基于Mojette频域最小冗余覆盖的有限角度计算机层析成像重建 
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Limited-Angle Computed Tomography Reconstruction Based on Mojette Minimal Redundancy Coverage in Frequency Domain
大连理工大学信息与通信工程学院, 辽宁 大连 116024
图 & 表
图 1. 傅里叶切片定理。(a)投影方向与水平方向夹角为θ时获得的投影R(s,θ);(b)夹角θ所对应的傅里叶切线
Fig. 1. Fourier slice theorem. (a) Projection R(s,θ) when angle between projection direction and horizontal direction is θ; (b) Fourier slice corresponding to angle θ
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图 2. 离散傅里叶切片定理。(a)不同方向下离散Radon投影数据;(b)离散Radon投影数据的一维傅里叶变换在图像频域上的映射
Fig. 2. Discrete Fourier slice theorem. (a) Discrete Radon projection data in different angles; (b) projection of one-dimensional Fourier transform of discrete Radon projection data in Fourier domain
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图 3. 3×3图像的Mojette正变换示例。(a)投影矢量(p,q);(b)不同投影矢量方向上Mojette投影数据
Fig. 3. Example of Mojette transform for 3×3 image. (a) Projection vector (p,q); (b) Mojette projection data in different projection vector directions
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图 4. 频域中不同投影方向上频点集的示意图
Fig. 4. Diagram of sets of discrete frequency points along different projection directions in Fourier domain
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图 5. 不同投影方向上的频点。(a) θ=arctan 3的切线所经过的一组频点;(b) θ=arctan(5/7)的切线所经过的一组频点
Fig. 5. Discrete frequency points in different projection positions. (a) Frequency points along tangential corresponding to θ=arctan 3; (b) frequency points along tangential corresponding to θ=arctan(5/7)
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图 6. 所提算法的重建结果。(a)~(d)原始图像;(e)~(h)重建图像的二维频域;(i)~(l)所提算法的重建结果
Fig. 6. Reconstruction results of proposed algorithm. (a)-(d) Original images; (e)-(h) two-dimensional frequency domain of reconstruction images; (i)-(l) reconstruction results of proposed algorithm
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图 7. 几种算法的重建结果对比。(a)(e)(i)原始图像;(b)(f)(j)文献[
14]中的重建结果;(c)(g)(k)所提算法的重建结果;(d)(h)(l) SART算法的重建结果
Fig. 7. Comparison of reconstruction results of several algorithms. (a)(e)(i) Original images; (b)(f)(j) reconstruction results in Ref. [14]; (c)(g)(k) reconstruction results of proposed algorithm; (d)(h)(l) reconstruction results of SART algorithm
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图 8. 水平方向28行切面上像素灰度值的变化曲线。(a)图7中第1排图像灰度值;(b)图7中第2排图像灰度值;(c)图7中第3排图像灰度值
Fig. 8. Image gray values in 28-th row of images in horizontal directions. (a) Image gray value in 1st row of Fig. 7; (b) image gray value in 2nd row of Fig. 7; (c) image gray value in 3rd row of Fig. 7
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表 1Radon变换和Mojette变换之间的差别
Table1. Difference between Radon transform and Mojette transform
| Transform method | Radon transform | Mojette transform |
|---|
| Difference | Fixed detector resolutionFixed angular steppingAccurate reconstruction in continuous domain, approximate reconstruction in discrete domain | Detector resolution varies with projection vectorsVariable angular steppingAccurate reconstruction in discrete domain |
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表 2任意10个投影等效后的角度范围
Table2. Angular ranges of ten equivalent projections
| Angular range of 172.38° | Equivalent angular range of 104.25° | Equivalent angular range of 31.42° | | |
|---|
| (16-order sets of Mojette projections) | (16-order sets of Mojette projections) | (48-order sets of Mojette projections) |
|---|
| Projection vector | Angle /(°) | Equivalent projection vector | Equivalent projection angle /(°) | Equivalent projection vector | Equivalent projection angle /(°) | | (11,1) | 5.19 | (9,11) | 50.71 | (1,3) | 71.57 | | (15,1) | 3.81 | (9,7) | 37.88 | (3,13) | 77.01 | | (11,5) | 24.43 | (7,9) | 52.13 | (7,41) | 80.31 | | (9,7) | 37.87 | (5,11) | 65.56 | (5,43) | 83.37 | | (1,15) | 86.19 | (1,15) | 86.19 | (1,47) | 88.78 | | (-1,15) | 93.81 | (-1,15) | 93.81 | (-1,47) | 91.22 | | (-5,11) | 114.44 | (-3,13) | 103.00 | (-1,15) | 93.81 | | (-9,7) | 142.13 | (-5,11) | 114.45 | (-5,43) | 96.63 | | (-13,3) | 167.01 | (-7,9) | 127.88 | (-7,41) | 99.69 | | (-15,1) | 176.19 | (-9,7) | 142.13 | (-3,13) | 102.99 |
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表 3几种算法的扫描参数区别
Table3. Comparison of scanning parameters for several algorithms
| Algorithm | Proposed algorithm | Literature [14] | SART |
|---|
| Reconstruction results | Figs. 6(c)(g)(k) | Figs. 6(b)(f)(j) | Figs. 6(d)(h)(l) | | Angular range | [86.37°,94.01°] | [0°,10°] | [86.37°,94.01°] | | Number of projections | 18 | 384 | 18 |
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表 4对比实验结果的量化指标
Table4. Quantitative indicators for comparison results
| Result | SSIM | PSNR /dB |
|---|
| Original image in Fig. 7(a) | Mojette result in Fig. 7(c) | 0.9803 | 28.53 | | | | SART result in Fig. 7(d) | 0.9445 | 28.34 | | Original image in Fig. 7(e) | Mojette result in Fig. 7(g) | 1.0000 | >50.00 | | | | SART result in Fig. 7(h) | 0.9419 | 34.73 | | Original image in Fig. 7(i) | Mojette result in Fig. 7(k) | 1.0000 | >50.00 | | | | SART result in Fig. 7(l) | 0.9337 | 36.13 |
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蒋敏, 曲芝萍, 孙怡. 基于Mojette频域最小冗余覆盖的有限角度计算机层析成像重建[J]. 光学学报, 2019, 39(7): 0711001. Min Jiang, Zhiping Qu, Yi Sun. Limited-Angle Computed Tomography Reconstruction Based on Mojette Minimal Redundancy Coverage in Frequency Domain[J]. Acta Optica Sinica, 2019, 39(7): 0711001.
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