正电子发射断层图像重建算法研究
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摘要
正电子发射断层扫描(Positron Emission Tomography,PET)是当今最高层次的核医学影像技术,它在肿瘤学,心血管疾病学,神经系统疾病研究,以及新药开发研究等领域中显示出卓越的性能。由投影数据进行图像重建在PET研制中占有重要的地位,随着先进的核医学断层影像设备的广泛应用和计算机技术的迅速发展,其研究越来越受到人们的重视。本文讨论了正电子发射成像的重建问题。正电子成像过程是对正负电子湮灭产生的光子对进行计数,而重建的目的是要恢复放射性核素的位置分布信息。一般来说,重建算法总是将连续的密度分布离散化为像素,且认为每个像素的密度分布均匀,它的正电子发射是空间Poisson过程。根据成像系统的线性,投影值也服从Poisson分布。
本文首先论述了各种PET图像重建算法,主要有解析法和迭代法。解析法主要是滤波反投影重建,它的特点是算法简单,速度快但重建图像的质量不好。迭代法分为代数迭代重建算法和统计迭代重建算法。代数迭代重建算法是一类以代数方程理论为基础的算法。而统计迭代重建算法是以各种统计规则为基础的算法,主要有基于最小均方误差准则的图像重建方法,基于最大似然估计的图像重建方法(Maximum Likelihood Expectation Maximization,MLEM)以及基于最大后验概率(Maximum a posterior,MAP)的贝叶斯图像重建方法。迭代法由于能够在重建的过程中考虑各种物理和统计模型,就重建图像质量而言,它们要好于解析法。
有序子集可分离抛物线替代函数(Ordered Subsets Separable Paraboloidal Surrogate,OSSPS)算法认为是一种有效的PET图像重建算法,它能够重建出比传统的最大似然期望值最大化算法质量更好的图像,然而它在重建的过程中需要人为指定松弛参数来保证重建结果不发散,这必然降低图像重建的重复性。文中将增量优化传递算法应用于PET图像重建中,并采用替代函数来代替原有的目标函数得到一种新的PET图像重建算法,EMIOT(Emission Image Reconstruction based on Incremental Optimization Transfer),它不需要人为指定松弛参数,提高了重建实验的重复性,同时它的重建性能和OSSPS的相当。
基于中值先验分布函数的PET图像重建算法(Median Root Prior Image Reconstruction,MRP)重建的图像仍然含有大量噪声,这是因为中值滤波器不能够有效去除PET图像中的高斯和Poisson噪声。文中提出将基于偏微分方程的各向异性扩散滤波器和中值滤波器结合起来用于PET图像重建中,新算法重建的图像无论是从视觉效果还是从客观标准都要好于传统的MAP和MRP。
基于偏微分方程的各向异性扩散滤波器能够有效去除核医学图像中的噪声,为此我们将它应用于MLEM每次迭代中间得到一种简单的正则化的重建算法,MLEM-PDE。大量的实验表明,它重建的图像质量较好,能够用于PET图像重建中,尤其适用于低剂量放射性药物的情况。
在图像复原中广泛使用的ROF模型基础上,结合PET投影数据的统计特性以及具有边界保持的先验函数,我们采用变分的方法得到一种新的PET图像重建算法,它重建的效果要好于MLEM。最后,我们将随迭代次数增加而逐渐减少的松弛参数应用于MLEM中,加快了MLEM重建速度。
PET (Positron Emission Tomography) is one of the most advanced techniques in the world. It could be used to study the metabolism and the function activity of human body in molecular level with the predominant performances in the research fields of oncology, cardiology, neurology and new medicine exploitation. Image reconstruction from projections is very important in designing PET instrumentation. It has become more and more important in PET research with wide application of PET and fast development of computer technology. The thesis is dealt with PET image reconstruction. The imaging of PET is to count the photon events emitted by the annihilation. The purpose of reconstruction is to recover the density distribution of isotopy. In reconstruction practice, we have to discrete density distribution into pixels. The isotopy distribution is uniformly through the whole pixel, and it emits positron follow a space Poisson point process. Due to the linearity of imaging system, the measured values are i. i. d. Poisson distribution.
PET image reconstruction algorithms available are introduced in the first part of the thesis. Currently, it can be classified into analytical and iterative method. Filtered back projection is one of the famous analytical methods which are characteristic of simplicity and fastness. However, its reconstructed images are very noisy. Iterative methods can be divided into algebraic iteration and statistical iteration method. The former is based on algebraic equation theory, while the latter is based on kinds of statistical rules, such as minimum least square, maximum likelihood and maximum a posterior. Because of introducing all kinds of physical and statistical models into the image reconstruction, iterative method can produce better images than filtered back projection.
Ordered subsets separable paraboloidal surrogate algorithm (OSSPS) is considered as a better PET image reconstruction than classical maximum likelihood expectation maximization (MLEM). However, it lacks repetition because of its requirement for designated relaxation parameters. In this thesis, we developed a new image reconstruction algorithm named as EMIOT by introducing incremental optimization transfer theory into PET combined with replacing original function by a surrogate function. It does not need designated relaxation parameters but also is comparable to OSSPS in image reconstruction.
The images reconstructed by median root prior image reconstruction (MRP) are still noisy due to median filter’s poor performance in removing Gaussian and Poisson noise. We proposed a new algorithm which combines anisotropic diffusion filter with median filter in PET. The new algorithm is better than traditional MAP and MRP both from subjective and objective perspective.
We developed a new PET image reconstruction algorithm named as MLEM-PDE, which is an improved MLEM regularized by interiteration filter. Because anisotropic diffusion filter based on partial differential equation can remove noise in nuclear medicine images, we used it in this thesis. The results showed that it can be used in PET imaging, especially in clinical situation.
ROF is an extensive used model in image restoration community, however, it can not be used in PET imaging directly. In this thesis, we considered the statistics in PET projection data and employed edge-preserving prior function, which can modify ROF. From the variational method not expectation maximization method, we get a new good image reconstruction framework. At last, we deduce an accelerated algorithm by introducing a relaxation parameter into MLEM which decrease with iteration.
本文首先论述了各种PET图像重建算法,主要有解析法和迭代法。解析法主要是滤波反投影重建,它的特点是算法简单,速度快但重建图像的质量不好。迭代法分为代数迭代重建算法和统计迭代重建算法。代数迭代重建算法是一类以代数方程理论为基础的算法。而统计迭代重建算法是以各种统计规则为基础的算法,主要有基于最小均方误差准则的图像重建方法,基于最大似然估计的图像重建方法(Maximum Likelihood Expectation Maximization,MLEM)以及基于最大后验概率(Maximum a posterior,MAP)的贝叶斯图像重建方法。迭代法由于能够在重建的过程中考虑各种物理和统计模型,就重建图像质量而言,它们要好于解析法。
有序子集可分离抛物线替代函数(Ordered Subsets Separable Paraboloidal Surrogate,OSSPS)算法认为是一种有效的PET图像重建算法,它能够重建出比传统的最大似然期望值最大化算法质量更好的图像,然而它在重建的过程中需要人为指定松弛参数来保证重建结果不发散,这必然降低图像重建的重复性。文中将增量优化传递算法应用于PET图像重建中,并采用替代函数来代替原有的目标函数得到一种新的PET图像重建算法,EMIOT(Emission Image Reconstruction based on Incremental Optimization Transfer),它不需要人为指定松弛参数,提高了重建实验的重复性,同时它的重建性能和OSSPS的相当。
基于中值先验分布函数的PET图像重建算法(Median Root Prior Image Reconstruction,MRP)重建的图像仍然含有大量噪声,这是因为中值滤波器不能够有效去除PET图像中的高斯和Poisson噪声。文中提出将基于偏微分方程的各向异性扩散滤波器和中值滤波器结合起来用于PET图像重建中,新算法重建的图像无论是从视觉效果还是从客观标准都要好于传统的MAP和MRP。
基于偏微分方程的各向异性扩散滤波器能够有效去除核医学图像中的噪声,为此我们将它应用于MLEM每次迭代中间得到一种简单的正则化的重建算法,MLEM-PDE。大量的实验表明,它重建的图像质量较好,能够用于PET图像重建中,尤其适用于低剂量放射性药物的情况。
在图像复原中广泛使用的ROF模型基础上,结合PET投影数据的统计特性以及具有边界保持的先验函数,我们采用变分的方法得到一种新的PET图像重建算法,它重建的效果要好于MLEM。最后,我们将随迭代次数增加而逐渐减少的松弛参数应用于MLEM中,加快了MLEM重建速度。
PET (Positron Emission Tomography) is one of the most advanced techniques in the world. It could be used to study the metabolism and the function activity of human body in molecular level with the predominant performances in the research fields of oncology, cardiology, neurology and new medicine exploitation. Image reconstruction from projections is very important in designing PET instrumentation. It has become more and more important in PET research with wide application of PET and fast development of computer technology. The thesis is dealt with PET image reconstruction. The imaging of PET is to count the photon events emitted by the annihilation. The purpose of reconstruction is to recover the density distribution of isotopy. In reconstruction practice, we have to discrete density distribution into pixels. The isotopy distribution is uniformly through the whole pixel, and it emits positron follow a space Poisson point process. Due to the linearity of imaging system, the measured values are i. i. d. Poisson distribution.
PET image reconstruction algorithms available are introduced in the first part of the thesis. Currently, it can be classified into analytical and iterative method. Filtered back projection is one of the famous analytical methods which are characteristic of simplicity and fastness. However, its reconstructed images are very noisy. Iterative methods can be divided into algebraic iteration and statistical iteration method. The former is based on algebraic equation theory, while the latter is based on kinds of statistical rules, such as minimum least square, maximum likelihood and maximum a posterior. Because of introducing all kinds of physical and statistical models into the image reconstruction, iterative method can produce better images than filtered back projection.
Ordered subsets separable paraboloidal surrogate algorithm (OSSPS) is considered as a better PET image reconstruction than classical maximum likelihood expectation maximization (MLEM). However, it lacks repetition because of its requirement for designated relaxation parameters. In this thesis, we developed a new image reconstruction algorithm named as EMIOT by introducing incremental optimization transfer theory into PET combined with replacing original function by a surrogate function. It does not need designated relaxation parameters but also is comparable to OSSPS in image reconstruction.
The images reconstructed by median root prior image reconstruction (MRP) are still noisy due to median filter’s poor performance in removing Gaussian and Poisson noise. We proposed a new algorithm which combines anisotropic diffusion filter with median filter in PET. The new algorithm is better than traditional MAP and MRP both from subjective and objective perspective.
We developed a new PET image reconstruction algorithm named as MLEM-PDE, which is an improved MLEM regularized by interiteration filter. Because anisotropic diffusion filter based on partial differential equation can remove noise in nuclear medicine images, we used it in this thesis. The results showed that it can be used in PET imaging, especially in clinical situation.
ROF is an extensive used model in image restoration community, however, it can not be used in PET imaging directly. In this thesis, we considered the statistics in PET projection data and employed edge-preserving prior function, which can modify ROF. From the variational method not expectation maximization method, we get a new good image reconstruction framework. At last, we deduce an accelerated algorithm by introducing a relaxation parameter into MLEM which decrease with iteration.
引文
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