# Direct Fan-Beam Reconstruction Algorithm via  Filtered Backprojection for Differential  Phase-Contrast  Computed Tomography

Authors: Zhihua Qi; Guang-Hong Chen
Journal: X-Ray Optics and Instrumentation
 (2008)

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## Abstract

Recently, a novel data acquisition method has been proposed and experimentally implemented for differential phase-contrast computed tomography (DPC-CT), in which a conventional X-ray tube and a Talbot-Lau-type interferometer were utilized in data acquisition. The divergent nature of the data acquisition system requires a divergent-beam image reconstruction algorithm for DPC-CT.  This paper focuses on addressing this image reconstruction issue.  We have developed a filtered backprojection algorithm to directly reconstruct the DPC-CT images from acquired fan-beam data. The developed algorithm allows one to directly reconstruct the decrement of the real part of the refractive index from the measured data. In order to accurately reconstruct an image, the data need to be acquired over an angular range of at least180∘ plus the fan angle. As opposed to the parallel beam data acquisition and reconstruction methods, a 180∘-rotation angle for the data acquisition system does not provide sufficient data for an accurate reconstruction of the entire field of view. Numerical simulations have been conducted to validate the image reconstruction algorithm.


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## Body

## 1. Introduction

In recent years, X-ray phase-contrast imaging methods have
attracted significant interest. Based upon the phase retrieval method, phase-contrast
imaging can be classified into three categories: interferometric methods [1, 2], analyzer crystal methods [3, 4], and free-space propagation methods [5–7]. These methods
generally require excellent spatial coherence and/or excellent spatial
resolution of the detectors. Phase-contrast imaging is thus normally conducted
using high-brilliance synchrotron beam lines or microfocus X-ray tubes. It is
difficult to adapt these phase retrieval methods to a clinical setting for
medical imaging.The shearing interferometry method has been introduced to
measure the gradient of the local phase shift distribution using synchrotron
beam lines [8–11]. The key
theoretical foundation is the Talbot effect [11–14] and Moiré
interferometry [15]. Computed tomography imaging methods have
been developed [11, 16] based on
the phase-contrast projection images.The shearing interferometry method only requires a mild
spatial coherence length, which opens up a new opportunity to implement phase-contrast
imaging using a conventional X-ray tube. A method which takes advantage of
these characteristics was proposed and experimentally implement in X-ray
differential phase-contrast imaging [17] and differential
phase-contrast computed tomography (DPC-CT) [18, 19].
In this scheme, Pfeiffer et al. [17–19] proposed to
mount an absorption grating in front of a low-brilliance X-ray source to
generate an array of equidistant secondary line sources. The coherence length
of the beam is determined not by the focal spot size of the X-ray tube, but
rather by the width of the opening of the absorption grating, which is about
20∼50 μm. This allows the use of a conventional X-ray tube, where the high
X-ray output can be turned into many mutually incoherent line sources. In this
paper, we refer to this grating as a beam slicer. Using this method, the
spatial coherence length of each line source has been improved by a factor of
ten or more over that of the X-ray tube without the grating, thus enabling
significant reduction of the distance between the X-ray source and the image
object. As a consequence, this method enables construction of an X-ray phase-contrast
imaging system with compact size, which is often a requirement in medical and
nondestructive inspection applications.For a given X-ray detector with rectangular geometry, the fan
angle is determined by the angle given by the following formula:(1)γm=2tan⁡−1(H2D), where H is the width of the detector and D is the distance
from the X-ray focal spot to the detector. Therefore, when the source to
detector distance D is very long, the divergence of the X-ray beam is
negligible, which is often the case for synchrotron beam line experiments and
inline holography phase-contrast imaging.
In this case, the beams are well approximated as plane wave, and a
parallel-beam image reconstruction method is sufficient for image
reconstruction in DPC-CT.However, using the new phase retrieval method by Pfeiffer et
al. [17–19], the distance
D is significantly reduced (D  160 cm in [19]). Note that the effective detector size is
limited by the size of the third grating in front of the detector, about 6 cm
in [19]. Using these parameters, the fan angle γm in [19] can be estimated as(2)γm=2tan⁡−1(H2D)~2°. In this case, the parallel-beam approximation is still
acceptable (see Section 4.2). Thus, the
well-known parallel-beam image reconstruction method [20] can be adapted to reconstruct
DPC-CT images [18, 21, 22].When a larger image object is scanned with the same
detector-to-source distanceD, the fan angle γm is increased and the parallel-beam
approximation becomes less accurate. When the fan angle is increases beyond 5°,
the divergent nature of the data acquisition system is no longer negligible (see
Section 4.2). For example, if the size of the image object is doubled in the
above estimation, the fan angle is nearly 5°. For larger objects, a divergent beam
image reconstruction algorithm would be desirable for this new DPC-CT data
acquisition method. In this paper, we present an image reconstruction formula
for fan-beam DPC-CT which can be utilized to accurately reconstruct DPC-CT
images at any fan angle provided that the data is acquired within the angular
range of 180° plus fan angle.


## 2. Derivation of the Fan-Beam Image Reconstruction Algorithm

In general, the experimental results should be explained by
wave optics. However, for convenience,
an effective geometrical model is utilized in this paper to develop the image
reconstruction algorithm.


### 2.1. Brief Review of the Parallel-Beam  Reconstruction of the Differential  Phase-Contrast X-Ray CT

As explained by Pfeiffer et al. [18], the fundamental idea in
differential phase retrieval is to locally measure the angular change Θ(ρ,θ) of the incident X-ray beam (Figure 1). Here,
the measured data is labeled by the distance ρ, which is measured from the origin of the coordinate system to
the incident ray direction and the angle θ, which specifies the direction of the incident ray.Figure 1
A geometric model of the parallel-beam
data acquisition model.The quantityΘ(ρ,θ) is related to the local gradient of the
object’s phase shift can be written as follows [15, 23]:(3)Θ(ρ,θ)=∂∂ρ∫ldlδ(x,y), where δ(x,y) is the decrement of the real part of the
object’s refractive index, n=1−δ+iβ. The letter l labels the incident ray;
therefore, the above line integral is performed along the incident ray
direction. Note that the integral in (3) is simply the Radon transform Rδ(ρ,θ) of the target function δ(x,y). Thus, (3) can be rewritten as(4)Θ(ρ,θ)=∂∂ρRδ(ρ,θ). After a Fourier transform with respect to the variable ρ,
(4) is recast into the following form:(5)Θ(ω,θ)=−i2πωRδ(ω,θ). Using the inverse Radon transform in Fourier space, one can
easily obtain the following image reconstruction formula [19, 21, 22]:(6)δ(x,y)=12π2∫0πdθ∫−∞+∞dω[iπsgn(ω)]Θ(ω,θ)ei2πω(xcos⁡θ+ysin⁡θ). Therefore, after the data are acquired from view angles in
the angular range [0,π], a one-dimensional Fourier transform is performed with respect to the Radon
distance ρ to obtain Θ(ω,θ). A filtering kernel iπsgn(ω) is used to filter the data. Finally, an
integral over the view angles is performed to reconstruct images.


### 2.2. Effective Fan-Beam Data Acquisition for Differential Phase Retrieval Method

Note that the image reconstruction formula (6), used by
Pfeiffer et al. [19], makes sense only when the acquired data are written in
terms of the Radon distance ρ and the view angle θ. Namely, it is only convenient for the parallel-beam data acquisition geometry.For the effective fan-beam data acquisition geometry as shown
in Figure2, the X-ray tube and detector are assumed to rotate around the image
object along a circle with radius R and source-to-detector distance D.
The acquired data are naturally labeled by the angular position, t, of the X-ray focal spot and the
detector position which is labeled by a projection distance u. We refer to this measured data as Θ(t,u) in this paper. The half width of the detector
is um, which implies that
the fan angle γm=2tan−1(um/D).Figure 2
Geometrical model for fan-beam DPC-CT data acquisition.R is the radius of the source
trajectory, and D is
source-to-detector distance.For fan-beam data, one way to reconstruct an image is to rebin
the acquired data into parallel-beam projection data and then utilize (7) to
reconstruct the image. In this paper, we develop new image reconstruction
formulae which directly reconstruct images from the acquired data Θ(t,u) without requiring the rebinning step.


### 2.3. Fan-Beam Reconstruction Formula for Full Scan Case

In order to directly reconstruct the image without the
rebinning step, we rewrite (6) in the real domain by taking the inverse Fourier
transform:(7)δ(x,y)=14π2∫02πdθ∫−∞+∞dρ1xcos⁡θ+ysin⁡θ−ρΘ(ρ,θ). This formula can simply be derived by noting that the product
in Fourier space is a convolution in real domain. This formula dictates how
each individual projection datum Θ(ρ,θ) contributes to the final image. Note that we have
extended the integral limit from [0,π] to [0,2π] and introduced an extra factor of 1/2 to account for the redundancy.We next derive image reconstruction formulae for the fan-beam
data acquisition geometry. Each individual projection datumΘ(t,u) can also be relabeled in terms of the Radon
distance ρ and angle θ by the following nonlinear coordinate transforms:(8)ρ=−Ruu2+D2,θ=π2+t+arctan⁡uD. This transform introduces an extra Jacobian factor:(9)dρdθ=RD2(u2+D2)3/2dudt,xcos⁡θ+ysin⁡θ−ρ=L(x,y;t)u2+D2[u−U(x,y;t)], where the functions L(x,y;t) and U(x,y;t) are given by(10)L(x,y;t)=R−xcos⁢⁡t−ysin⁡t,U(x,y;t)=Dxsin⁡t−ycos⁡tR−xcos⁡t−ysin⁡t. Therefore, (7) can now be written as(11)δ(x,y)=14π2∫02πdtRL(x,y;t)F[t,U(x,y)], where the filtered function F[t,U(x,y)] is given by(12)F(t,U)=∫−∞+∞du1u−UΘ¯(t,u). In this equation, the preweighted projection data Θ¯(t,u) is defined as(13)Θ¯(t,u)=D2u2+D2Θ(t,u). Equations (11)–(13) are the main
results of the paper. They give a direct image reconstruction from the measured
projection data Θ(t,u).
The algorithm for fan-beam DPC-CT is summarized in the following steps:(1)
preweight the
acquired projection data using (13);(2)
filter the preweighted data using the Hilbert
filtering kernel in (12);(3)
back-project the filtered data to reconstruct an
image using (11).Note that these formulae require that the data are acquired
along a full circle. This is referred to as the full scan data acquisition
mode. In the next subsection, we extend the above result to the short scan case,
where the data are acquired from less than a full circle.


### 2.4. Fan-Beam Reconstruction Formula for Short Scan Case

In absorption X-ray CT, it has been proven that the data are
sufficient to reconstruct the entire image object when the angular range is greater
than180° + fan angle, where the fan angle is defined as the entire angle
subtended by the detector from the source position. The same condition is also
true for differential phase-contrast CT, we refer to this case as the short
scan mode. When the short scan data acquisition mode is utilized, some incident
rays will have a conjugate ray, while others will be measured only once.
Therefore, a weighting function is needed for the redundantly measured data. It
is not difficult to convince oneself that the following two data points should
be considered as the same:(14)(t,u)⟺(t′=t+π−2arctan⁡(uD),u′=−u). Thus, similar to absorption CT [24], the following weighting
function can be utilized to weight the data before the filtering step:(15)W(t,u)={sin⁡2(π2tγm−2γ),t∈[0,γm−2γ),1,t∈[γm−2γ,π−2γ),sin⁡2(π2π−t+γmγm+2γ),t∈[π−2γ,π+γm], where the variable γ is defined as γ=arctan(u/D). Thus, by incorporating the normalized weighting function W(t,u) into (13), the preweighted projection data become(16)Θ¯(t,u)=W(t,u)D2u2+D2Θ(t,u). The image reconstruction formulae (11)–(13) together
with (16) can be used to reconstruct the image.


## 2.1. Brief Review of the Parallel-Beam  Reconstruction of the Differential  Phase-Contrast X-Ray CT

As explained by Pfeiffer et al. [18], the fundamental idea in
differential phase retrieval is to locally measure the angular change Θ(ρ,θ) of the incident X-ray beam (Figure 1). Here,
the measured data is labeled by the distance ρ, which is measured from the origin of the coordinate system to
the incident ray direction and the angle θ, which specifies the direction of the incident ray.Figure 1
A geometric model of the parallel-beam
data acquisition model.The quantityΘ(ρ,θ) is related to the local gradient of the
object’s phase shift can be written as follows [15, 23]:(3)Θ(ρ,θ)=∂∂ρ∫ldlδ(x,y), where δ(x,y) is the decrement of the real part of the
object’s refractive index, n=1−δ+iβ. The letter l labels the incident ray;
therefore, the above line integral is performed along the incident ray
direction. Note that the integral in (3) is simply the Radon transform Rδ(ρ,θ) of the target function δ(x,y). Thus, (3) can be rewritten as(4)Θ(ρ,θ)=∂∂ρRδ(ρ,θ). After a Fourier transform with respect to the variable ρ,
(4) is recast into the following form:(5)Θ(ω,θ)=−i2πωRδ(ω,θ). Using the inverse Radon transform in Fourier space, one can
easily obtain the following image reconstruction formula [19, 21, 22]:(6)δ(x,y)=12π2∫0πdθ∫−∞+∞dω[iπsgn(ω)]Θ(ω,θ)ei2πω(xcos⁡θ+ysin⁡θ). Therefore, after the data are acquired from view angles in
the angular range [0,π], a one-dimensional Fourier transform is performed with respect to the Radon
distance ρ to obtain Θ(ω,θ). A filtering kernel iπsgn(ω) is used to filter the data. Finally, an
integral over the view angles is performed to reconstruct images.


## 2.2. Effective Fan-Beam Data Acquisition for Differential Phase Retrieval Method

Note that the image reconstruction formula (6), used by
Pfeiffer et al. [19], makes sense only when the acquired data are written in
terms of the Radon distance ρ and the view angle θ. Namely, it is only convenient for the parallel-beam data acquisition geometry.For the effective fan-beam data acquisition geometry as shown
in Figure2, the X-ray tube and detector are assumed to rotate around the image
object along a circle with radius R and source-to-detector distance D.
The acquired data are naturally labeled by the angular position, t, of the X-ray focal spot and the
detector position which is labeled by a projection distance u. We refer to this measured data as Θ(t,u) in this paper. The half width of the detector
is um, which implies that
the fan angle γm=2tan−1(um/D).Figure 2
Geometrical model for fan-beam DPC-CT data acquisition.R is the radius of the source
trajectory, and D is
source-to-detector distance.For fan-beam data, one way to reconstruct an image is to rebin
the acquired data into parallel-beam projection data and then utilize (7) to
reconstruct the image. In this paper, we develop new image reconstruction
formulae which directly reconstruct images from the acquired data Θ(t,u) without requiring the rebinning step.


## 2.3. Fan-Beam Reconstruction Formula for Full Scan Case

In order to directly reconstruct the image without the
rebinning step, we rewrite (6) in the real domain by taking the inverse Fourier
transform:(7)δ(x,y)=14π2∫02πdθ∫−∞+∞dρ1xcos⁡θ+ysin⁡θ−ρΘ(ρ,θ). This formula can simply be derived by noting that the product
in Fourier space is a convolution in real domain. This formula dictates how
each individual projection datum Θ(ρ,θ) contributes to the final image. Note that we have
extended the integral limit from [0,π] to [0,2π] and introduced an extra factor of 1/2 to account for the redundancy.We next derive image reconstruction formulae for the fan-beam
data acquisition geometry. Each individual projection datumΘ(t,u) can also be relabeled in terms of the Radon
distance ρ and angle θ by the following nonlinear coordinate transforms:(8)ρ=−Ruu2+D2,θ=π2+t+arctan⁡uD. This transform introduces an extra Jacobian factor:(9)dρdθ=RD2(u2+D2)3/2dudt,xcos⁡θ+ysin⁡θ−ρ=L(x,y;t)u2+D2[u−U(x,y;t)], where the functions L(x,y;t) and U(x,y;t) are given by(10)L(x,y;t)=R−xcos⁢⁡t−ysin⁡t,U(x,y;t)=Dxsin⁡t−ycos⁡tR−xcos⁡t−ysin⁡t. Therefore, (7) can now be written as(11)δ(x,y)=14π2∫02πdtRL(x,y;t)F[t,U(x,y)], where the filtered function F[t,U(x,y)] is given by(12)F(t,U)=∫−∞+∞du1u−UΘ¯(t,u). In this equation, the preweighted projection data Θ¯(t,u) is defined as(13)Θ¯(t,u)=D2u2+D2Θ(t,u). Equations (11)–(13) are the main
results of the paper. They give a direct image reconstruction from the measured
projection data Θ(t,u).
The algorithm for fan-beam DPC-CT is summarized in the following steps:(1)
preweight the
acquired projection data using (13);(2)
filter the preweighted data using the Hilbert
filtering kernel in (12);(3)
back-project the filtered data to reconstruct an
image using (11).Note that these formulae require that the data are acquired
along a full circle. This is referred to as the full scan data acquisition
mode. In the next subsection, we extend the above result to the short scan case,
where the data are acquired from less than a full circle.


## 2.4. Fan-Beam Reconstruction Formula for Short Scan Case

In absorption X-ray CT, it has been proven that the data are
sufficient to reconstruct the entire image object when the angular range is greater
than180° + fan angle, where the fan angle is defined as the entire angle
subtended by the detector from the source position. The same condition is also
true for differential phase-contrast CT, we refer to this case as the short
scan mode. When the short scan data acquisition mode is utilized, some incident
rays will have a conjugate ray, while others will be measured only once.
Therefore, a weighting function is needed for the redundantly measured data. It
is not difficult to convince oneself that the following two data points should
be considered as the same:(14)(t,u)⟺(t′=t+π−2arctan⁡(uD),u′=−u). Thus, similar to absorption CT [24], the following weighting
function can be utilized to weight the data before the filtering step:(15)W(t,u)={sin⁡2(π2tγm−2γ),t∈[0,γm−2γ),1,t∈[γm−2γ,π−2γ),sin⁡2(π2π−t+γmγm+2γ),t∈[π−2γ,π+γm], where the variable γ is defined as γ=arctan(u/D). Thus, by incorporating the normalized weighting function W(t,u) into (13), the preweighted projection data become(16)Θ¯(t,u)=W(t,u)D2u2+D2Θ(t,u). The image reconstruction formulae (11)–(13) together
with (16) can be used to reconstruct the image.


## 3. Numerical Simulations and Results

In order to validate both the full scan and short scan DPC-CT
reconstruction formulae derived in this paper, numerical simulations were
conducted using the fan-beam data acquisition geometry as shown in Figure2.
The mathematical phantom consisted of a uniform elliptical disk and two uniform
circular disks. The projection data were numerically generated using Snell’s
law [23]. The radius of the circular disks was 0.07 m. The semimajor axis and
semiminor axis of the elliptical disk were 0.35 m and 0.175 m, respectively. The
decrement of the real part of the refractive index was assumed to be 1.0×10−6 and 0.5×10−6 for the circular and elliptical disks,
respectively. The radius of the source trajectory R was assumed to be 1.4 m. The distance from the
source to the detector D was selected to be 2.1 m. The sampling rate of
the view angle was Δt=0.5°.
The detector was assumed to be collinear, with total width of 1.13 m. The
detector sampling rate Δu was 1.13 m/600 ≈ 2 mm.
The image matrices were 256×256.


### 3.1. Full Scan Data Acquisition Mode

A fan angle of 30° was used in order to cover the entire
image object. The view angle sampling range was [0°,360°]. The results are
presented in Figure 3. From Figure 3(a) and the corresponding image intensity
plots, it is evident that the image was accurately reconstructed using the new
algorithm.Reconstruction results for a fan-beam full scan. (a): Reconstructed image. (b), (c), and (d) Image intensity plots corresponding to lines 1, 2, and 3, respectively. The solid line represents the reconstructed image, and the dashed line represents the ground truth. The abscissas of (b), (c), and (d) are normalized to 0.35 m.
(a)(b)(c)(d)


### 3.2. Short Scan Data Acquisition Mode

In the short scan mode, all parameters stay the same as the
full scan mode with the following exceptions: (1) the decrement of the real
part of the refractive index of the circular objects is changed to 0, and (2)
the angular range of the data acquisition is reduced from360° to 210°, which
is sufficient for reconstruction according to X-ray absorption CT image
reconstruction theory [20]. The numerical results are presented in Figure 4. Again,
the image is faithfully reconstructed with very high accuracy as indicated in
image intensity plots.Numerical results for short scan data acquisition mode. (a) Reconstructed image. (b), (c), and (d) Image intensity plots corresponding to lines 1, 2 and 3, respectively. The solid line represents the reconstructed image, and the dashed line represents the ground truth. The abscissas of (b), (c), and (d) are normalized to 0.35 m.
(a)(b)(c)(d)


### 3.3. Super Short Scan Data Acquisition Mode

In this subsection, we numerically demonstrate that the
angular range for data acquisition should be at leastπ + fan angle for the presented new algorithm.
When the scanning angular range is shorter than that, significant image
artifacts may occur. In numerical simulations, all of the parameters are the
same as the short scan case except that the source angle is reduced to from
210° to 180°, which is less than a short scan.
As shown in Figure 5, image artifacts appear in the reconstructed image,
and signal drop is clearly observed in image intensity plots. This numerical
experiment also indicates that the short scan angular range is necessary for
the presented fan-beam DPC-CT reconstruction.The reconstruction result for angular range less than a short scan. Figure6(a) is the reconstructed image. Figures (b), (c), and (d) are profile plots corresponding to lines 1, 2, and 3, respectively, in Figure 6(a). The solid line represents the reconstructed image, and the dashed line represents the ground truth. The abscissas of (b), (c), and (d) are normalized to 0.35 m.
(a)(b)(c)(d)


## 3.1. Full Scan Data Acquisition Mode

A fan angle of 30° was used in order to cover the entire
image object. The view angle sampling range was [0°,360°]. The results are
presented in Figure 3. From Figure 3(a) and the corresponding image intensity
plots, it is evident that the image was accurately reconstructed using the new
algorithm.Reconstruction results for a fan-beam full scan. (a): Reconstructed image. (b), (c), and (d) Image intensity plots corresponding to lines 1, 2, and 3, respectively. The solid line represents the reconstructed image, and the dashed line represents the ground truth. The abscissas of (b), (c), and (d) are normalized to 0.35 m.
(a)(b)(c)(d)


## 3.2. Short Scan Data Acquisition Mode

In the short scan mode, all parameters stay the same as the
full scan mode with the following exceptions: (1) the decrement of the real
part of the refractive index of the circular objects is changed to 0, and (2)
the angular range of the data acquisition is reduced from360° to 210°, which
is sufficient for reconstruction according to X-ray absorption CT image
reconstruction theory [20]. The numerical results are presented in Figure 4. Again,
the image is faithfully reconstructed with very high accuracy as indicated in
image intensity plots.Numerical results for short scan data acquisition mode. (a) Reconstructed image. (b), (c), and (d) Image intensity plots corresponding to lines 1, 2 and 3, respectively. The solid line represents the reconstructed image, and the dashed line represents the ground truth. The abscissas of (b), (c), and (d) are normalized to 0.35 m.
(a)(b)(c)(d)


## 3.3. Super Short Scan Data Acquisition Mode

In this subsection, we numerically demonstrate that the
angular range for data acquisition should be at leastπ + fan angle for the presented new algorithm.
When the scanning angular range is shorter than that, significant image
artifacts may occur. In numerical simulations, all of the parameters are the
same as the short scan case except that the source angle is reduced to from
210° to 180°, which is less than a short scan.
As shown in Figure 5, image artifacts appear in the reconstructed image,
and signal drop is clearly observed in image intensity plots. This numerical
experiment also indicates that the short scan angular range is necessary for
the presented fan-beam DPC-CT reconstruction.The reconstruction result for angular range less than a short scan. Figure6(a) is the reconstructed image. Figures (b), (c), and (d) are profile plots corresponding to lines 1, 2, and 3, respectively, in Figure 6(a). The solid line represents the reconstructed image, and the dashed line represents the ground truth. The abscissas of (b), (c), and (d) are normalized to 0.35 m.
(a)(b)(c)(d)


## 4. Discussion

In this section, several aspects of the newly developed image
reconstruction algorithm will be discussed. First, fan-beam DPC-CT image
reconstruction and fan-beam absorption CT image reconstruction algorithms are
compared. Next, the differences between the fan-beam and known parallel-beam reconstruction
algorithms are discussed. Finally, the angular range requirements for fan-beam
DPC-CT data acquisition are examined.


### 4.1. Comparison with Fan-Beam Absorption CT  Image Reconstruction Algorithm

In this paper, the fan-beam image reconstruction algorithm
was derived using a collinear detector geometry. It is easy to change to the
curved detector case using the relationγ=arctan(u/D). For absorption CT, fan-beam image reconstruction algorithms have been derived
for both full and short scan cases [20]. The DPC-CT fan-beam image
reconstruction algorithm is different from its absorption counterpart in three ways.
First, in DPC-CT, the preweighting factor is D2/(D2+u2). For absorption CT, the preweighting factor is a square root, that is, D2/(D2+u2)=D/D2+u2. Second, the distance weighting function in the backprojection step in DPC-CT is R/L(x,y;t).
While for absorption CT, the weighting function is R2/L2(x,y;t), the square of the DPC-CT expression. Finally, the filtering kernel for DPC-CT
image reconstruction is a Hilbert filter, whereas a ramp filter is used in
absorption CT image reconstruction.


### 4.2. Parallel-Beam Approximation

When the source to detector distance is much larger than the
size of the image object, one can expect that the difference between the
parallel-beam and fan-beam image reconstruction methods is negligible. The
numerical results demonstrate that when the fan angle is smaller than5°, the
parallel-beam approximation is valid, and parallel-beam image reconstruction
can be utilized to reconstruct DPC-CT images. However, for larger fan angles, a
fan-beam image reconstruction algorithm, such as the one presented in this
paper, is needed for an accurate image reconstruction.In the first numerical experiment, we applied the parallel-beam
reconstruction algorithm [19, 21, 22] to the case of a
30° fan angle, the same angle used in the reconstruction presented in Figure 3.
The parallel-beam reconstruction algorithm generates significant image
artifacts as shown in Figure 6(a). In the second numerical experiment, the
radius of the source trajectory is increased to 4 m and the fan angle of the divergent
beam is reduced to 10° in order to cover the entire image object.The parallel-beam reconstruction result for fan-beam projection data. (a), (b), (c), and (d) correspond to fan angles30°, 10°, 5°, and 2.5°, respectively.
(a)(b)(c)(d)As shown in Figure6(b), the image quality is improved
comparing with Figure 6(a), but blurring still exists at the boundaries. When
the source to detector distance increases such that the fan angle is reduced to
5°, as shown in Figure 6(c), the image artifacts become increasingly
negligible, indicating that the parallel beam approximation is becoming more acceptable. In the last numerical experiment, the fan
angle of the divergent beam is reduced to 2.5° for the same image object. In
this case, as shown in Figure 6(d), the parallel-beam reconstruction method is
sufficient to reconstruct DPC-CT images. Using the experimental data presented
in [19], the fan angle can be estimated to be less than 2.5°. Thus, parallel-beam
reconstruction would be sufficient to reconstruct near artifact-free DPC-CT
images [19].


## 4.1. Comparison with Fan-Beam Absorption CT  Image Reconstruction Algorithm

In this paper, the fan-beam image reconstruction algorithm
was derived using a collinear detector geometry. It is easy to change to the
curved detector case using the relationγ=arctan(u/D). For absorption CT, fan-beam image reconstruction algorithms have been derived
for both full and short scan cases [20]. The DPC-CT fan-beam image
reconstruction algorithm is different from its absorption counterpart in three ways.
First, in DPC-CT, the preweighting factor is D2/(D2+u2). For absorption CT, the preweighting factor is a square root, that is, D2/(D2+u2)=D/D2+u2. Second, the distance weighting function in the backprojection step in DPC-CT is R/L(x,y;t).
While for absorption CT, the weighting function is R2/L2(x,y;t), the square of the DPC-CT expression. Finally, the filtering kernel for DPC-CT
image reconstruction is a Hilbert filter, whereas a ramp filter is used in
absorption CT image reconstruction.


## 4.2. Parallel-Beam Approximation

When the source to detector distance is much larger than the
size of the image object, one can expect that the difference between the
parallel-beam and fan-beam image reconstruction methods is negligible. The
numerical results demonstrate that when the fan angle is smaller than5°, the
parallel-beam approximation is valid, and parallel-beam image reconstruction
can be utilized to reconstruct DPC-CT images. However, for larger fan angles, a
fan-beam image reconstruction algorithm, such as the one presented in this
paper, is needed for an accurate image reconstruction.In the first numerical experiment, we applied the parallel-beam
reconstruction algorithm [19, 21, 22] to the case of a
30° fan angle, the same angle used in the reconstruction presented in Figure 3.
The parallel-beam reconstruction algorithm generates significant image
artifacts as shown in Figure 6(a). In the second numerical experiment, the
radius of the source trajectory is increased to 4 m and the fan angle of the divergent
beam is reduced to 10° in order to cover the entire image object.The parallel-beam reconstruction result for fan-beam projection data. (a), (b), (c), and (d) correspond to fan angles30°, 10°, 5°, and 2.5°, respectively.
(a)(b)(c)(d)As shown in Figure6(b), the image quality is improved
comparing with Figure 6(a), but blurring still exists at the boundaries. When
the source to detector distance increases such that the fan angle is reduced to
5°, as shown in Figure 6(c), the image artifacts become increasingly
negligible, indicating that the parallel beam approximation is becoming more acceptable. In the last numerical experiment, the fan
angle of the divergent beam is reduced to 2.5° for the same image object. In
this case, as shown in Figure 6(d), the parallel-beam reconstruction method is
sufficient to reconstruct DPC-CT images. Using the experimental data presented
in [19], the fan angle can be estimated to be less than 2.5°. Thus, parallel-beam
reconstruction would be sufficient to reconstruct near artifact-free DPC-CT
images [19].


## 5. Conclusions

In this paper, an image reconstruction formula for fan-beam
DPC-CT was derived and validated using numerical simulations. It has been
demonstrated that when the image object is relatively large, the fan angle must
increase to cover the entire image object, and a parallel-beam approximation
cannot be directly applied to reconstruct images. In this case, the fan-beam
image reconstruction formula presented in this paper can be directly applied to
accurately reconstruct DPC-CT images. The presented image reconstruction
formula is also useful for other DPC-CT imaging method [18] provided that the
divergent beam data acquisition geometry is utilized.As a final remark, as shown in Figure5, one limitation of
the presented algorithm in this paper is that the data acquisition angular
range must be longer than π + fan angle. Otherwise, significant image
artifacts are observable in the reconstructed image. In order to enable
accurate image reconstruction when the angular range is shorter than π + fan angle, some other new image
reconstruction must be developed [25, 26]. Note that it is straightforward to
extend the fan-beam reconstruction algorithms such as the one presented in this
paper and others [25, 26] to cone-beam data acquisition systems using the
so-called FDK approximation [27], where the only change is to replace the
fan-beam data by the projected cone-beam data, that is, the multiplication of
cone-beam data and a cosine weighting factor.



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*Source: 835172-2008-06-22.xml*