# Forming Mechanism and Correction of CT Image Artifacts Caused by the Errors of Three System Parameters

Authors: Ming Chen; Gang Li
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We know that three system parameters, a center of X-ray source, an isocenter, and a center of linear detectors, are very difficult to be calibrated in industrial CT system. So there are often the offset of an isocenter and the deflection of linear detectors. When still using the FBP (filtered backprojection) algorithm under this condition, CT image artifacts will happen and then can seriously affect test results. In this paper, we give the appearances and forming mechanism of these artifacts and propose the reconstruction algorithm including a deflection angle of linear detectors. The numerical experiments with simulated data have validated that our propose algorithm can correct CT images artifacts without data rebinning.


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## Body

## 1. Introduction

We usually adopt the FBP (filtered backprojection) algorithm in industrial CT. This algorithm requires two necessary conditions [1, 2]: (i) the isoray (an imaginary ray that connects a center of X-ray source with an isocenter) is perpendicular to linear detectors; (ii) an insection point where the isoray and linear detectors cross is the center of linear detectors. However, a center of X-ray source, an isocenter, and a center of linear detectors are difficult to be calibrated in industrial CT. When still using the FBP algorithm under the errors of three system parameters, the image artifacts will happen and then can seriously affect test results, especially for these reconstructed points away from the center of CT images.When there happens the offset of an isocenter, people usually take the projection point of the isocenter on linear detectors as the center of linear detectors and then translate the projections. For parallel beam, this translation can recalibrate the isocenter [3]. However for fan beam, this translation is impossible unless fan beam projections are rebinned as parallel beam projections [4]. For measuring and correcting the CT system parameters, there are some methods proposed. Gullberg et al. [5] proposed the method to correct the isocenter for fan beam; however, the involved parameters are difficult to obtain. Sun et al. [6] assumed that the plane of four small balls is perpendicular to the turn table and then measured cone-beam CT system parameters by use of projection data under one angle. For micro-CT, Patel et al. [7] proposed the autocalibration method without model measurement and measured and corrected some system parameters. For cone-beam CT, Chen et al. [8] estimated some parameters by obtaining the barycenter under the condition that the plane detector is parallel to the axis of rotation.The remainder of this paper is organized as follows. In Section2, we introduce the FBP algorithm for fan beam and point out its necessary conditions. In Section 3, we give the appearances of three image artifacts caused by the offset of an isocenter and the deflection of linear detectors. In Section 4, we analyze the forming mechanism of three artifacts. In Section 5, we propose the FBP algorithm including a deflection angle of linear detectors. Finally, numerical experiments and conclusions are presented in Section 6.


## 2. The FBP Algorithm for Fan Beam

For convenience of the formula derivation in Section5, we introduce the FBP algorithm for equal-spaced fan beam in this section.A simple geometric relationship with no errors of CT system parameters is shown in Figure1. We define a right-handed coordinate system Ox1x2, where the origin O is an isocenter, x1 axis is parallel to linear detectors (the bold line in Figure 1), and x2 axis is parallel to the isoray. Let R1 denote the distance from X-ray source to x1 axis (if the isoray is perpendicular to linear detectors, R1 is also the distance between X-ray source S and the isocenter O), and let R2 denote the distance between S and the center OD of linear detectors. Let p(β,s) denote the equal-spaced fan beam projection data, where β is the angle of the isoray formed with the x2 axis and s is a sample on the imaginary detectors which are through the isocenter O and parallel to the actual linear detectors. Making use of the FBP reconstruction algorithm, the image function, f(x)=f(x1,x2), can be shown to be
(1)f(x)=∫02πR12(R1-x·β⊥)2×{R1R12+s2p(β,s)*h(s)}|s=s0dβ,
where s0 is the projection address of a reconstructed point x on the imaginary detectors, β=(cosβ,sinβ), β⊥=(-sinβ,cosβ), s0=(R1x·β)/(R1-x·β⊥), and h(s)=∫-∞+∞|ω|ei2πωsdω is a filter function.Figure 1
A simple geometric relationship of an equal-spaced fan beam.According to a geometric relationship in Figure1, the FBP algorithm requires two necessary conditions: (i) the isoray is perpendicular to linear detectors; (ii) an insection point where the isoray and linear detectors cross is the center of linear detectors.


## 3. Appearances of Three Image Artifacts

We give a test phantom, which is comprised of eleven circles, and ten circles are distributed evenly among the center of the phantom, as shown in Figure2.Figure 2
Phantom.The offset of an isocenter may cause CT image artifacts [9–12]. This offset is divided into two cases: along linear detectors or along the direction which is perpendicular to linear detectors. The latter is equivalent to error of R1. When R1 is much larger than the field of view (FOV), we can still reconstruct a satisfying CT image, even if R1 remains some errors [13]. For this reason, we only consider the offset of an isocenter along linear detectors.We give a simple geometric relationship of CT scanning system with the offset of an isocenter in Figure3, where O1 is the isocenter. Let x2 axis denote the direction which is through the center OD of linear detectors and perpendicular to linear detectors. Let x1 axis denote the direction which is through O1 and perpendicular to x2 axis. Let O denote the insection point where the x1 axis and the x2 axis cross. Let γ denote the angle contained by the line OS and the line O1S.Figure 3
A simple geometric relationship of CT scanning system with the offset of an isocenter.We perform numerical experiments with the simulated data to show the appearance of image artifacts caused by the offset of an isocenter. CT scanning system parameters are as follows: the distance from X-ray sourceS to x1 axis R1=550.000 mm, the distance from X-ray source S to linear detectors R2=905.000 mm, linear detectors are composed of 1024 cells, with the size of each cell 0.4 mm. We assume that there is the offset of an isocenter |OO1|=5.0mm. Each detector takes 720 projections in 2π. The image matrix is 1024 × 1024. For the phantom in Figure 2, we reconstruct CT images using the FBP formula (1), as shown in Figure 4, where the artifacts nonuniformly spread to all directions. And the reconstructed points away from the center of CT images are comparatively worse.Figure 4
CT image artifacts caused by the offset of an isocenter.For this offset of an isocenter, we may obtain the projection point of the isocenter on linear detectors by many experiments and then translate the projection data. The reconstructed images from the translated projection data are shown in Figure5, where the artifacts obviously reduce. In fact, an isoray is not perpendicular to linear detectors when the offset of an isocenter happens. So there still exist some image artifacts caused by the deflection of linear detectors in Figure 5. That is, linear detectors deflect to the dotted line from x1 axis in Figure 3.Figure 5
Reconstruction images from the translated projection data, or CT image artifact caused by the deflection of linear detectors.We also give a simple geometric relationship of CT scanning system with the deflection of linear detectors in Figure6, where linear detectors deflect to the heavy continuous line from x1 axis. Let γ denote the clockwise deflection angle. Making use of the same previous parameters, we can calculate γ=0.52°. We can reconstruct CT images using the FBP formula (1) from the projection data with the deflection of linear detectors, as shown in Figure 5, which is exactly same with the reconstructed image from the translated projections data with the offset of isocenter.Figure 6
A simple geometric relationship of CT scanning system with the deflection of linear detectors.Similarly, when the offset of an isocenter and the deflection of linear detectors simultaneously happen, we also obtain CT image artifacts, as shown in Figure7, where the isocenter offset is 2.0 mm and γ=0.52°, and the other parameters are same as above mentioned.Figure 7
CT image artifacts caused by the offset of an isocenter and the deflection of linear detectors.


## 4. Forming Mechanism of Three Image Artifacts

We give the forming mechanism of three image artifacts in this section. For a reconstructed pointx, we analyze a reconstruction process of x and give a minimum bias expression under every projection angle.For ease of the following analysis, let a polar coordinate(r,θ)  denote x, and let s0 denote its projection address on linear detectors. If there is no error in CT system, we can calculate s0=R2×rcos(β-θ)/(R1+rsin(β-θ)). From Figure 3, the projection point of x is a point E on linear detectors, and a projection address is s=O1DE, where O1D is the projection point of the isocenter O1 on linear detectors. However, we still take O as an isocenter in image reconstruction when using the FBP formula (1). So the other point F is regarded as the projection point of x where ODF=O1DE. Under this condition, x will be reconstructed on the line SF. Now, we draw a vertical line which is through x and perpendicular to the line SF, and let M(β) denote the insection point. The trajectory of M(β) can approximately describe the reconstruction result of x when β ranges from 0 to 2π. Now, firstly we calculate the distance R(β) between x and M(β) as follows:
(2)R(β)=||OS|2×rcos(β-θ)-s×|ODS|×|OS|-s×|ODS|×rsin(β-θ){|OS|2|OS|2}|×(s2×|ODS|2+|OS|4)-1,
where s=s0-(|ODS|×|OO1|/|OS|).So, we can obtain a coordinate ofM(β) as follows:
(3)M(β)=(rcosθ-R(β)×cos(β+tan-1s×|ODS||OS|2),rsinθ-R(β)×sin(β+tan-1s×|ODS||OS|2)).We choose a reconstructed pointx0=(90,π/4) and assume that the offset |OO1| of an isocenter is 0.744 mm, that is, 2.15 pixel. According to formula (3), we may draw the trajectory of M(β) by Mathematica, where β ranges from 0 to 2π, as shown in Figure 8(a). The reconstruction image of x0 using the FBP formula (1) is shown in Figure 8(b), which explain the artifacts in Figure 4.Analysis of artifacts caused by the offset of an isocenter: (a) the trajectory ofM(β); (b) the reconstruction image.
(a)
(b)Similarly, for the linear detectors deflection, we may calculate the same previous expressions (2) and (3) of R(β) and M(β), where s=s0sinα/sin(γ+α), α=tan-1(s0/|OS|). We choose γ=0.52°. The trajectory of M(β) and the reconstruction image of x0 are as shown in Figure 9, which explain the artifacts in Figure 5.Analysis of artifacts caused by the deflection of linear detectors: (a) the trajectory ofM(β); (b) the reconstruction image.
(a)
(b)Similarly, for the offset of an isocenter and the deflection of linear detectors, we also calculate the previous expressions (2) and (3) of R(β) and M(β), where s=(s0×|OS|-|ODS|×|OO1|)×sinα/|OS|×sin(γ+α),α=tan-1(s0/|OS|). We choose the offset of an isocenter |OO1|=0.5 mm and γ=0.6°. The trajectory of M(β) and the reconstruction image of x0 are as shown in Figure 10, which explain the artifacts in Figure 7.Analysis of artifacts caused by the offset of an isocenter and the deflection of linear detectors: (a) the trajectory ofM(β); (b) the reconstruction image.
(a)
(b)


## 5. Derivation of FBP Formula Including a Deflection Angle of Linear Detectors

In this section, we describe a new coordinate system and derive the FBP formula including a deflection angle of linear detectors, where the offset of an isocenter is attributed to the deflection of linear detector.Referring to Figure11, we establish the coordinate system Ox1x2, where the origin O is the isocenter, x2 axis is parallel to the isoray and points to X-ray source S, and x1 axis and x2 axis form right-handed coordinate system. Let φ denote the angle contained by the x1 axis and linear detectors. Obviously, x2 axis is not perpendicular to linear detectors, and there is a deflection of linear detectors and no offset of an isocenter in this system.Figure 11
A geometric relationship of FBP formula derivation including a deflection angle of linear detectors.For convenience of derivation, let the polar coordinatef(r,θ) denote the image function. Let O′ denote the projection point of the isocenter O on linear detectors, R1=|OS| and R2=|O′S|. We use the imaginary detectors in formula derivation. Let d,q, and s denote three projection points of the reconstructed point x=(r,θ) on linear detectors, the imaginary detectors, and x1 axis, respectively. Let p(d,β), p1(q,β), and p2(s,β) denote the corresponding projection data. For a reconstructed point x0=(r0,θ0), and let d0, q0, and s0 denote three projection points corresponding to x0, respectively.From Figure11, we can obtain the relationship between s0 and q0, s, and q as follows:
(4)s0=R1q0cosφR1-q0sinφ,(5)s=R1qcosφR1-qsinφ.Now, we rewrite the FBP formula (1) as follows:
(6)f(r0,θ0)=12∫02πR12(R1-r0sin(θ0-β))2×∫-∞∞‍R1R12+s2p2(s,β)h(s0-s)dsdβ,
where h(s)=∫-∞∞|ω|ei2πωsdω, s0=R1r0cos(θ0-β)/(R1-r0sin(θ0-β)).From formula (4), (5), and (6), we may obtain
(7)q0=R1r0cos(θ0-β)R1cosφ-r0sin(θ0-β-φ).From formula (5) and h(s), we can obtain
(8)dsdq=R12cosφ(R1-qsinφ)2,h(s0-s)=1C2h(q0-q),
where C=R12cosφ/(R1-q0sinφ)(R1-qsinφ).Finally, we substitute formulae (5) and (8) into (6) and obtain after simplifying
(9)f(r0,θ0)=12∫02π(R1-q0sinφ)2(R1-r0sin(θ0-β))2cosφ×∫-∞∞‍R1-qsinφR12+q2-2R1qsinφ×p1(q,β)h(q0-q)dqdβ,
where p1(q,β)=p(R2q/R1,β).The proposed previous formula can directly reconstruct CT image without data rebinning. The formula includes three parametersR1, R2, and φ, which are unknown, independence from the inspected objects, and identified by CT system. For obtaining three parameters, we have designed the model with a dense matter such as iron or steel, by a row of mutual parallel width and of the slit spacing formed. By super precise scanning for the model in 2π, we could make use of the geometric relationship of these slit spacing projection and estimate three parameters. But, this method is very sensitive to a deflection angle of linear detectors φ. We can improve measurement precision by averaging the testing values of repeated measurements.


## 6. Numerical Simulation Experiment and Conclusion

In this section we perform numerical experiments with simulated data to demonstrate our formula (9). We choose the phantom in Figure 2 and the system parameters in Figure 4. We can estimate R1=550.023 mm, R2=905.014 mm, and φ=0.52° in the formula (9). The reconstruction results are shown in Figure 12 using the formula (9). Obviously, the results validate our formula, which can correct the image artifacts caused by the offset of an isocenter and the deflection of linear detectors.Figure 12
Reconstruction images using the FBP formula (9) including a deflection angle of linear detectors.We have given the appearances of three image artifacts caused by the offset of an isocenter and the deflection of linear detectors and analyzed the forming mechanism, which can provide reference for three artifacts identification. The correction method of the image artifacts is also proposed. Our FBP algorithm including a deflection angle of linear detectors can effectively correct three artifacts in CT images.



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