Stochastic Aspect of the Tomographic Reconstruction Problems in a Transport Model (1003.2591v1)

Abstract: 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.