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基于相对TV最小的CT图像重建算法

张峰嘉 乔志伟

张峰嘉, 乔志伟. 基于相对TV最小的CT图像重建算法[J]. CT理论与应用研究, 2022, 32(0): 1-17. DOI: 10.15953/j.ctta.2022.190
引用本文: 张峰嘉, 乔志伟. 基于相对TV最小的CT图像重建算法[J]. CT理论与应用研究, 2022, 32(0): 1-17. DOI: 10.15953/j.ctta.2022.190
ZHANG F J, QIAO Z W. Computed Tomography Reconstruction Algorithm Based on Relative Total Variation Minimum[J]. CT Theory and Applications, 2022, 32(0): 1-17. DOI: 10.15953/j.ctta.2022.190. (in Chinese)
Citation: ZHANG F J, QIAO Z W. Computed Tomography Reconstruction Algorithm Based on Relative Total Variation Minimum[J]. CT Theory and Applications, 2022, 32(0): 1-17. DOI: 10.15953/j.ctta.2022.190. (in Chinese)

基于相对TV最小的CT图像重建算法

doi: 10.15953/j.ctta.2022.190
基金项目: 国家自然科学基金面上项目(模型与数据耦合驱动的快速四维EPRI肿瘤氧成像(62071281));中央引导地方科技发展资金项目(新型TV和学习先验联合约束的快速四维EPRI成像方法(YDZJSX2021A003));山西省重点研发计划(电子顺磁共振成像(EPRI)中美联合实验室平台建设(201803D421012));山西省留学人员科技活动择优资助项目(基于压缩感知的四维EPRI成像方法研究(2018-172));山西省回国留学人员科研资助项目(基于新型四维TV正则机理的快速EPRI肿瘤氧成像方法研究(2020-008))。
详细信息
    作者简介:

    张峰嘉:男,山西大学计算机科学与技术专业硕士研究生,主要从事医学图像重建、图像处理等方面的研究,E-mail:821967394@qq.com

    乔志伟:男,博士,山西大学计算机与信息技术学院教授、博士生导师,主要从事医学图像重建、信号处理、大规模最优化等方面的研究,E-mail:zqiao@sxu.edu.cn

    通讯作者:

    乔志伟*,

  • 中图分类号: O  242;TP  391.41

Computed Tomography Reconstruction Algorithm Based on Relative Total Variation Minimum

  • 摘要: 总变差(TV)最小算法是一种有效的CT图像重建算法,可以对稀疏或含噪投影数据进行高精度重建。然而,在某些情况下,TV算法会产生阶梯状伪影。在图像去噪领域,相对TV算法展现了优于TV算法的性能。鉴于此,将相对TV模型引入图像重建,提出相对TV最小优化模型,并在自适应梯度下降-投影到凸集(ASD-POCS)框架下设计对应的求解算法,以进一步提升重建精度。以Shepp-Logan、FORBILD及真实CT图像仿真模体进行重建实验,验证了该算法的正确性并评估了算法的稀疏重建能力和抗噪能力。实验结果表明,相对TV算法可以实现逆犯罪,可以对稀疏或含噪投影数据进行高精度重建,与TV算法相比,该算法可以取得更高的重建精度。

     

  • 图  1  噪声图像的偏导数特点

    Figure  1.  Partial derivative characteristics of noisy images

    图  2  RTV算法的正确性研究关于Shepp-Logan模体的重建结果

    Figure  2.  Correctness research of RTV algorithm on reconstruction results of Shepp-Logan phantom

    图  3  RTV算法的正确性研究关于FORBILD模体的重建结果

    Figure  3.  Correctness research of RTV algorithm on reconstruction results of FORBILD phantom

    图  4  RTV算法重建Shepp-Logan模体收敛行为的展示

    Figure  4.  Display of convergence behavior when reconstructing Shepp-Logan phantom by RTV algorithm

    图  5  RTV算法重建FORBILD模体收敛行为的展示

    Figure  5.  Display of convergence behavior when reconstructing FORBILD phantom by RTV algorithm

    图  6  不同模体在20个投影角度下使用RTV和TV最小化重建算法的RMSE趋势曲线的比较

    Figure  6.  Comparison of RMSE trend curves of the same phantom using RTV and TV minimization reconstruction algorithms at 20 projection angles

    图  7  RTV算法和TV算法对于FORBILD模体重建结果的比较:图像上方的数字表示投影个数;左边文字表示使用的算法

    Figure  7.  Comparison of FORBILD phantom reconstruction results between RTV algorithm and TV algorithm: the number above the image represents the number of projections; The text on the left indicates the algorithm used

    图  8  RTV算法和TV算法对于Shepp-Logan模体重建结果的比较:图像上方的数字表示投影个数;左边文字表示使用的算法

    Figure  8.  Comparison of Shepp-Logan phantom reconstruction results between RTV algorithm and TV algorithm: The number above the image represents the number of projections; The text on the left indicates the algorithm used

    图  9  RTV算法和TV算法对于真实CT图像重建结果的比较:图像上方的数字表示投影个数;左边文字表示使用的算法

    Figure  9.  Comparison of CT-phantom reconstruction results between RTV algorithm and TV algorithm: the number above the image represents the number of projections; The text on the left indicates the algorithm used

    图  10  在50个投影角度并于投影数据中加入方差为0.05的高斯白噪声条件下分别使用RTV和TV重建算法进行重建的实验结果

    Figure  10.  Under the condition of 50 projection angles and adding Gaussian white noise with variance of 0.05 to the projection data, RTV and TV minimization reconstruction algorithms are used to reconstruct the results respectively

    图  11  不同模体在50个投影角度并于投影数据中加入方差为0.05的高斯白噪声条件下使用RTV和TV最小化重建算法重建结果的RMSE趋势曲线的比较

    Figure  11.  Comparison of RMSE trend curves of different phantom reconstruction results using RTV and TV minimization reconstruction algorithm under the condition of 50 projection angles and adding Gaussian white noise with variance of 0.05 to the projection data

    图  12  λ不同取值的实验结果和其各自对应的中心线波形图

    Figure  12.  Experimental results with different λ values and their corresponding centerline waveforms

    图  13  ε不同取值的实验结果和其各自对应的中心线波形图

    Figure  13.  Experimental results with different ε values and their corresponding centerline waveforms

        1. Repeat main loop     13. $for{\text{ } }i = 1{\text{:} }{N_{{\rm{grad}}} }{\text{ do RTV - ASD loop} }$:
        2. $ {f}_{p}={f}^{(n)} $       ${\boldsymbol{d} }{ {\boldsymbol{f} }^{(k)} } = {\nabla _f}\left\| { { {\boldsymbol{f} }^{(n)} } } \right\|_{{\rm{RTV}}}^{(k)}$
        3. ${\text{for } }i = 1{\text{ : } }{N_d}{\text{ do } }{\boldsymbol{f} } = {\boldsymbol{f} } + \beta { {\boldsymbol{A} }_{\boldsymbol{i} } }\displaystyle \frac{ { {g_i} - { {\boldsymbol{A} }_{\boldsymbol{i} } }{\boldsymbol{ \times f} } } }{ { { {\boldsymbol{A} }_{\boldsymbol{i} } }{\boldsymbol{ \times } }{ {\boldsymbol{A} }_{\boldsymbol{j} } } } }$       $ {\boldsymbol{d}}{{\boldsymbol{f}}^{(k)}} = {{{\boldsymbol{d}}{{\boldsymbol{f}}^{(k)}}} \mathord{\left/ {\vphantom {{{\boldsymbol{d}}{{\boldsymbol{f}}^{(k)}}} {{{\left\| {{\boldsymbol{d}}{{\boldsymbol{f}}^{(k)}}} \right\|}_2}}}} \right. } {{{\left\| {{\boldsymbol{d}}{{\boldsymbol{f}}^{(k)}}} \right\|}_2}}} $
        4. ${\text{for }}i = 1{\text{ : }}{N_i}{\text{ do if }}{f_i} < 0{\text{ then }}{f_i} = 0$       $ {{\boldsymbol{f}}^{(n)}} = {{\boldsymbol{f}}^{(n)}} - {d_\alpha }*{\boldsymbol{d}}{{\boldsymbol{f}}^{(k)}} $
        5. ${ {\boldsymbol{f} }_{{\rm{res}}} } = { {\boldsymbol{f} }^{(n)} }$       end for
        6. ${{\boldsymbol{g}}^{(n)}} = {\boldsymbol{A}} \times {{\boldsymbol{f}}^{(n)}}$     14. $ \nabla {{\boldsymbol{f}}_{{\text{RTV}}}} = {\left\| {{{\boldsymbol{f}}^{(n)}} - {{\boldsymbol{f}}_p}} \right\|_2} $
        7. ${d_p} = {\left\| {{{\boldsymbol{g}}^{(n)}}{\boldsymbol{ - }}{{\boldsymbol{g}}_{\boldsymbol{0}}}} \right\|_2}$     15. ${\rm{if}}{\text{ } }\nabla { {\boldsymbol{f} }_{ {\text{RTV} } } } > {r_{\max} }*\nabla { {\boldsymbol{f} }_{ {\text{POCX} } } }{\text{ } }{\rm{and}}{\text{ } }{d_p} > \in {\text{ then} }$
        10. $\nabla {{\boldsymbol{f}}_{{\text{POCX}}}} = {\left\| {{{\boldsymbol{f}}^{(n)}} - {{\boldsymbol{f}}_p}} \right\|_2}$       ${d_\alpha } = {d_\alpha }*{\alpha _{{\rm{red}}} }$
        11. ${\text{if first iterarion , then }}{d_\alpha } = \alpha \cdot \nabla {{\boldsymbol{f}}_{{\text{POCX}}}}$     16. $\beta = \beta \cdot {\beta _{{\rm{red}}} }$
        12. ${{\boldsymbol{f}}_p} = {{\boldsymbol{f}}^{(n)}}$     17. ${\text{until}}\left\{ {{\text{stopping criteria}}} \right\}$
        18.${\text{return } }\;{f_{ {\rm{res} } } }$
    下载: 导出CSV

    表  1  RTV算法和TV算法重建FORBILD模体的RMSE和SSIM比较

    Table  1.   RMSE and SSIM comparison of RTV and TV algorithms for FORBILD phantom reconstruction

    投影个数20304050
    RMSERTV30.0×10-59.96×10-57.11×10-55.60×10-5
    TV5130×10-52530×10-5620×10-5590×10-5
    SSIMRTV0.95870.97920.99720.9992
    TV0.92930.95830.99430.9986
    下载: 导出CSV

    表  2  RTV算法和TV算法重建Shepp-Logan模体的RMSE和SSIM比较

    Table  2.   RMSE and SSIM comparison of RTV and TV algorithms for Shepp-Loganphantom reconstruction

    投影个数20304050
    RMSERTV8.010×10-54.791×10-53.349×10-52.586×10-5
    TV1610×10-5530×10-5160×10-580×10-5
    SSIMRTV0.98660.99480.99780.9995
    TV0.98120.99230.99560.9991
    下载: 导出CSV

    表  3  RTV算法和TV算法重建真实CT模体的RMSE和SSIM比较

    Table  3.   RMSE and SSIM comparison of RTV and TV algorithms for CT phantom reconstruction

    投影个数20304050
    RMSERTV0.04030.02940.02100.0120
    TV0.04670.03020.02180.0168
    SSIMRTV0.92130.95460.97430.9956
    TV0.89770.93870.95420.9842
    下载: 导出CSV

    表  5  RTV算法和TV算法在不同等级噪声下重建Shepp-Logan模体的RMSE和SSIM比较

    Table  5.   Comparison of RMSE and SSM of RTV and TV algorithms for reconstructing Shepp- Logan phantom under different levels of noise

    噪声方差0.010.020.030.040.05
    RMSERTV0.00150.00280.00350.00480.0055
    TV0.00650.00810.01020.01260.0148
    SSIMRTV0.99430.98230.97810.96530.9526
    TV0.99140.98140.97270.95940.9512
    下载: 导出CSV

    表  4  RTV算法和TV算法在不同等级噪声下重建FORBILD模体的RMSE和SSIM比较

    Table  4.   Comparison of RMSE and SSM of RTV and TV algorithms for reconstructing FORBILD phantom under different levels of noise

    噪声方差0.010.020.030.040.05
    RMSERTV0.00210.00290.00320.00430.0050
    TV0.00850.01070.01340.01620.0184
    SSIMRTV0.99230.98430.98120.97560.9712
    TV0.98360.97930.97540.96980.9652
    下载: 导出CSV

    表  6  RTV算法和TV算法在不同等级噪声下重建真实CT模体的RMSE和SSIM比较

    Table  6.   Comparison of RMSE and SSM of RTV and TV algorithms for reconstructing CT phantom under different levels of noise

    噪声方差0.010.020.030.040.05
    RMSERTV0.01880.01980.02050.02250.0233
    TV0.01940.02150.02330.02470.0286
    SSIMRTV0.92580.90420.89380.88140.8715
    TV0.91230.89260.86260.86120.8523
    下载: 导出CSV
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  • 收稿日期:  2022-09-25
  • 修回日期:  2022-10-13
  • 录用日期:  2022-10-17
  • 网络出版日期:  2022-11-04

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