Reconstruction Method for Nonconcyclic Dual-Source Circular Cone-Beam CT with a Large Field of View

In industrial computed tomography (CT), it is often required to inspect large objects whose size is beyond a reconstructed field of view (FOV). Somemultiscanmodes have been developed to acquire the complete CT projection data for a larger object using small panel detectors. In this paper, we give a non-concyclic dual-source circular cone-beam scanning geometry based on the idea of multiscan modes and propose a backprojection-filtration-based (BPF) reconstruction algorithm without data rebinning. Since the FOV calculated according to this nonconcyclic dual-source circular CT scanning geometry is larger than cardiac dual-source CT scanning geometry, our method can reconstruct larger horizontal slices (i.e., the slices perpendicular to rotation axis) than cardiac dual-source CT.The quality of CT images is expected to be superior to those obtained using larger panel detectors. The simulation results have indicated that CT images obtained by the proposed method are satisfying.


Introduction
In industrial CT, it is often required to inspect large objects whose size is beyond a field of view (FOV) calculated according to CT scanning geometry, where the maximum horizontal (i.e., perpendicular to rotation axis) FOV is usually fixed in CT system.When a slice of the inspected object cannot be completely covered within the FOV of the cone beam, complete projection data passing through the object cannot be obtained.However, conventional CT reconstruction algorithm, for example, filtered backprojection (FBP), requires the complete projection data.Therefore, those conventional scan modes and reconstruction methods can not be used.To solve this problem, several multiscan modes have been developed to acquire the complete projection data.Of them, the rotation-translation-translation (RTT) multiscan mode [1,2] and rotation-translation (RT) multiscan mode [3,4] are typical.In fact, RT and RTT multiscan modes not only acquire completed projection data, but can also reduce the differences of the flux intensities detected by different detector cells.
For fan-beam RT multiscan modes, the reconstruction algorithm based on backprojection-filtration (BPF) was proposed by Chen et al. [5,6], which can exactly reconstruct the images and does not introduce data rebinning.BPF algorithm, as the basis of our work, was first proposed by Zou and Pan [7] to exactly reconstruct the images for helical cone beam CT.After that, an explicit BPF formula for 2D image reconstruction was proposed by Noo et al. [8] using the finite inversion formula of Hilbert transform.
In this work, we give a nonconcyclic dual-source circular cone beam scanning geometry based on the idea of RT multiscan scan mode and propose a BPF-based reconstruction formula.Since The FOV calculated according to this nonconcyclic dual-source circular CT scanning geometry is larger than cardiac dual-source CT scanning geometry, our method can reconstruct larger horizontal slices.The quality of CT images reconstructed is expected to be superior to those obtained using larger panel detectors.The simulation results have indicated that CT images obtained by the proposed algorithm are satisfying.Since we make use of the approximate idea of Feldkamp (FDK) algorithm [9] in reconstructing formula deduction, reconstruction images are satisfying for a small cone angle, which are usually better for static objects.For dynamic process, there are some results in [10,11].

Nonconcyclic Dual-Source Circular Scan
Mode with a Larger FOV In this section, we give a nonconcyclic dual-source circular cone beam scan mode with a larger FOV based on RT dualscan modes.
As shown in Figure 1, there are two pairs of X-ray sources and panel detectors.Let  1 and  2 denote the focuses of X-ray sources.Let  1 and  2 denote panel detectors.In the nonconcyclic dual-source circular scan mode, two pairs of X-ray sources and panel detectors are fixed, and the inspected object is placed on the turntable.Now, we give this nonconcyclic dual-source scanning geometry.Firstly, the center of the turntable  is located in the cone beam formed by  1 and  1 .For some large inspected objects, one cone beam formed by  1 and  1 cannot completely cover its horizontal slice.So, secondly, we set  2 and  2 making  1 ,  2 and  1 ,  2 located at the same side of , and two pairs of cone beams are parallel, as shown in Figure 1.Now, we give some relevant parameters to describe this scanning geometry.Let  2 denote the distance from  1 to  1 (equal to the distance from  2 to  2 ).Let  1 denote the distance from  1 to the virtual panel detectors which is through  and parallel to panel detectors  1 .Two pairs of panel detectors have the same size, where  and  denote the length and width of the panel detectors, respectively.To obtain sufficient projection data, the distance from  2 to  1 must be less than  1 /(2 2 ) [6].According to the location of the three points ,  1 , and  2 , we know that distance from  2 to  is larger than the distance from  1 to .So two X-ray sources are nonconcyclic in this scanning geometry.The maximum horizontal FOV is 4( 2 − )/(4 + √4 2  2 +  2 ) in nonconcyclic scanning geometry [6], where  is the distance from collimator of the detector to the crystal of the detector.

Reconstruction Formula
For the nonconcyclic dual-source circular scan mode above, we give the reconstruction formula in this section.We know that the key point to BPF algorithm is to obtain the Hilbert image from projections along some directions.The problem we face is how to weigh each set of the derivatives of two pairs of projection data to obtain two differentiated backprojection (DBP) images, and how to merge them into an entire DBP image that is related to the Hilbert image.
For the deduction of the reconstruction formula, we need give some denotations for the scan mode in Section 2. As shown in Figure 2, we establish 3D Cartesian coordinate system , where -axis is the rotating axis.Like the deduction of many CT reconstruction formulas, we also adopt two virtual panel detectors whose centers are  1 and  2 , where the three points  1 ,  2 , and  are collinear.Let   1 ,   2 denote the projection point of  1 ,  2 to the upper side line of two virtual panel detectors, as shown in Figure 2. We need to establish rotating coordinate system  in the deduction, and let  denote the rotating angle that is a clockwise angle from -axis to -axis.We set the 2D coordinate system     V  ( = 1,2) in two virtual panel detectors, where the direction vector of   in  coordinate system is  = (cos , sin , 0), and the direction vector of V  is the same direction with -axis.
Let (x 0 ) denote the density function of the inspected object where x 0 = ( 0 ,  0 ,  0 ) is a reconstructed point on the inspected object.Let   (x 0 ) denote the DBP image of (x 0 ), where  is an angle measured from the -axis anticlockwise in the plane .Let   (,   , V  ) denote the projection data under X-ray source   , and  , (x 0 ) denote the DBP image from   (,   , V  ).Let ℎ  denote the distance between the points  and   , and ℎ  must satisfy 0 < ℎ 1 <  1 /(2 2 ) and ℎ 2 < 3 1 /(2 2 ) [6] that indicates that the projection area of the inspected objects under two X-ray sources is some what overlap.Let ( 0, V 0, ) denote the projection address of x 0 in     V  system under X-ray source   .Let  = 1, 2 above.Now, using the idea of FDK algorithm, we deduct the formula of DBP image   (x 0 ) from the fan-beam DBP formula for RT multiscan modes [5,6].The formula deduction is divided into two kinds of situations: (i) x 0 = ( 0 ,  0 ,  0 ) is on the middle plane where  0 = 0; (ii) x 0 = ( 0 ,  0 ,  0 ) is on the off-middle plane where  0 ̸ = 0. (i) If x 0 is on the middle plane, we may directly obtain the cone beam DBP image   (x 0 ) from the fan-beam DBP [6] as follows: where where where  is a small positive number that is determined by the projection overlapping area, and () is a mollification kernel function as follows: (ii) If x 0 is on an off-middle plane, for obtaining   (x 0 ) we need to define the oblique surface which is through the points  1 ,  2 , and x 0 .Let   denote the intersection point between -axis and the oblique surface, and we know   = (0, 0, V 1 ) from Figure 2 above.Now, we establish the coordinate system and give the relevant parameters in the oblique surface.Let    denote the coordinate system in the oblique surface, where ,  is the same direction with the vectors u 1 and O   1 S 1 , respectively.Let ,   denote a vector from   to x 0 and an angle variable in the oblique surface, respectively, and   1 denote the distance from  1 to   in the oblique surface.
We know that X-ray source is regarded as a fulcrum in FDK algorithm, and an off-middle plane is approximately obtained by inclining the middle plane.Now, based on the idea of FDK algorithm, we give the steps of deduction of   (x 0 ) in the off-middle plane as follows: (i) writing   (x 0 ) =  ,1 (x 0 ) +  ,2 (x 0 ) using the variables in the oblique surface; (ii) finding the relation between  and   , where  is the rotation angle increment in the middle plane, and   is the rotation angle increment in the oblique plane; (iii) finding the relation between x 0 and ; (iv) calculating   1 ; (v) obtaining   (x 0 ) by accumulating all   (x 0 ) for the angle variable .
Without loss of generality, we give the deduction of  ,1 (x 0 ) referring to the steps above.From the formula (2), making use of the parameters in the oblique surface   , we obtain. where We can obtain the relation between  and   from Figure 2 as and the relation between x 0 and  as Since  is in the plane   , we can obtain We easily calculate the distance   1 from  1 to the detectors in the oblique surface, Now, we substitute formulas ( 7), ( 8), (9), and ( 10) to ( 6) and obtain  ,1 (x 0 ) by accumulating all   (x 0 ) for the angle variable  as where Similarly, we can obtain where where V 0, = ( 1 ×  0 )/( 1 − x 0 ×  ⊥ ).When  0 = 0, the formulas ( 11) and ( 13) are ( 2) and (3), respectively.We need two steps for reconstructing (x).Firstly, we need to obtain the Hilbert image   (x) of each layer according to the reference [8].Using the relation of the Hilbert image and the DBP image, we can obtain from the formulas (1), ( 2), ( 3), (11), and ( 13), where x = (, , ) is an arbitrary point on the inspected object.Secondly, making use of the virtual trajectories and virtual PI-lines in [12,13], we can obtain the following formula based on the finite inversion formula of Hilbert transform: where   = (cos , sin , 0),  ⊥ = (− sin , cos , 0), and the constants  , ,  , , and  , relate to x ⋅   =  and .We can obtain  , from the integral of (x) along the virtual PI-lines [14].However, since the PI-lines are virtual except the middle plane, we cannot obtain an accurate  , .So the proposed reconstruction formula is approximated on the offmiddle planes.

Numerical Experiments
To validate our algorithm, we perform some numerical experiments with simulated data in this section.3D Shepp-Logan  phantom is used to get the simulated data.The parameters of the phantom in 3D Cartesian coordinate system  are listed in Table 1, where (  ,   ,   ) is the center coordinate of an ellipsoid, and three variables ,  and  denote the length of three half axis of an ellipsoid, and  is the rotating angle of an ellipsoid in the plane .The parameters of two pairs of X-ray sources and panel detectors in the scanning geometry are the same as follows:  1 = 1426.0mm,  2 = 1800.0mm, and the length and width of the panel detectors  =  = 77.1 mm, which is composed of 257 × 257 detector cells.
According to the parameters above, we can calculate the horizontal diameter of the maximum FOV formed by a set of X-ray source and panel detector is 60.82 mm.From the parameters of 3D Shepp-Logan phantom in Table 1, we calculate the longest axis of ellipsoid along -axis to be 97.9 mm which is greater than 60.82 mm.So we cannot reconstruct its CT image using Cardiac dual-source CT scanning geometry.For acquiring the complete CT data of

Conclusion
In this work we give nonconcyclic dual-source circular cone beam scan mode with a larger FOV, and deduce the reconstruction formula.Using two pairs of projections obtained from this nonconcyclic dual-source scanning geometry, the FOV is enlarged effectively in the same equipment condition without data rebinning.It is because of the fact that for the real CT system, the flux output from X-ray source is not isotropic, and then the data acquired in this scanning mode using small panel detectors are relative to more uniform intensity of the flux output than a large one.The experiment confirmed that our reconstruction method is effective, when a cone angle is small.

Figure 1 :Figure 2 :
Figure 1: The nonconcyclic dual-source circular cone beam scanning geometry with a larger FOV.

Figure 3 :
Figure 3: Two DR images under the 100th project angle, (a) DR image from the cone beam formed by the X-ray source  1 and the panel detectors  1 ; (b) DR image from the cone beam formed by the X-ray S 2 and the panel detectors  2 .