图像重建的迭代算法Iterative procedure for image reconstruction and OSEM

除了简单的back projection 和 改进的filtered back projection,用的更多的图像重建方是迭代法，其中包括EM法。

 

简单迭代法：

 The task in iterative reconstruction in emission tomography is to attempt to solve a linear set of hundreds of thousands of equations with typically hundreds of thousands of unknowns. Such a set of equations
is shown in Fig. 6a: the unknown numbers Ai (together forming a so-called vector A, with element indexes i ranging from 1 to V ) are the amounts of radioactive tracer present in each tiny volume element (voxel) within the object.

At the end of a reconstruction, each of the estimated numbers can be transformed to colour- or grey-
scale pixels for visualisation.

 

Before the iterative calculations can start, one needs a set of numbers (matrix elements) in which each of the
elements Mji represents a probability that gamma quanta emitted by an amount of tracer Ai present in a voxel i will be detected in a pixel j at one of the detectors.

Mji together with P determine the set of equations from which the activity distribution A has to be solved. For example, during an iteration of the ML-EM algorithm, the actual estimate of A (which we call Ae ) is used to generate an estimate of the projection, denoted with vector Pe , simply by carrying out the summations, as presented in Fig. 6a, but with Ae instead of A. Next,ML-EMuses the relative differences between P and Pe
, to calculate an object error map. The error map is used to update Ae with a simple equation. The basic idea
behind all iterative methods is that when Pe is very close to P, Ae must be close to the reality A, because it produces almost the same projection as does A.

 

下面是Medical imaging 里面用的较多的OSEM

The generation of a new Pe and the updating of Ae often need to be repeated hundreds of times to obtain a good solution. Because of the many iterations required, acceleration methods to speed up the algorithms have been developed. The ordered subset expectation maximisation (OS-EM, [36]) is currently the most popular method.