单光子发射计算机断层成像算法研究
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摘要
单光子发射计算机断层成像(以下简称SPECT)是重要的核医学成像技术之一,在临床上有重要应用。常用的SPECT图像重建方法有解析方法和代数方法两类。解析重建方法依赖于指数(或衰减)Radon变换的反演公式,其优点是计算复杂度低,缺点是对退化数据成像效果不理想。代数重建方法易于将数据退化因素内蕴于成像系统矩阵中,重建图像的质量较高,其缺点是计算复杂度高。
在第三章和第四章,我们研究了180°投影数据下SPECT成像的解析重建方法。前人的工作己经指出：对指数Radon变换的投影数据进行加权微分反投影运算可以将SEPCT重建问题转化为一维双曲余弦Hilbert变换(以下简称CHT)的反演。尽管前人己经证明了在一定条件下CHT反演的唯一性,但目前并没有得到CHT的精确反演公式。在第三章,我们根据Hilbert变换的Tricomi反演公式,利用双曲余弦函数Taylor展开的特性,提出了反演CHT的矩方法。在第四章,我们还提出了一种基于CHT的半解析SPECT图像重建方法。由于解析方法缺乏抗噪性,使用矩方法重建的图像存在很多伪影,为此,根据反投影运算中权函数的特点,我们设计了基于CHT的SPECT正则化重建模型,其平衡参数随着位置的变化而改变。我们通过数值实验验证了上述两种算法的有效性。
在第五章,我们研究了SPECT成像的代数重建方法。我们研究了EM迭代superiorization后的收敛性,并应用于SPECT成像。由于SPECT投影数据服从Poisson分布,所以EM迭代是SPECT成像的常用算法。但是由于投影数据的噪声和问题的欠定性,EM算法重建的图像往往含有很多伪影。正则化方法是提高重建图像质量的常用技术,这意味着我们需要求解最优化问题。由于成像问题的规模比较大,目前没有高效的求解算法。对迭代算法采用superiorization技术是解决最优化成像问题的新思路。我们首先证明了当扰动满足一定条件时扰动EM迭代仍然收敛。其次,基于扰动EM迭代的收敛条件,我们设计了EM迭代的superiorization算法,并针对全变差函数和小波系数的l1范数极小化问题讨论了算法实现的细节。最后,数值实验验证了superiorized EM算法的有效性。
Single photon emission computed tomography (SPECT), which is widely used in clin-ical applications, is one of the important nuclear imaging techniques. The variety ofexisting SPECT reconstruction algorithms can be split into a family of analytical meth-ods and a wide class of iterative techniques. The analytical methods are based on theinversion of the exponential or attenuated Radon transform. The main advantage ofthem is low computational cost, and the main disadvantage of them is that the recon-structed image is undesired for the noised projection data. The iterative methods takea lot of degraded factors of projection data into system matrix, and the qualities of re-constructed images are higher. However, the computational cost of them is very high.
In the third and forth chapters of this thesis, we studied the analytical methods for180projection data. It had been pointed out that the weight-differentiated backprojec-tion (WDBP) of projection data of the exponential Radon transform could reduce theSPECT reconstruction to inverting a one-dimensional cosh-Hilbert transform (CHT).Although the uniqueness of the inversion of CHT had been proved under some condi-tions, there is no analytically and accurately inverse formula of it. Based on the Tricomiinversion formula for Hilbert transform and the characteristics of the Taylor expansionof hyperbolic cosine function, we proposed a moment-based method for the inversionof CHT numerically in the third chapter. Furthermore, Based on the CHT, we proposeda semi-analytical method for SPECT image reconstruction in the forth chapter. Dueto the lack of noise immunity, the reconstructed images by the moment-based methodhave a lot of distortions. In order to suppress the distortions, we proposed a regulariza-tion model based on the CHT, in which the balance parameter varies according to theweight function of WDBP. We validated the performances of the proposed methods bynumerical simulations.
In the fifth chapter of the thesis, we investigated the algebraic methods of SPECTimage reconstruction. We studied the convergence of the superiorized EM algorithm,and applied to SPECT image reconstruction. Because the projection data of SPECTobey the Poisson distribution, EM iteration is widely used in the field of SPECT im- age reconstruction. However, the reconstructed images have a lot of distortions dueto the noise of projection data and uncertainty of the system matrix. Regularizationmethods are the common techniques to improve the qualities of reconstructed images,which implies that we have to solve optimal problems. Due to the large scale of imag-ing problems, there is no efficient algorithm at present. The superiorization of iterativealgorithms is a new idea to handle the optimal problems. Firstly, we proved the con-vergence of perturbed EM iteration under some conditions. Secondly, we designed thesuperiorized EM algorithm based on the convergent conditions, and discussed the de-tails of implementations for total variation and l1-norm minimization problems. Lastly,the numerical experiments were conducted to validate the efficiency of the proposedalgorithms.
在第三章和第四章,我们研究了180°投影数据下SPECT成像的解析重建方法。前人的工作己经指出：对指数Radon变换的投影数据进行加权微分反投影运算可以将SEPCT重建问题转化为一维双曲余弦Hilbert变换(以下简称CHT)的反演。尽管前人己经证明了在一定条件下CHT反演的唯一性,但目前并没有得到CHT的精确反演公式。在第三章,我们根据Hilbert变换的Tricomi反演公式,利用双曲余弦函数Taylor展开的特性,提出了反演CHT的矩方法。在第四章,我们还提出了一种基于CHT的半解析SPECT图像重建方法。由于解析方法缺乏抗噪性,使用矩方法重建的图像存在很多伪影,为此,根据反投影运算中权函数的特点,我们设计了基于CHT的SPECT正则化重建模型,其平衡参数随着位置的变化而改变。我们通过数值实验验证了上述两种算法的有效性。
在第五章,我们研究了SPECT成像的代数重建方法。我们研究了EM迭代superiorization后的收敛性,并应用于SPECT成像。由于SPECT投影数据服从Poisson分布,所以EM迭代是SPECT成像的常用算法。但是由于投影数据的噪声和问题的欠定性,EM算法重建的图像往往含有很多伪影。正则化方法是提高重建图像质量的常用技术,这意味着我们需要求解最优化问题。由于成像问题的规模比较大,目前没有高效的求解算法。对迭代算法采用superiorization技术是解决最优化成像问题的新思路。我们首先证明了当扰动满足一定条件时扰动EM迭代仍然收敛。其次,基于扰动EM迭代的收敛条件,我们设计了EM迭代的superiorization算法,并针对全变差函数和小波系数的l1范数极小化问题讨论了算法实现的细节。最后,数值实验验证了superiorized EM算法的有效性。
Single photon emission computed tomography (SPECT), which is widely used in clin-ical applications, is one of the important nuclear imaging techniques. The variety ofexisting SPECT reconstruction algorithms can be split into a family of analytical meth-ods and a wide class of iterative techniques. The analytical methods are based on theinversion of the exponential or attenuated Radon transform. The main advantage ofthem is low computational cost, and the main disadvantage of them is that the recon-structed image is undesired for the noised projection data. The iterative methods takea lot of degraded factors of projection data into system matrix, and the qualities of re-constructed images are higher. However, the computational cost of them is very high.
In the third and forth chapters of this thesis, we studied the analytical methods for180projection data. It had been pointed out that the weight-differentiated backprojec-tion (WDBP) of projection data of the exponential Radon transform could reduce theSPECT reconstruction to inverting a one-dimensional cosh-Hilbert transform (CHT).Although the uniqueness of the inversion of CHT had been proved under some condi-tions, there is no analytically and accurately inverse formula of it. Based on the Tricomiinversion formula for Hilbert transform and the characteristics of the Taylor expansionof hyperbolic cosine function, we proposed a moment-based method for the inversionof CHT numerically in the third chapter. Furthermore, Based on the CHT, we proposeda semi-analytical method for SPECT image reconstruction in the forth chapter. Dueto the lack of noise immunity, the reconstructed images by the moment-based methodhave a lot of distortions. In order to suppress the distortions, we proposed a regulariza-tion model based on the CHT, in which the balance parameter varies according to theweight function of WDBP. We validated the performances of the proposed methods bynumerical simulations.
In the fifth chapter of the thesis, we investigated the algebraic methods of SPECTimage reconstruction. We studied the convergence of the superiorized EM algorithm,and applied to SPECT image reconstruction. Because the projection data of SPECTobey the Poisson distribution, EM iteration is widely used in the field of SPECT im- age reconstruction. However, the reconstructed images have a lot of distortions dueto the noise of projection data and uncertainty of the system matrix. Regularizationmethods are the common techniques to improve the qualities of reconstructed images,which implies that we have to solve optimal problems. Due to the large scale of imag-ing problems, there is no efficient algorithm at present. The superiorization of iterativealgorithms is a new idea to handle the optimal problems. Firstly, we proved the con-vergence of perturbed EM iteration under some conditions. Secondly, we designed thesuperiorized EM algorithm based on the convergent conditions, and discussed the de-tails of implementations for total variation and l1-norm minimization problems. Lastly,the numerical experiments were conducted to validate the efficiency of the proposedalgorithms.
引文
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[3]高上凯,医学成像系统.北京：清华大学出版社,2ed.,2010.
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