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CT image reconstruction model and algorithm from few views
LIN Luping, WANG Yongge
School of Mathematics and Systems Science, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2016-03-24; Accepted: 2016-09-09; Published online: 2016-10-28 15:15
Foundation item: National Natural Science Foundation of China (91538112)
Corresponding author. WANG Yongge, E-mail: wangyongge@buaa.edu.cn
Abstract: To improve the accuracy and efficiency of few-view computed tomography (CT) image reconstruction, CT image reconstruction is studied from limited view and sparse view, and a novel objective function of total variation norm is proposed. According to the newly-developed objective function, the next iteration is based on the information acquired in the previous one, through which the updated sparse representation model is achieved at each iteration. Additionally, the constrained optimization problem is converted to unconstrained optimization one by adopting the augmented Lagrangian method. Then it can be equally expressed by three sub-problems which can be solved by the alternating minimization scheme. The experimental results using the proposed strategy show that it can attain higher quality CT images which possess integral information, clear detail and high precision. Furthermore, the relative root mean square error can be reduced by 42.1%-98.5% and the streak indicator 42.8%-98.5%, compared with those using Split Bregman-based algorithm.
Key words: computed tomography (CT)     image reconstruction     compressed sensing     total variation regularization     augmented Lagrangian method

1 建立目标函数

 (1)

 (2)

 (3)

 (4)

 (5)

2 增广Lagrangian方法交替求解 2.1 增广Lagrangian函数

 (6)

 (7)

 (8)

 (9)

 (10)

 (11)
 (12)
 (13)

2.2 u-子问题

 (14)

 (15)

 (16)

 (17)

 (18)
 (19)

 (20)
2.3 w-子问题

 (21)

 (22)

 (23)

 (24)
2.4 z-子问题

 (25)

 (26)

 (27)

 (28)
3 算法总结

While停止迭代条件不满足时Do

p=uk

While不满足Armijo条件时Do

αk=ραk

End Do

End Do

4 仿真实验

4幅用于测试的医学原始图像如图 1所示。

 图 1 4幅医学的原始图像 Fig. 1 Four original medical images

 (29)
 (30)

4.1 有限角度下的重建

 图 2 有限角度下3种算法的重建图像 Fig. 2 Reconstruction images of three algorithms in limited view

 重建算法 Phantom 肺部 上腹部 肝部 RRSME SI 重建时间/s RRSME SI 重建时间/s RRSME SI 重建时间/s RRSME SI 重建时间/s FBP 0.77 14.41 1.18 0.74 10.76 1.89 0.77 12.99 5.24 0.76 13.55 1.49 SpBr 0.48 10.41 155.28 0.41 8.13 149.06 0.41 9.90 202.54 0.37 10.69 201.22 UIAL 0.06 1.54 29.22 0.17 3.81 29.69 0.13 3.78 48.62 0.14 4.80 44.27

 图 3 有限角度下重建图像与原始图像的像素比较 Fig. 3 Pixel comparison of reconstruction images with original images in limited view

4.2 稀疏角度下的重建

 图 4 稀疏角度下3种算法的重建图像 Fig. 4 Reconstruction images of three algorithms in sparse view

 重建算法 Phantom 肺部 上腹部 肝部 RRSME SI 重建时间/s RRSME SI 重建时间/s RRSME SI 重建时间/s RRSME SI 重建时间/s FBP 0.38 12.13 1.19 0.28 9.47 1.29 0.19 8.87 1.52 0.26 12.49 1.73 SpBr 0.20 6.15 148.19 0.19 6.43 129.84 0.10 5.08 178.15 0.20 9.66 183.95 UIAL 0.003 0.09 23.60 0.11 3.68 24.06 0.05 2.38 28.02 0.07 3.17 17.49

 图 5 稀疏角度下重建图像与原始图像的像素比较 Fig. 5 Pixel comparison of reconstruction images with original images in sparse view
4.3 参数的选取

θ的选取：以肝部图像稀疏角度为例，当θ变化时，指标RRSME、SI的变化如图 6所示。

 图 6 根据RRSME和SI的值找到最优的θ Fig. 6 Find optimum θ according to RRSME and SI

θ=0.9时，RRSME和SI的值最小，所以取θ=0.9。

vβζλμ的选取是根据式 (11)~式 (13) 来更新计算。其中0＜βkβ, 0＜μkμ，并且Hestenes[19]证明了该方法的收敛性。

5 结论

1) 在有限角度下重建，UIAL算法重建图像信息恢复完整且没有伪影。与SpBr算法相比，RRSME下降了58.5%~87.5%，SI下降了53.1%~85.2%，重建效率提高了4倍多。

2) 在稀疏角度下重建，UIAL算法重建图像没有伪影，细节清晰。与SpBr算法相比，RRSME下降了42.1%~98.5%，SI下降了42.8%~98.5%，重建效率提高了许多。

3) 重建图像与原图像的横纵剖线图比较，也可以看出UIAL算法具有很高的精度。

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#### 文章信息

LIN Luping, WANG Yongge

CT image reconstruction model and algorithm from few views

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(4): 823-830
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0232