x ) =
−1
8π
2
_
2π
0
_ _
S
p


(

n)h(g)J d

n dλ. (24)
In the following, we substitute the expansion (8) into (24) and use the fact that p


lm
= 0
for even l, and we have
f (

x ) =
−1
8π
2
_
2π
0
_ _
S
N
l

l=1
l odd
l

m=−l
p


lm
Y
lm
(

n)h(g)J d

n dλ (25)
or
f (

x ) =
−1
8π
2
_
2π
0
N
l

l=1
l odd
l

m=−l
p


lm
_ _
S
h(g)JY
lm
(

n) d

n dλ. (26)
Equation (26) can be further written as
f (

x ) =
−1
8π
2
_
2π
0
N
l

l=1
l odd
l

m=−l
p


lm
h
∗
lm
dλ (27)
where h
∗
lm
is the complex conjugate of the spherical harmonic expansion coefficient of h(g)J
h
lm
=
_ _
S
h(g)JY
∗
lm
(

n) d

n. (28)
Equation (27) is the spherical harmonic cone-beam image reconstruction formula in the form
of an inner product.
3. Special case: central slice reconstruction
If a circular scanning orbit is used, the central slice of the image volume can be reconstructed
more efficiently. The central slice is the 2D image in the plane that contains the circular
cone-beam orbit. In this special case, the function g in (23) can be further simplified because
z = 0 for the central plane:
g =

x ·

k − t =
_
_
x
y
0
_
_
·
_
_
−sin θ cos(ϕ − λ)
sin θ sin(ϕ − λ)
cos θ
_
_
+ R
0
sin θ cos ϕ (29)
= [(−R
0
+ x cos λ) cos ϕ + (x sin λ − y cos λ) sin ϕ] sin θ (30)
= [Acos ϕ + B sin ϕ] sin θ (31)
where
A = −R
0
+ x cos λ and B = x sin λ − y cos λ. (32)
Let g = 0, we have two solutions for ϕ
ϕ
1
= tan
−1
_
−A
B
_
and ϕ
2
= tan
−1
_
A
−B
_
(33)
and
_
∂
∂ϕ
g
¸
¸
¸
¸
ϕ=ϕ
1
_
2
= [−Asin ϕ
1
+ B cos ϕ
1
]
2
sin
2
θ = (A
2
+ B
2
) sin
2
θ (34)
2246 G L Zeng and G T Gullberg
_
∂
∂ϕ
g
¸
¸
¸
¸
ϕ=ϕ
2
_
2
= (A
2
+ B
2
) sin
2
θ. (35)
Sine A < 0 in (32), we always have sin ϕ
1
= −A/
√
A
2
+ B
2
> 0 and sin ϕ
2
=
A/
√
A
2
+ B
2
< 0.
Recall from (12) that h(t ) = δ

(t ), a delta function property gives
h(g) = δ

(g) =
δ

(ϕ − ϕ
1
)
[g

(ϕ
1
)]
2
+
δ

(ϕ − ϕ
2
)
[g

(ϕ
2
)]
2
=
δ

(ϕ − ϕ
1
) + δ

(ϕ − ϕ
2
)
(A
2
+ B
2
) sin
2
θ
. (36)
Therefore, (28) can be further evaluated as follows, using (21) and (2):
h
lm
=
_ _
S
h(g)JY
∗
lm
(

n) d

n (37)
=
_
π
0
_
2π
0
δ

(ϕ − ϕ
1
) + δ

(ϕ − ϕ
2
)
(A
2
+ B
2
) sin \
2
θ
(R
0
sin \ θ|sin ϕ|)
_
1
2π

lm
(cos θ) e
−imϕ
_
sin \ θ dϕ dθ
(38)
=
R
0
2π(A
2
+ B
2
)
_
π
0

lm
(cos θ) dθ
_
2π
0
[δ

(ϕ − ϕ
1
) + δ

(ϕ − ϕ
2
)]|sin ϕ| e
−imϕ
dϕ (39)
=
R
0
2π(A
2
+ B
2
)
q
lm
_
2π
0
[δ

(ϕ − ϕ
1
) + δ

(ϕ − ϕ
2
)]|sin ϕ| e
−imϕ
dϕ (40)
=
−R
0
2π(A
2
+ B
2
)
q
lm
_
2π
0
[δ(ϕ − ϕ
1
) + δ(ϕ − ϕ
2
)][|sin ϕ| e
−imϕ
]

dϕ (41)
=
−R
0
2π(A
2
+ B
2
)
q
lm
B + imA
√
A
2
+ B
2
[e
−imϕ
1
− (−1)
m
e
−imϕ
1
] (42)
=
_
_
_
−R
0
π(A
2
+ B
2
)
q
lm
B + imA
√
A
2
+ B
2
e
−imϕ
1
if m is odd
0 if m is even
(43)
where
q
lm
=
_
π
0

lm
(cos θ) dθ (44)
is independent of the projection measurements and can be pre-calculated. Finally, the
reconstruction algorithm (27) for the central slice is reduced to
f (

x ) =
R
0
8π
3
_
2π
0
1
(A
2
+ B
2
)
3/2
l

m=−l
modd
(B − imA)
_
_
_
N
l

l=1
l odd
p


lm
q
lm
_
_
_ dλ. (45)
Algorithm (45) is more efficient than algorithm (27). In (27), a different spherical harmonic
expansion of h is required for every reconstruction point

x and every focal point position (λ),
while the expansion of h is not required in (45). The coefficients q
lm
of (44) only needs to be
evaluated once.
Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics 2247
x
y
ψ
ϕ
λ
φ
Orbit
φ
t
Projection ray
R
0
x
g – t x φ ⋅ ( ) – =
Figure 4. A fan-beam imaging coordinate system. Here ϕ is locally defined and φ is globally
defined.
4. 2D case: fan-beam reconstruction
Following the same procedure as in section 2, a similar 2D fan-beam reconstruction algorithm
can be obtained. We point out to readers that we use the same notation as in section 2. For
example, h in this section represents a ramp filter, while h represents a derivative filter in
section 2. Let us start with the 2D Radon inversion formula for the parallel-beam data p(t, φ),
and the reconstructed image f (

x ) is given by
f (

x ) =
1
2
_
π
−π
_
∞
−∞
p(t, φ)h(

x ·

φ − t ) dt dφ (46)
where h is the ramp-filter kernel. For a circular fan-beam vertex orbit

= R
0
(cos λ, sin λ) (47)
we can change the parallel-beam variables (t, φ) to fan-beam variables (λ, ϕ) (see figure 4):
p(t, φ) = p


(ϕ) (48)
where
t = R
0
sin ϕ (49)
φ =
π
2
+ λ − ϕ. (50)
The transformation Jacobian is
J = R
0
|cos ϕ|. (51)
Thus (46) becomes
f (

x ) =
1
2
_
2π
0
_
π
−π
p


(ϕ)h(g)R
0
|cos ϕ| dϕ dλ (52)
2248 G L Zeng and G T Gullberg
x
y
ψ
ϕ
λ
Orbit
Fan-beam projection ray
Focal point
R
0
x
ξ
α
τ = −g
x ψ – ( ) ∠
Figure 5. The fan-beam ray is parametrized with a locally defined angle ξ.
with
g =

x ·

φ − t =
_
x
y
_
·
_
_
_
cos
_
π
2
+ λ − ϕ
_
sin
_
π
2
+ λ − ϕ
_
_
¸
_ − R
0
sin ϕ. (53)
Let the local circular harmonic expansion of p


(ϕ)R
0
| cos ϕ| be
p


(ϕ)R
0
|cos ϕ| =

m
p


m
e
imϕ
(54)
with
p


m
=
_
2π
0
p


(ϕ)R
0
|cos ϕ| e
−imϕ
dϕ (55)
then (52) becomes
f (

x ) =
1
2
_
2π
0
_
π
−π

m
p


m
e
imϕ
h(g) dϕ dλ (56)
or
f (

x ) =
1
2
_
2π
0

m
p


m
h
∗
m
dλ (57)
where h
∗
m
is the complex conjugate of the circular harmonic expansion coefficient h
m
:
h
m
=
_
π
−π
h(g) e
−imϕ
dϕ. (58)
Equation (57) is the fan-beamimage reconstruction algorithmin the Fourier coefficient domain
in the form of an inner product. In 2D, the circular harmonic expansion is the same as the
Fourier expansion.
Equation (57) is not efficient for computer implementation, because the function g depends
on both

x and

. In the following we derive an algorithm that is more efficient than (57). For
a fixed

x and

, we introduce two angles ξ and α as defined in figure 5 with
Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics 2249
ϕ = ξ + α (59)
α = λ ± π −

(

x −

) (60)
and
τ =

x −

sin ξ. (61)
In figure 4, −g is the distance from the point

x to the fan-beam projection ray. With the
local coordinate system defined in figure 5, τ is the distance from the point

x to the fan-beam
projection ray. Thus
−g = τ =

x −

sin ξ (62)
and
dt = −d(

x ·

φ − t ) = −dg = dτ =

x −

cos ξ dξ. (63)
It is known that the ramp-filter kernel satisfies (Horn 1979)
h(−τ) = h(τ) and h(aτ) =
1
a
2
h(τ) (64)
which give
h(

x ·

φ − t ) = h(g) = h(−τ) = h(τ) = h(

x −

sin ξ) =
1


x −


2
h(sin ξ). (65)
Using (48), (59), (63) and (65), (46) becomes
f (

x ) =
1
2
_
2π
0
_
π
−π
p


(ξ + α)
1


x −


2
h(sin ξ)

x −

cos ξ dξ dλ. (66)
Let the local Fourier expansion of p


(ϕ)R
0
be
p


(ϕ)R
0
=

m
ˆ p


m
e
imϕ
=

m
ˆ p


m
e
im(ξ+α)
(67)
with
ˆ p


m
=
_
p


(ϕ)R
0
e
−imϕ
dϕ (68)
then (66) becomes
f (

x ) =
1
2
_
2π
0
_
π
−π

m
ˆ p


m
e
imα
e
imξ
1


x −


h(sin ξ) cos ξ dξ dλ (69)
or
f (

x ) =
1
2
_
2π
0
1


x −



m

p


m
e
imα
ˆ
h
∗
m
dλ (70)
where
ˆ
h
m
=
_
π
−π
h(sin ξ) cos ξ e
−imξ
dξ. (71)
Algorithm (70) is more efficient for computer implementation than (57), because
ˆ
h
m
of (71)
only needs to be evaluated once, while h
m
of (58) must be evaluated for each

x and

.
2250 G L Zeng and G T Gullberg
Table 1. 2D Shepp–Logan phantom (scaling factor = 128).
Ellipse index Half-axis ( a) Half-axis (b) Centre (x) Centre (y) Tilt angle Density
1 0.69 0.92 0 0 0 1.5
2 0.6624 0.874 0 −0.0184 0 −0.98
3 0.11 0.31 0.22 0 −18
◦
−0.2
4 0.16 0.41 −0.22 0 18
◦
−0.2
5 0.21 0.25 0 0.35 0 0.1
6 0.046 0.046 0 0.1 0 0.1
7 0.046 0.046 0 −0.1 0 0.1
8 0.046 0.023 −0.8 −0.605 0 0.1
9 0.023 0.023 0 −0.605 0 0.1
10 0.023 0.046 0.06 −0.605 0 0.1
11 0.0333 0.206 0.5538 −0.3858 −18
◦
0.03
Table 2. 3D Shepp–Logan phantom (scaling factor = 2).
Ellipsoid Half-axis Half-axis Half-axis Centre Centre Centre Tilt Tilt
index (a) (b) (c) (x) (y) (z) angle (α) angle (β) Density
1 15.43 20.574 27.093 0 0 0 0 0 2
2 14.95 19.725 26.114 0 −0.393 −0.393 0 0 −0.98
3 2.709 2.709 2.709 5.51 16.073 0 0 0 −1
4 2.709 2.709 2.709 −5.51 16.073 0 0 0 −1
5 9.76 13.011 10.837 0 0 −16.256 0 0 −1
6 0.981 0.491 0.491 −1.707 −12.907 8.128 0 0 0.48
7 0.491 0.491 0.981 0 −12.907 8.128 0 0 0.48
8 0.491 0.981 0.491 1.28 −12.907 8.128 0 0 0.48
9 0.981 0.981 0.981 0 2.133 8.128 0 0 0.48
10 5.506 5.506 5.506 0 −2.133 2.709 0 0 0.48
11 4.48 5.53 4.907 0 7.467 8.128 0 0 0.48
12 2.347 6.613 5.419 4.693 0 8.128 18
◦
0 −0.52
13 3.413 8.747 8.128 −4.693 0 8.128 −18
◦
0 −0.52
14 0.64 4.267 4.267 11.947 −8.533 8.128 18
◦
0 0.48
5. Computer simulations
Computer simulations were performed to verify the feasibility of the proposed image
reconstruction algorithms for both fan-beam and cone-beam geometries. For both imaging
geometries the fan-beam or cone-beam vertex orbit was a circle with a radius of 100 units.
The fan-beam and cone-beam focal length was 100 units. A slice of the image volume was
100 × 100 units
2
. In both imaging geometries, equiangular detectors were used. The angular
sampling interval on the detector was 0.35
◦
which was approximately the resolution of a
SPECT measurement. The fan-beam and cone-beam detectors rotated around the object 360
◦
with 120 stops. Computer-generated Shepp–Logan head phantoms were used in computer
simulations. The phantom parameters are given in tables 1 and 2 for the fan-beam and
cone-beam simulations, respectively.
The reconstruction code was written in Matlab. The fan-beam code involved the fast
Fourier transform. A 1024-point FFT was used.
The cone-beamcode involved both the fast Fourier transformand the Legendre transform.
The spherical coefficients were evaluated first by performing the Legendre transformation in
Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics 2251
Figure 6. Fan-beam reconstruction of a computer-generated head phantom using algorithm (70).
the θ direction and by performing the FFT in the ϕ direction. A 512-point FFT was used for
the ϕ angle transformation and associated Legendre polynomials of the highest degree of 512
were used for the θ angle transformation. The proposed algorithms are numerically stable.
Figure 6 shows the fan-beam reconstruction of the 2D Shepp–Logan head phantom; the
reconstruction algorithm was (70). Figure 7(a) shows the central slice of the cone-beam
reconstruction of the 3D Shepp–Logan head phantom; the reconstruction algorithm was (45).
The display window was linear and was from the minimal value to maximal value in the image
slice.
For the cone-beamdata, an image (see figure 7(b)) was also reconstructed using Grangeat’s
algorithm for comparison purpose. The projection data was almost the same as that described
above, except that an equispaced (instead of an equiangular) cone-bean detector was used.
The sampling interval was chosen such that the detector solid angles for both detectors were
the same for the central ray. For other cone-beam projection rays, the equispace-detector
had a finer angular sampling than the angular sampling in the equiangular-detector. Our
implementation details of Grangeat’s algorithm can be found in Weng et al (1993).
In order to compare these two cone-beam images, line profiles are shown in figure 7(c),
which indicate that the proposed algorithmgives more accurate reconstruction than Grangeat’s
algorithm. This can be easily seen at the thin skull regions.
6. Discussion
In the cone-beam algorithm derivation, we assumed for simplicity that the cone-beam focal
point orbit is a circle in a plane. In fact, this planar circular orbit does not satisfy Tuy’s
cone-beam data sufficiency condition (Tuy 1983). A non-planar orbit such as a helix or one
consisting of a circle and lines should be used in practice. The orbit must be parametrized
using a single parameter similar to (15). The remainder of the derivation in section 2 is
relatively unchanged for this case. However, when a set of complete data is used, the multiply
measured data must be handled properly. One way is to modify (28) by introducing a function
M(

x ,

,

n) so that
h
lm
=
_
S
_
1
M(

x ,

,

n)
h(g)JY
∗
lm
(

n) d

n (72)
where M(

x ,

,

n) is the number of times the planar integral is measured for the plane with
normal direction

n containing the points

x and

.
One point of caution is that the function p(t,

k ) in (11) is different from the function
p


(

n) in (8). The function p


(

n) represents the derivative of the Radon planar integral with
2252 G L Zeng and G T Gullberg
(a) (b)
Ideal
(a)
(b)
(c)
Figure 7. Cone-beam reconstructions (central slice) of a computer-generated Shepp–Logan head
phantom: (a) with algorithm(45) and (b) with Grangeat’s algorithm. Line profiles along the broken
line in (a) and (b) are shown in (c).
respect to the variable normal to the plane for the plane passing the point

with the normal
direction

n, while p(t,

k ) in (11) represents the derivative of the Radon planar integral for the
plane with respect to the variable t for the plane with the normal direction

k and a distance
t from the (global) origin. The expression in (19) and (20) gives the transformation between
the global coordinates (t,

k ) and the local coordinates

n.
There are several methods that can be used to reconstruct a 3D image from the Radon
projection data. The method presented in the appendix is one that relates the spherical harmonic
expansion of the image to a spherical harmonic expansion of the Radon planar projections.
However, the integrals in (A.2) and (A.3) are numerically unstable, because the evaluation
of these integrals requires the calculation of Legendre polynomials outside the interval
[−1, 1]. Outside this interval the Legendre polynomials have large values, which can result
Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics 2253
in numerical overflow. Our numerical simulations verified this, which caused us to direct our
attention towards the development of the algorithm in (27).
The main differences between our proposed cone-beam method and other existing
analytical cone-beam reconstruction methods (e.g., Grangeat’s (1991) or Katsevich’s (2003)
methods) are (i) the method does not involve performing a backprojection, (ii) the method
does not require forming line integrals on the projection plane or evaluating derivatives along
some certain directions and (iii) the method does not require any interpolation. Due to (iii),
a potential advantage of the algorithm is that it may provide higher resolution cone-beam
reconstructions than other approaches.
If one would apply the direct Fourier method to fan-beamdata, first the fan-beamdata must
be rebinned (re-sorted) into parallel-beamdata. This rebinning step involves data interpolation.
A 1D Fourier transform is taken for the rebinned parallel-beam data at each projection angle.
According to the central section theorem, the Fourier transformed data is mapped on the
2D Fourier space. The 2D Fourier domain data are then re-sampled (i.e. interpolated) into
uniformly spaced grid points. Finally, an inverse 2D Fourier transform is used to obtain the
reconstructed image.
On the other hand, our fan-beam algorithm (70) does not require rebinning the fan-beam
projection data into parallel-beam data before performing the 1D Fourier transform. The
transformed data are not mapped into the 2D Fourier domain. Instead, an inner product is
calculated between the 1D Fourier data and a location (

x ) dependent function. Finally, a
weighted sum is calculated over all projection angles. No re-sampling of the data and no
inverse 2D Fourier transform are utilized during image reconstruction.
The fan-beam algorithm (70) does not involve any rebinning or interpolation; however, it
has a higher computational cost than the direct Fourier method. The calculation complexity
of a direct Fourier method is at the order of O(N
2
log N), where N is the size in an N × N
2D image array and is proportional to the number of projection angles. The computational
complexity for the fan-beam algorithm (70) is at the order of O(N
4
).
In general the computation burden in a reconstruction algorithm is dominated by the
backprojector. A direct implementation of a backprojector for the 3D Radon data has a
computational complexity of O(N
5
), where N is the size in an N × N × N 3D image
array and is proportional to the number of projection angles. Marr’s method implements a
3D parallel planar backprojector as two sequential 2D parallel linear backprojectors (Marr
1974). As a result, the computational complexity is reduced to O(N
4
). If linogram method
(Axelsson and Danielsson 1994) is adapted the computational complexity can be further
reduced to O(N
3
log N). The proposed cone-beam algorithm (45), on the other hand, has a
poor computational efficiency, and the computational complexity is at the order of O(N
6
).
The proposed fan-beam algorithm (70) is equivalent to the conventional fan-beam filtered
backprojection (FBP) algorithm if the data sampling is continuous. In (70) if one replaces

m
ˆ p


m
e
imα
ˆ
h
∗
m
by p


(ϕ) ∗ h(ϕ), where ∗ is a convolution operator, (70) becomes the
conventional fan-beam FBP algorithm. In fact, substituting the Fourier expansion (67)
(ignoring the constant R
0
) into p


(ϕ) gives
p


(ϕ) ∗ h(ϕ) =

m
ˆ p


m
[e
imϕ
∗ h(ϕ)] =

m
ˆ p


m
e
imα
ˆ
h
∗
m
. (73)
In discrete implementation these two methods are not equivalent. In the FBP algorithm the
discrete samples of the convolution p


(ϕ) ∗h(ϕ) need some interpolation in the backprojection
step. On the other hand in the Fourier expansion, a shift from a sampling point to any location
is an exact phase shift in the Fourier expansion. Thus, data interpolation can be avoided using
the expansion method.
2254 G L Zeng and G T Gullberg
Fan-beam FBP algorithms for equiangular and equispaced measurements are almost the
same except for some weighting factors. However, for the circular expansion method (70),
the measurements are assumed to be equiangular sampled. If the measurements are equispace
sampled, one cannot use the circular expansion directly. Instead, the angular variable (such
as ϕ) must first be changed into the linear variable (say, t) along the detector surface, the
Jacobian factor is then evaluated to relate these two variables, and the expansion is finally
carried out in terms of t. In other words, the expansion method (70) can still be used
for equispaced fan-beam measurements, except that an extra Jacobian factor needs to be
introduced.
The aim of this paper was to investigate a different approach to reconstructing cone-
beam and fan-beam images using orthogonal polynomials. The algorithms still need to be
carefully evaluated and compared with existing algorithms in terms of propagation of noise and
reconstruction accuracy. The idea of using orthogonal polynomials in image reconstruction
is not new and can be extended to other complex cone-beam geometries and potentially other
more general imaging geometries as well. The goal is to some day be able to specify sets of
orthogonal polynomials that represent singular value decompositions of complex cone-beam
geometries.
Unlike Grangeat’s algorithm, the proposed algorithm processes data one cone-beam
projection at a time, and the algorithm could be executed while the data acquisition is in
progress. Our preliminary comparison study shows that the proposed spherical harmonics
expansion cone-beam image reconstruction method provides better spatial resolution than the
image reconstructed with Grangeat’s algorithm. More careful and systematic comparison
studies will be carried out in future work. The main drawback of our proposed algorithms is
the poor computational efficiency. Our future work will focus on improving the computational
efficiency of the reconstruction algorithms that use spherical and circular harmonic expansions.
Acknowledgments
This work was supported by the National Institute of Biomedical Imaging and Bioengineering,
and National Cancer Institute of the National Institutes of Health under grants R21-CA100181,
R01-EB00121 and by the Director, Office of Science, Office of Biological and Environmental
Research, Medical Sciences Division of the US Department of Energy under contract
DE-AC03-76SF00098. We thank Dr Katsuyuki Taguchi of Medical Systems Company,
Toshiba Corporation for assistance in this project.
Appendix
In this appendix, an alternative approach of a spherical harmonic cone-beam image
reconstruction algorithm is presented. This approach is in line with the traditional circular
harmonic image reconstruction methods.
Step 1. Convert the cone-beam line integral to the first derivative of the Radon planar integral
via spherical harmonic expansion.
See (1) and (10). The first derivative of the Radon planar integrals is p


(

n) which has a
spherical harmonic expansion (8) with coefficients p


lm
.
Step 2. Rebin the convergent data format p


(

n) to parallel data format p(t,

k ).
Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics 2255
The rebinning equation is given by (17). In the Radon space we have the first derivative
of the planar integral p(t,

k ) using (18)–(20).
Step 3. Evaluate the spherical harmonic expansion of p(t,

k ):
p(t,

k ) =
N
l

l=0
l

m=−l
p
lm
(t )Y
lm
(

k ). (A.1)
It is possible to evaluate the coefficients p
lm
(t ) directly from the coefficients p


lm
using
(18)–(20).
Step 4. Evaluate the Legendre integrals
q
lm
(r) =
1
2πr
_
∞
r
p

lm
(t )P
l
_
t
r
_
dt. (A.2)
This integral can also be evaluated via using integration by parts
q
lm
(r) = −
1
2πr
p
lm
(r) −
1
2πr
2
_
∞
r
p
lm
(t )P

l
_
t
r
_
dt. (A.3)
Thus calculating the derivative of p
lm
(t ) numerically can be avoided, while an exact expression
of P

l
_
t
r
_
is available.
Step 5. Sum the harmonic expansion to obtain the reconstruction.
The 3D image is finally reconstructed by evaluating the following summation:
f (r,

ω) =
N
l

l=0
l

m=−l
q
lm
(r)Y
lm
(

ω) (A.4)
where

ω is a unit vector in the 3D image space, and the point, (r

ω), is reconstructed.
Steps 4 and 5 are based on relations given by Deans (1983). If a 3D object f (r,

ω) can
be expressed as a spherical harmonic expansion
f (r,

ω) =

l

m
q
lm
(r)Y
lm
(

ω) (A.5)
then the Radon transform (i.e. the planar integrals) of f (r,

ω) is
ˆ
f (t,

k) =

l

m
ˆ q
lm
(t )Y
lm
(

k ). (A.6)
Here q
lm
(r) and ˆ q
lm
(t ) are related by the following Legendre transform pair:
ˆ q
lm
(t ) = 2π
_
∞
t
rq
lm
(r)P
l
_
t
r
_
dr (A.7)
q
lm
(r) =
1
2πr
_
∞
r
ˆ q

lm
(t )P
l
_
t
r
_
dt (A.8)
where
ˆ q

lm
(t ) =
d
2
dt
2
ˆ q
lm
(t ).
2256 G L Zeng and G T Gullberg
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