**1. EM algorithm**

frequencies of the image solution. The results of the first iterations are smooth because of the predominance of low frequencies (internal structure) of the object. Subsequently, more iteration are applied more high frequencies (overall shape and background noise) are repre‐ sented. The image produced by iterations approximate gradually the image solution (the algorithm converges) [6]. However, we show that when using such iterative methods and after a number of iterations, the process begins to diverge (under the influence of noise) and the image moves away from the true solution. To overcome this inconvenient, we impose a constraint to the reconstruction process to interrupt the iteration after a certain number of iterations. This is equivalent to use a low pass filter as in the case of filtered back-projection

100 Imaging and Radioanalytical Techniques in Interdisciplinary Research - Fundamentals and Cutting Edge Applications

**fk** 

**Figure 7.** Illustration of projection and back-projection processes.

**2.1. Projection modeling**

following points:

**pi Projection** 

rectangular regardless of the direction of projection [6].

The projection process is modeled by considering the coefficients of the projection matrix R that generate the acquisition data. Some special geometric and physical considerations are necessary for this modeling. In the iterative reconstruction method the modeling concerns the

**1. Modeling of the pixel (detector) intensity distribution**: it is necessary to specify the real conditions of image pixels projection [6]. It is based on the evaluation of the contribution of each pixel in the corresponding projection line (ray). Knowing that the most accurate model (perfect) and the more complicated to be applied consists in considering square pixel (uniform); simpler models are possible. Among these simple models, there is the model called Dirac model in which all the pixel intensity is concentrated in the centre of the pixel. Thus, the whole intensity of the pixel contrib‐ utes to the projection line (ray) if and only if it passes through the Dexel. There is also another model called the model of concave disc which is considered as a compromise between the two previous models. In this model, the intensity of the pixel is geometri‐ cally limited to a disk included in the pixel and distributed so that its projection is

**fk** 

**pi Back-projection** 

method.

This algorithm was developed and proposed by Lange and Carson. The formula of this algorithm is the following [6]:

$$f^{n+1} = f^n \mathbb{R}^t \frac{p}{\mathbb{R}.f^n} \tag{24}$$

Where n is the number of the actual iteration. This algorithm is characterized by the fact that it keeps the number of iterations for each projection. Moreover its multiplicative form gives a positivity constraint although it implies a slow convergence.
