**The centroid method and the Anger camera**

Below, we show how the analytical position estimation algorithm called centroid method, which was described for a 1D position-sensitive detector model in Chapter 2.3.1., can be applied in case of a 2D position-sensitive detector.

Let the surface of the NaI(Tl) scintillation single crystal be covered optimally with a set A consisting of a PMT block (Figures 1. and 2.), which contains n elements (PMTs). Let there be the same “m” number of PMTs arranged symmetrically in all of the quarter-planes defined by the directions (+x,-x) and (+y,-y). The position estimation of a gamma event in the individual quarter-planes is performed as follows. Let the output signals of the PMTs taking part in the position estimation of direction +x be denoted by

- the set A
^{+}_{x}= (A^{+}_{x1}, A^{+}_{x2}, ..., A^{+}_{xm}),and let the individual weighting factors belonging to the corresponding signals be denoted by - W
^{+}_{1x}, W^{+}_{2x}, ..., W^{+}_{mx}.

Analogously for the other directions:

- direction -x: A
^{-}_{x}= (A^{-}_{x1}, A^{-}_{x2}, ..., A^{-}_{xm}); W^{-}_{1x}, W^{-}_{2x}, ..., W^{-}_{mx} - direction +y: A
^{+}_{y}= (A^{+}_{y1}, A^{+}_{y2}, ..., A^{+}_{ym}); W^{+}_{1y}, W^{+}_{2y}, ..., W^{+}_{my} - direction –y: A
^{-}_{y}= (A^{-}_{y1}, A^{-}_{y2}, ..., A^{-}_{ym}); W^{-}_{1y}, W^{-}_{2y}, ..., W^{-}_{my}

With the aid of **equations F4 and F5**, with respect to the different directions and based on the PMT subsets A^{+}_{x}, A^{-}_{x}, A^{+}_{y}, A^{-}_{y}, the position estimations can be written as follows:

It is already known that the scintillation light collected by all the PMTs, which is proportional to the energy of the gamma photon, can be obtained by giving the algebraic sum **(formula F7)** of all the PMTs:

where:

By applying **formula F3** for **equations F10, ..., F14,** an X and Y coordinate based on the centroid method algorithm can be given in the case of a 2D position-sensitive detector:

The physical meaning of normalization by the energy signal E in **formulae F6, F16 and F17** and its effects on imaging can be interpreted as follows. It is already known that the scintillation detector yields an energy-dependent signal, thus the values W^{+}_{x}, W^{+}_{y} and W^{-}_{x}, W^{-}_{y} carry not only position information, but also energy information. The position estimated by W^{+}_{x}, W^{+}_{y} and W^{-}_{x}, W^{-}_{y} is true only in the case of a given, known gamma photon energy. In order to obtain the real position information of the energy-independent X, Y coordinates, the division with the value E – normalization – must be carried out, as shown in **formulae F6, F16 and F17**. This also carries the following physical meaning: the position information – image size – obtained this way for a given detector will be energy-independent – i.e. isotope independent – in the gamma photon energy range of 60 keV ≤ E_{γ} ≤ 600 keV γ after selecting an appropriate energy window. The basic requirement of this is the following: the energy function E(X,Y) has to be considered constant as a function of position along the surface of the detector, within the applied gamma energy range (see the theoretical basis of the Anger camera, criterion 3).