Fast Backprojection Filtration Algorithm in Circular Cone-Beam Computed Tomography

In order to meet the heavy backprojection calculation of the backprojection fltration (BPF) algorithm in circular cone-beam computed tomography (CT), a fast backprojection method is developed, which uses the integral operator of the fxed scanning angle integral interval. Te proposed method combines with Hilbert fltration and the flter of the Blackman window, and then, the fast BPF (F-BPF) algorithm is obtained. Te experimental results of simulation data and real data demonstrate that the proposed algorithm is fast enough for high-quality reconstructed images only with half-circle projection data.


Introduction
Cone-beam computed tomography (CT) technologies have been widely applied in diferent applications [1][2][3], such as micro-CT and cone-beam tomography used in radiotherapy [4][5][6].It is common that the size of the feld of view (FOV) is smaller than that of an imaging object, which leads to transverse truncation in projection data.From such truncated data, conventional reconstruction algorithms will generate various truncation artifacts, such as Feldkamp-Davis-Kress (FDK) algorithm [7].Even the iterative algorithm also gives inaccurate images [8][9][10].
In recent years, the backprojection fltration (BPF) algorithm for the reconstruction of CT images was proposed based on PI-line segments in helical cone-beam CT [11][12][13], which could realize the accurate reconstructed images of the region of interest (ROI) [14,15].Due to the wide application of circular cone-beam CT [16,17], the BPF algorithm is extended to the image reconstruction of circular cone-beam CT on account of virtual PI-line segments [18].Te extended BPF algorithm reconstructs the exact reconstructed images in the middle plane and the approximate reconstructed images in the nonmiddle planes.It provides a way to solve the image reconstruction of truncation data caused by limited FOV size [19].
In the process of the implementation of the BPF algorithm in circular cone-beam CT, the scanning angles of the two end points of each PI-line segment are diferent and need to be calculated.Te scanning angle integration interval of all reconstructed points in the backprojection calculation of the FDK algorithm is the same.However, the scanning angle integration interval of the points on diferent PI-line segments in the backprojection calculation of the BPF algorithm is diferent.It causes the heavy cost of the backprojection calculation and the difculty of implementation using parallel computation [20].Te problem obstructs the use of the BPF algorithm in practical circular cone-beam CT, although the BPF algorithm can realize the image reconstruction with truncation data in a short scan [21].To solve it, the fast BPF (F-BPF) algorithm was developed and verifed with numerical simulation and the real data reconstruction in this paper.
Te rest of the paper is arranged as follows: Te theory of the conventional BPF algorithm and the F-BPF algorithm is described in Section 2. Both numerical simulation and real data reconstruction are carried out to demonstrate the proposed algorithm in Section 3. Finally, a conclusion is given in Section 4.

Theory
In the following, we briefy introduce the conventional BPF algorithm and then propose the F-BPF algorithm in circular cone-beam CT.
2.1.Conventional BPF Algorithm.Te implementation of the conventional BPF algorithm mainly includes two steps.Te frst step is to calculate the backprojection of the derivatives of the weighted projection data on PI-line segments.Te second step is to perform fnite Hilbert fltering on the backprojection data along the PI-line segments.Te reconstruction formula of the BPF algorithm is given as follows [22]: where f( r → ) is the image function, r → is the vector of a reconstruction point, K( r → , r → ′) expresses the integral kernel, and g( r →′ ) is the backprojection function.Its formula is given as follows: s) expresses the X-ray source, β → is the unit vector along the direction from r → 0 (s) to r →′ , and λ 1 and λ 2 are the projection angles of the two end points on the PI-line segment.
Combining (1) and ( 2), the reconstruction formula of the conventional BPF algorithm can be written as follows: where (x c , λ 1 , λ 2 ) is another representation of r → with its coordinates on the PI-line segment, C denotes the projection value along the PI-line segment, and (x c1 , λ 1 , λ 2 ) and (x c2 , λ 1 , λ 2 ) are the representations of two end points of the PI-line segment.
Generally, a group of equally spaced parallel segments on diferent planes are selected as PI-line segments in circular cone-beam CT, where the backprojection formula is written as follows: where R is the distance from the X-ray source to the rotation axis, S is the distance from the X-raysource to the detector,

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, and P(u d , v d , λ) is the projection value of the coordinate position (u d , v d ) at the scanning angle λ.

F-BPF Algorithm.
In circular cone-beam CT, the conventional BPF algorithm must calculate the projection angles of the two end points of every PI-line segment, which leads to lower reconstruction efciency and is not conducive to parallel computing.Tus, a fast approximate backprojection method is proposed by use of the integral function with a fxed integral interval.
Te proposed backprojection function is where the line connecting the X-ray source at the views of 0 and π is supposed to be parallel to the PI-line and in the center of the imaging region, which implies that λ 1 is close to 0 and λ 2 is close to π.From ( 2) and ( 5), the absolute uncertainty of new backprojection is obtained by Let R 0 be the radius of the imaging object and R r � | r →′ |.
It is obvious that R r ≤ R 0 < R. Te source trajectory and r →′ can be expressed as r We set Note that |∆g( r →′ )| ⟶ 0 if λ 1 ⟶ 0. Actually λ 1 is close to 0, so the new backprojection method is feasible.
By (4), the backprojection of the F-FBP algorithm can be written as International Journal of Optics It is clear that the proposed algorithm possesses three major advantages.First, it just needs the projection views from 0 to π. it reduces the time of data acquisition and reconstruction greatly.Tird, similar to the backprojection calculation of the FDK algorithm, the scanning angle integration interval in the backprojection operation of the BPF algorithm is fxed, so the proposed F-BPF algorithm is suitable for parallel computing.
In the F-BPF algorithm, the Hilbert transform is implemented in the frequency domain, which can avoid the oddity in the time domain and improve reconstruction efciency.Te Hilbert transform of x(t) can be written as and then, Tus, the Hilbert transform of x(t) is To reduce or eliminate spectral energy leakage and barrier efects, diferent window functions are used to truncate the signal.Te three commonly used window functions in CT image reconstruction are the Blackman window, the Hanning window, and the Hamming window.Te expression for the Blackman window function is as follows: Te expression for the Hanning window function is as follows: Te expression for the Hamming window function is as follows: Te time-domain diagrams of the Blackman window, the Hanning window, and the Hamming window are shown in Figure 1.It can be seen that the Blackman window function has the fastest sidelobe decrease and can efectively suppress truncation artifacts.Te Blackman window is added as the flter to improve the imaging quality in the process of the Hilbert transform [23].

Experimental Results and Analysis
In order to verify the efectiveness of the proposed F-BPF algorithm, we used simulation data and real data for experiments.Te CPU used in the experiments is Intel(R) Core(TM) i5-10210U CPU @ 1.60 GHz.

Simulation Experiments.
In the following, the simulation data were used to verify the efectiveness of the proposed algorithm in this paper.We realized the proposed F-BPF algorithm and the conventional BPF algorithm in MATLAB and analyzed the experimental results of 256 × 256 × 256 standard 3D Shepp-Logan head phantom [24].Te projection data of 360 angles are uniformly generated on the circular scanning trajectory.Te conventional BPF algorithm used all the conebeam projection data, while the F-BPF algorithm used only the data of 180 projection views from 0 to π.
In Figure 2, we display 2D slices in 3D images reconstructed by use of the conventional BPF algorithm and the F-BPF algorithm.Figure 2 Te mean square error of the conventional BPF algorithm is 0.0646 and that of the F-BPF algorithm is 0.0694.Te time to reconstruct the slice using the conventional BPF    algorithm is 111.7 seconds, while the F-BPF algorithm only costs 6.6 seconds.Te simulation experiments prove that the proposed algorithm is efective and that its reconstruction efciency has been greatly improved in contrast with the conventional BPF algorithm.
Te comparative experiments on the parallel performance of the algorithm and the F-BPF algorithm are also conducted, in which the program development software is CUDA 9.0 and the GPU is NVIDIA Tesla K20M.Te experimental results are shown in Table 1.It can be seen that the acceleration ratio of the BPF algorithm is 289 and that the acceleration ratio of the F-BPF algorithm is 997.Te parallel performance of the F-BPF algorithm is improved compared to that of the BPF algorithm.

Experiments of Real Data.
Next, the real data were used to further verify the efectiveness of the proposed F-BPF algorithm.Te real data were obtained using the cone-beam CT system, which used the X-ray source (Hawkeye 130, Tales, France) and the fat-panel detector (4343F, Tales, France) with a pixel size 0.148 mm.

Reconstruction of the Printed Circuit Board.
In order to test the performance of F-BPF algorithm, we carried out the experiments to reconstruct the ROI of the printed circuit board (PCB) by the FDK algorithm with the complete data and the F-BPF algorithm with the truncated data.Te relevant parameters for experimental data collection were as  International Journal of Optics follows: Te trajectory radius was 477 mm, the distance from the X-ray source to the fat-panel detector was 1265 mm, the size of the ROI was 542 × 474 × 284 voxels, and the projection data of 360 angles were uniformly generated on the circular scanning trajectory.Te F-BPF algorithm used 180 projection views from 0 to π, and the FDK algorithm used 180 projection views evenly distributed over the 2π circular trajectory.
Figures 4(a) and 4(b) display 2D slices in 3D images reconstructed by use of the FDK algorithm with nontruncated data and the proposed F-BPF algorithm with truncated data.Te reconstruction time of the FDK algorithm is 62 minutes, and the proposed algorithm only costs 41 minutes.From the reconstruction results, the two algorithms reconstruct images with little diference, and the correctness of the proposed algorithm is also demonstrated by real data.

Reconstruction of the Spark Plug.
We performed the experiments on a spark plug to compare the reconstruction results with and without the Blackman window in the process of Hilbert fltration.In the circular cone-beam confguration, the trajectory has a radius of 477 mm and a source-detector distance of 1265 mm.Te size of the reconstructed ROI is 512 × 512 × 512 voxels.Te cone-beam data were generated with 180 projection views uniformly distributed over the π circular trajectory.
In Figure 5, we display the 2D slices in 3D images reconstructed with and without the Blackman window.In Figure 6, we also show the profles in the images reconstructed by the F-BPF algorithm along the middle horizontal line with and without the Blackman window.Reconstruction quality gets improved after adding the Blackman window.So adding the Blackman window as a flter in Hilbert fltration is efective in reducing noise in real data reconstruction.

Discussion and Conclusion
In this paper, we have developed the F-BPF algorithm for image reconstruction in circular cone-beam CT and then performed simulation experiments and real data experiments to validate the proposed algorithm.Te results of simulation show that the proposed algorithm is reliable and that its reconstruction efciency has been greatly improved in contrast with the conventional BPF algorithm.Te verifcation is also demonstrated with real data of PCB, and the images reconstructed by the FDK algorithm and the proposed algorithm are comparable.Considering the noise inevitably exists in real data, the reconstruction results of a spark plug with and without the Blackman window imply that adding the Blackman window is efective in reducing the noise in Hilbert fltration.
Te experimental results show that the reconstruction efciency of the proposed algorithm is a great improvement over the conventional BPF algorithm, and a remarkable merit of the proposed algorithm is that it can be perfectly carried out with only half-circle projection data.Terefore, if the proposed algorithm is used in an industrial cone-beam CT system, it will shorten data acquisition and reconstruction time.If the algorithm is used in medical devices, it will perform well in both saving data acquisition and reconstruction time and reducing their radiation dose to patients.International Journal of Optics (a) shows the slice of the 3D Shepp-Logan model, and Figures 2(b) and 2(c) show the slices of the reconstruction results by use of the conventional BPF algorithm and F-BPF algorithm, respectively.In Figure 3, we also show the profles of the images shown in Figure 2. Figures 3(a) and 3(b) display the profles of Figure 2(b) on the middle horizontal line and the middle vertical line, respectively.Figures 3(c) and 3(d) show the profles of Figure 2(c) on the middle horizontal line and the middle vertical line, respectively.Te reconstruction and the corresponding real profles are expressed by the dotted and solid curves, respectively.

Figure 1 :
Figure 1: Time-domain diagrams of the Blackman window, the Hanning window, and the Hamming window.

Figure 3 :
Figure 3: Profles of the reconstruction images in Figure 2. (a) Profle of Figure 2(b) on the middle horizontal line.(b) Profle of Figure 2(b) on the middle vertical line.(c) Profle of Figure 2(c) on the middle horizontal line.(d) Profle of Figure 2(c) on the middle vertical line.

Figure 2 :
Figure 2: Reconstruction images by use of the conventional BPF algorithm and the F-BPF algorithm.(a) Shepp-Logan model.(b) Reconstruction image using the conventional BPF algorithm.(c) Reconstruction image using the F-BPF algorithm.

Figure 6 :
Figure 6: Profles of the reconstructed images in Figure 5 along the middle horizontal line.Te solid and dotted curves represent the results obtained with and without the Blackman window, respectively. q�s

Table 1 :
Te experimental results about the parallel performance of the BPF and F-BPF algorithms.