Positron Emission Tomography Reconstruction Techniques

The main subject of this paper is the reconstruction of positron emission tomography (PET). The imaging of PET is to count the photon events emitted by the rannihilation. The purpose of reconstruction is to recover the density distribution of isotopy. Essentially, the imaging of PET is finite photons counting procedure. The emitting photon pairs travel randomly in all direct. We think is unsuitable to consider is as attenuated Radon transform. In reconstruction practice, we have to discrete density distribution into pixels (voxels). 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. So PET is a very noisy compared to other imaging model. The imaging procedure of PET can be described with an linear equation. But due to the ill-posed nature of PET as well as the large dimension of system matrix, it is impossible to solve the equation. In general, the purpose of PET reconstruction is to find an algorithm to estimate the density distribution. The most frequently used algorithms are borrowed from signal estimation. Firstly, we discussed the detectors system, and the establishment of system matrix as well as the compensation of attenuation, scattering and all other measurement error in chapter two. Follow the attenuating Radon transform interpretation of PET, we can use the FBP algorithm of transmission tomography, but the results show it is too noise, c.f. chapter three. According to the estimation rules, the reconstruction algorithms can be divided into two categories: deterministic methods and probability based methods. Deterministic algorithms are those data matching methods based on measurement equation, such as least square error. Some other deterministic rules include: weighted least square error, which takes the measurement error into consideration; p -norm solution, which is the generalized distance measure; minimum Kullback-Leibler distance ride, this methods not suitable for large dimensional question. We have performed reconstruction based on these methods .To overcome the ill-posed problem, the regulation process is also introduced in those methods. In fact, the regulation can be thought as a special case of Bayesian reconstruction. We compared those methods in third chapter. The probability reconstruction methods are based the statistic nature of imaging procedure and the some prior estimation of density distribution. We know that the distributions of each pixel and each measurement are Poisson, then the likelihood function is the mutliplication of Gamma function. With the unobserved 揷omplete data? we can get the iteration formula of maximum expectation algorithm of maximum likelihood estimator. Tow obstacles of maximum expectation algorithm are the convergence rate and the noise in the reconstruction results of high iteration number. We considered some acceleration methods for maximum expectation algorithm. To overcome the reconstruction noise, one must induce some prior information of the density distribution. this is indeed Bayesian reconstruction. The fifth chapter is about Bayesian reconstruction. First, we discuss the Gibbs field priori and neighborhood Markov random field of image. The purpose of a priori is to restrain the reconstruction noise, but it always oversmooth the image. We have constructed several kinds of prior functions, which can preserve the edges of image. Due to the prio...