输运模型中层析重建问题的随机性
Stochastic Aspect of the Tomographic Reconstruction Problems in a Transport Model
摘要：
考虑了描述介质中粒子吸收和散射的相互间弱相互作用的粒子质点系系统的随机微分和积分方程。考虑核成像中参数的随机性，研究了含散射的含时输运方程。利用动态衰减Radon变换，可导出考虑散射作为微扰的输运方程的解。我们分析了随机变量对图像重建的影响，一般情况下，更详细的情况下，点源。滤波反投影（FBP）方法的例子表明，在单光子发射型计算机断层扫描（SPECT）中，未考虑的衰减系数的微小波动会引起图像纹理的实质性畸变和图像重建时分辨率的降低，而在X射线计算机断层扫描（CT）中则较少。分析了这些失真的机理。对于点源，给出了消除它们的方法。本文论证了当平均衰减系数与真实衰减系数之差具有一定的随机性时，就可以确定平均衰减系数。它表明，对于正电子发射断层成像（PET）的传输模型的参数中的随机组件没有散射可以考虑的投影校正。
英文摘要：
The stochastic differential and integral equations describing the system of particles weakly interacting among themselves which are absorbed and scattered by particles of a medium are considered. The time-dependent transport equation with scattering is studied taking into account stochastic nature of parameters in nuclear imaging. Using dynamic attenuated Radon transform the solution of transport equation may be derived taking into account of the scattering as perturbation. We analyze the influence of the random variables upon the image reconstruction both generally and in more details for the case of point source. It is shown by the example of the method of the filtered back projection (FBP) that unaccounted small fluctuations of attenuation coefficient can cause essential distortions of image texture and degradation of the resolution at image reconstruction in single-photon emission computerized tomography (SPECT) and less in X-ray computerized tomography (CT). The mechanism of these distortions is analyzed. The way for their elimination is shown for point sources. We demonstrate that for the practical purposes it is enough to define averaged attenuation coefficient in investigated area when its difference from true attenuation coefficient has certain stochastic properties. It is shown that for positron emission tomography (PET) stochastic components in parameters of a transport model without the scattering can be taken into account of the corrections of projections.
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