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Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics
This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2004 Phys. Med. Biol. 49 2239 (http://iopscience.iop.org/0031-9155/49/11/009) View the table of contents for this issue, issue, or go to the journal the journal homepage for more
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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 Zeng1 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 CT helical detector hardware. This technology has been, and remains, a fundamental component of the medical industry. Continuing advancements in conebeam 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 transform of 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 thedevelopment of thealgorithm based on sphericalharmonics for 3D cone-beam reconstruction. It is then shown how similar 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-beamprojectionsat 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
Cone-beam detector z
θ=0
ψ
g ( n)
Cone-beam ray n
Vertex orbit ψ
Ray with θ = π /2 and ϕ = 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 from the existing cone-beam reconstruction 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
=
g lm Y lm ( n)
(1)
l 0 m
=
=−l
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 1 (2) Y lm ( n) Y lm (θ,ϕ) lm (cos θ ) eimϕ 2π with 0 θ < π, π ϕ < π , and
=
−
lm (t )
=
= −
2l + 1 (l m)! m P (t ) 2 (l + m)! l
( 1)m
−
2l + 1 (l m )! |m| P (t ) 2 (l + m )! l
−| | | |
for m 0 (3) for m < 0
2242
G L Zeng and G T Gullberg
where P lm (t ) is the associated Legendre function and lm (t ) is the normalized associated Legendre polynomial. Both P lm (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
(4)
= δ δ
mm
ll
the expansion coefficients can be calculated as
glm
or
glm
= =
S
∗ ( n) d n g ( n)Y lm
π 1 2π −π
π
(5)
g (θ,ϕ)lm (cos θ ) e−imϕ sin θ dθ dϕ
(6)
0
where S represents a unit sphere. If the half-cone-angle of the cone-beam detector is α , (6) can be evaluated as
glm
=
π/ 2+α
α 1 2π −α
g (θ,ϕ)lm (cos θ ) e−imϕ sin θ dθ dϕ.
(7)
π/ 2 α
−
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: N l
p ( n)
l
=
p lm Y lm ( n)
(8)
l 0 m
=−l
=
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
lm
= −2π lP −1(0)g l
(9)
where P l−1 (.) is the Legendre function and P l−1 (0)
p ( n)
− = l odd m
= 0 if l is even. Therefore,
2π lP l−1 (0)glm 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
dk
p(t, k )
t x k
= ·
(11)
Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics y
2243
Radon plane with n as its normal vector
Focal point ψ n
ϕ Cone-beam ray t
λ
λ+π–ϕ
R
x
0
k
Orbit
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 thefirst-order derivative of theRadon projections with respect to t (t 0), and S represents a unit sphere. The derivative operation in the Fourier domain is a multiplication by i ν where ν is the frequency, and can be implemented as a convolution with a kernel h h(t)
∞
=
2π iν e2π itν dν.
(12)
−∞
For band-limited signals we can use
h(t)
= =− =
2π iν e2π itν dν
−
2π ν sin(2πtν) dν
−
0 2 cos(2πt)
if t
=0 = 0. t
(13) sin(2πt) − if 2 t π t 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)
=
= R0 (cos λ, sin λ, 0)
(15)
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G L Zeng and G T Gullberg
z
n
ψ
p ( n )
Vertex orbit ψ t
θg k y
ϕg
x
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 ϕg , sin θg sin ϕg , cos θg )
(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 0, n points to the centre of the detector. We have (see figure 2) ϕ
=
=
(17)
θg
=θ ϕ = λ+π −ϕ t = · k = −R0 sin θ cos ϕ.
(18)
g
(19)
p(t, k )
where
=p
( n)
(20)
If we change the variables (t,θg , ϕg ) to variables (λ,θ,ϕ), the transformation Jacobian is given by J
= R0 sin θ |sin ϕ|.
(21)
By changing the variables, (14) becomes
f (x )
where g is defined as g
=− − · − = · 4π 2
=x
2π
1
k
t
p ( n)h(g)J dλ d n
S
x y z
(22)
0
sin θ cos(ϕ λ) sin θ sin (ϕ λ) + R0 sin θ cos ϕ. cos θ
− −
(23)
Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics
2245
Changing the order of integrations, we have
−1 f (x ) = 8π 2
2π
0
p ( n)h(g)J d n dλ.
(24)
S
In the following, we substitute the expansion (8) into (24) and use the fact that p lm for even l, and we have N l
2π
l
−1 f (x ) = 8π 2
−1 f (x ) = 8π 2
0
S l 1 m l odd
=0
(25)
(26)
p lm Y lm ( n)h(g)J d n dλ
=−l
=
or
N l
2π
l
plm
0
l 1 m l odd
=−l
=
h(g)JY lm ( n) d n dλ.
S
Equation (26) can be further written as
−1 f (x ) = 8π 2
N l
2π
l
0
∗ plm hlm dλ
(27)
l 1 m l odd
=−l
=
where h∗lm is the complex conjugate of the spherical harmonic expansion coefficient of h(g)J hlm
= S
∗ ( n) d n. h(g)JY lm
(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
sin θ cos(ϕ λ) sin θ sin(ϕ λ) + R0 sin θ cos ϕ cos θ
− − = · t
x y
0
− −
(29)
= [(−R0 + x cos λ) cos ϕ + (x sin λ − y cos λ) sin ϕ]sin θ = [A cos ϕ + B sin ϕ ]sin θ
(30)
= −R0 + x cos λ
(32)
(31)
where A
Let g
= 0, we have two solutions for ϕ −A ϕ1 = tan−1 B
and
∂ϕ
B
= x sin λ − y cos λ.
and
ϕ2 = tan−1
A
−B
(33)
2
=− ∂
and
[ A sin ϕ1 + B cos ϕ1 ]2 sin2 θ
g
ϕ ϕ1
=
= (A2 + B 2 ) sin2 θ
(34)
2246
G L Zeng and G T Gullberg 2
= ∂
(A2 + B 2 ) sin2 θ.
g
∂ϕ
(35)
ϕ ϕ2
=
√
Sine A < 0 in (32), we always have sin ϕ1 A/ A2 + B 2 > 0 and sin ϕ2 A/ A2 + B 2 < 0. Recall from (12) that h(t) δ (t ), a delta function property gives
= −
√
=
=
h(g) = δ (g) =
δ (ϕ
− ϕ1) + δ (ϕ − ϕ2) = δ (ϕ − ϕ1) + δ (ϕ − ϕ2) .
[g (ϕ1 )]2
[g (ϕ2 )]2
(A2 + B 2 ) sin2 θ
(36)
Therefore, (28) can be further evaluated as follows, using (21) and (2): hlm
= = S
2π
π
0
∗ ( n) d n h(g)JY lm
(37)
δ (ϕ
− ϕ1) + δ (ϕ − ϕ2 ) (R sin 0 \ θ |sin ϕ |) \ 2θ (A2 + B 2 ) sin
0
2π
π
R0
= 2π(A 2 + B 2)
lm (cos θ ) dθ
0
R0
= 2π(A 2 + B 2) q
lm
[δ (ϕ
0
2π
[δ (ϕ
0
− ϕ1 ) + δ (ϕ − ϕ2)]|sin ϕ| e−i
mϕ
− ϕ1) + δ (ϕ − ϕ2)]|sin ϕ| e−i
2 = 2π(A−2R+0 B 2) q [δ(ϕ − ϕ1 ) + δ(ϕ − ϕ2 )][|sin ϕ | e−i 0 B + imA −i = 2π(A−2R+0 B 2) q √ − (−1) e−i ] [e 2 2 A +B −R0 q √ B + i mA −i if m is odd 2 2 = π(A + B ) A2 + B 2 e lm
where
mϕ
dϕ
(39)
dϕ
(40)
] dϕ
(41)
π
mϕ1
lm
\
1 lm (cos θ ) e−imϕ sin θ dϕ dθ 2π (38)
m
mϕ
mϕ1
(42)
mϕ1
lm
0
(43)
if m is even
π
qlm
=
lm (cos θ ) dθ
(44)
0
is independent of the projection measurements and can be pre-calculated. reconstruction algorithm (27) for the central slice is reduced to
f (x )
= 8Rπ03
2π
0
1 (A2 +
B 2 )3/2
m l m odd
=−
N l
l
(B
− imA)
l 1 l odd
=
p lm qlm
Finally, the
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 qlm of (44) only needs to be evaluated once.
Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics
2247
y
φ Projection ray
ϕ
– g = t – ( x ⋅ φ )
ψ
t
x φ
λ R
x
0
Orbit
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 1 π ∞ f (x ) p(t,φ)h( x φ t) dt dφ 2 −π −∞ where h is the ramp-filter kernel. For a circular fan-beam vertex orbit
=
· −
= R0 (cos λ, sin λ)
(46)
(47)
we can change the parallel-beam variables (t,φ) to fan-beam variables (λ,ϕ) (see figure 4):
(ϕ)
(48)
= R0 sin ϕ π + λ − ϕ. φ= 2
(49)
p(t,φ)
where
=p
t
(50)
The transformation Jacobian is J
= R0 |cos ϕ |.
(51)
Thus (46) becomes
f (x )
=
1 2
2π
π
0
−π
p (ϕ)h(g)R0 cos ϕ dϕ dλ
|
|
(52)
2248
G L Zeng and G T Gullberg y Fan-beam projection ray
∠( x – ψ ) ξ
τ = −g
Focal point ψ
α
ϕ x
λ R
x 0
Orbit
Figure 5. The fan-beam ray is parametrized with a locally defined angle ξ .
with g
= · − = | |= = | | = =
x φ
t
cos
x
·
y
sin
π
2
π
2
+λ +λ
− − − ϕ
R0 sin ϕ.
(53)
ϕ
Let the local circular harmonic expansion of p (ϕ)R0 cos ϕ be
imϕ p me
p (ϕ)R0 cos ϕ
|
|
(54)
m
with
2π
p m
p (ϕ)R0 cos ϕ e−imϕ dϕ
(55)
0
then (52) becomes
1 2
f (x )
or
1 2
f (x )
2π
0
π
−π
2π
0
imϕ e h(g) dϕ dλ pm
(56)
m
∗ p m hm dλ
(57)
m
where h∗m is the complex conjugate of the circular harmonic expansion coefficient hm : π
hm
=
−π
h(g) e−imϕ dϕ.
(58)
Equation (57) is the fan-beam image reconstruction algorithm in 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
=ξ +α α = λ ± π − ( x − )
(59)
ϕ
2249
(60)
and
= x − sin ξ . (61) In figure 4, −g is the distance from the point x to the fan-beam projection ray. With the τ
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) and
h( τ )
− = h(τ)
h(aτ)
= a12 h(τ)
(64)
which give
Using (48), (59), (63) and (65), (46) becomes
f (x )
=
1 2
2π
π
0
1
p (ξ + α)
h(sin ξ ) x
x − 2
− cos ξ dξ dλ.
2
x −
−π
1
· φ − t ) = h(g) = h(−τ ) = h(τ) = h( x − sin ξ ) =
h( x
h(sin ξ ).
(65)
(66)
Let the local Fourier expansion of p (ϕ)R0 be
p (ϕ)R0
=
with
pˆ m
then (66) becomes
f (x )
or
=
1 2
2π
=
f (x )
imα imξ e pˆ me
m
1 2
=
where ˆm h
m
=
im(ξ +α) pˆ me
2π
0
x − h(sin ξ ) cos ξ dξ dλ
1
(68)
1
x −
(67)
m
π
−π
imϕ pˆ me
p (ϕ)R0 e−imϕ dϕ
0
pm
ˆ ∗ dλ eimα h m
(69)
(70)
m
π
=
h(sin ξ ) cos ξ e−imξ dξ.
(71)
−π
ˆ m of (71) Algorithm (70) is more efficient for computer implementation than (57), because h only needs to be evaluated once, while hm 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)
1 2 3 4 5 6 7 8 9 10 11
0.69 0.6624 0.11 0.16 0.21 0.046 0.046 0.046 0.023 0.023 0.0333
0.92 0.874 0.31 0.41 0.25 0.046 0.046 0.023 0.023 0.046 0.206
0 0 0.22 0.22 0 0 0 0.8 0 0.06 0.5538
Table 2. 3D Shepp–Logan phantom (scaling factor
Tilt angle
0
0 0 18◦ 18 ◦ 0 0 0 0 0 0 18◦
−0.0184
− −
Centre ( y)
0 0 0.35 0.1 0.1 0.605 0.605 0.605 0.3858
− − − − −
−
Density 1.5
−0.98 −0.2 −0.2
−
0.1 0.1 0.1 0.1 0.1 0.1 0.03
= 2).
Ellipsoid index
Half- axis (a)
Half -axis (b)
Half- axis (c)
Centre ( x)
Centre ( y)
Centre (z)
Tilt angle (α )
Tilt angle (β )
Density
1 2 3 4 5 6 7 8 9 10 11 12 13 14
15.43 14.95 2.709 2.709 9.76 0.981 0.491 0.491 0.981 5.506 4.48 2.347 3.413 0.64
20.574 19.725 2.709 2.709 13.011 0.491 0.491 0.981 0.981 5.506 5.53 6.613 8.747 4.267
27.093 26.114 2.709 2.709 10.837 0.491 0.981 0.491 0.981 5.506 4.907 5.419 8.128 4.267
0 0 5.51 5.51 0 1.707 0 1.28 0 0 0 4.693 4.693 11.947
0 0.393 16.073 16.073 0 12.907 12.907 12.907 2.133 2.133 7.467 0 0 8.533
0 0.393 0 0 16.256 8.128 8.128 8.128 8.128 2.709 8.128 8.128 8.128 8.128
0 0 0 0 0 0 0 0 0 0 0 18 ◦ 18◦ 18 ◦
0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 0.98 1 1 1 0.48 0.48 0.48 0.48 0.48 0.48 0.52 0.52 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 units2. 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-beam code involved both the fast Fourier transform and 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
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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 thecone-beam data, an image (see figure 7(b)) was also reconstructed usingGrangeat’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 algorithm gives 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 1 ∗ (n) d (72) hlm h(g)JY lm n
= S
M( x , , n)
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
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(a)
(c)
(b)
Ideal (a) (b)
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. Themethod presentedin theappendixis onethat relates thespherical 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 onewould apply the direct Fourier method to fan-beam data, first the fan-beam data must be rebinned (re-sorted) into parallel-beam data. 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 imα ˆ ∗ ˆ is a convolution operator, (70) becomes the hm by p (ϕ) h(ϕ), where me mp conventional fan-beam FBP algorithm. In fact, substituting the Fourier expansion (67) (ignoring the constant R0) into p (ϕ) gives
×
× ×
∗
p (ϕ)
∗ h(ϕ) =
∗
imϕ pˆ m [e
m
∗ h(ϕ)] =
∗
imα ˆ pˆ hm . me
(73)
m
In discrete implementation these two methods are not equivalent. In the FBP algorithm the discrete samples of the convolution p (ϕ) h(ϕ) needsomeinterpolation 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.
∗
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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 conebeam 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 plm .
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 ): N l
p(t, k )
l
=
(A.1)
plm (t)Y lm ( k ).
l 0 m
=−l
=
It is possible to evaluate the coefficients plm (t ) directly from the coefficients p lm using (18) –(20). Step 4. Evaluate the Legendre integrals qlm (r)
∞
1
= 2π r
p
lm (t)P l
r
t
dt.
r
(A.2)
This integral can also be evaluated via using integration by parts qlm (r)
1
= − 2π r p
lm (r)
1
− 2πr 2
∞
r
plm (t)P l
t
r
dt.
(A.3)
Thus calculating the derivative of plm (t ) numerically can be avoided, while an exact expression of P l rt is available.
Step 5. Sum the harmonic expansion to obtain the reconstruction. The 3D image is finally reconstructed by evaluating the following summation: N l
f (r, ω)
l
=
(A.4)
qlm (r)Y lm ( ω)
l 0 m
=−l
=
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)
=
(A.5)
qlm (r)Y lm ( ω)
m
l
then the Radon transform (i.e. the planar integrals) of f (r, ω) is
ˆ k) f(t,
=
qˆ lm (t)Y lm ( k ).
(A.6)
m
l
Here qlm (r) and qˆ lm (t ) are related by the following Legendre transform pair: qˆ lm (t ) qlm (r)
= 2π 1
∞
rqlm (r)P l
t
= 2π r
∞
r
qˆ lm (t)P l
where 2
qˆ lm (t )
= ddt 2 qˆ
lm (t).
t
r
t
r
dr
(A.7)
dt
(A.8)
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