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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.
{img fileId="3329" thumb="mousesticky" width="400" styleimage="border" align="center" desc="Figure 1."}
{img src="http://oftankonyv.reak.bme.hu/tiki-download_file.php?fileId=3134&display" thumb="mousesticky" width="400" styleimage="border" align="center" desc="Figure 2."}
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{SUP()}+{SUP}{SUB()}x{SUB} = (A{SUP()}+{SUP}{SUB()}x1{SUB}, A{SUP()}+{SUP}{SUB()}x2{SUB}, ..., A{SUP()}+{SUP}{SUB()}xm{SUB}),and let the individual weighting factors belonging to the corresponding signals be denoted by
*W{SUP()}+{SUP}{SUB()}1x{SUB}, W{SUP()}+{SUP}{SUB()}2x{SUB}, ..., W{SUP()}+{SUP}{SUB()}mx{SUB}.
Analogously for the other directions:
* direction -x: A{SUP()}-{SUP}{SUB()}x{SUB} = (A{SUP()}-{SUP}{SUB()}x1{SUB}, A{SUP()}-{SUP}{SUB()}x2{SUB}, ..., A{SUP()}-{SUP}{SUB()}xm{SUB}); W{SUP()}-{SUP}{SUB()}1x{SUB}, W{SUP()}-{SUP}{SUB()}2x{SUB}, ..., W{SUP()}-{SUP}{SUB()}mx{SUB}
*direction +y: A{SUP()}+{SUP}{SUB()}y{SUB} = (A{SUP()}+{SUP}{SUB()}y1{SUB}, A{SUP()}+{SUP}{SUB()}y2{SUB}, ..., A{SUP()}+{SUP}{SUB()}ym{SUB}); W{SUP()}+{SUP}{SUB()}1y{SUB}, W{SUP()}+{SUP}{SUB()}2y{SUB}, ..., W{SUP()}+{SUP}{SUB()}my{SUB}
*direction –y: A{SUP()}-{SUP}{SUB()}y{SUB} = (A{SUP()}-{SUP}{SUB()}y1{SUB}, A{SUP()}-{SUP}{SUB()}y2{SUB}, ..., A{SUP()}-{SUP}{SUB()}ym{SUB}); W{SUP()}-{SUP}{SUB()}1y{SUB}, W{SUP()}-{SUP}{SUB()}2y{SUB}, ..., W{SUP()}-{SUP}{SUB()}my{SUB}
With the aid of __equations F4 and F5__, with respect to the different directions and based on the PMT subsets A{SUP()}+{SUP}{SUB()}x{SUB}, A{SUP()}-{SUP}{SUB()}x{SUB}, A{SUP()}+{SUP}{SUB()}y{SUB}, A{SUP()}-{SUP}{SUB()}y{SUB}, the position estimations can be written as follows:
::{EQUATION(size="90")}$
\begin{matrix}
W_{x}^{+}=\frac{\sum\limits_{i=1}^{m}{W_{ix}^{+}}A_{xi}^{+}}{\sum\limits_{i=1}^{m}{W_{ix}^{+}}} & W_{x}^{-}=\frac{\sum\limits_{i=1}^{m}{W_{ix}^{-}}A_{xi}^{-}}{\sum\limits_{i=1}^{m}{W_{ix}^{-}}} \\
W_{y}^{+}=\frac{\sum\limits_{i=1}^{m}{W_{iy}^{+}}A_{yi}^{+}}{\sum\limits_{i=1}^{m}{W_{iy}^{+}}} & W_{y}^{-}=\frac{\sum\limits_{i=1}^{m}{W_{iy}^{-}}A_{yi}^{-}}{\sum\limits_{i=1}^{m}{W_{iy}^{-}}} \\
\end{matrix} \Bigg\}
{EQUATION}::
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:
::{EQUATION(size="75")}$
E=\sum\limits_{i=1}^{n}{{{A}_{i}}}
{EQUATION}::
where:
::{EQUATION(size="75")}$
A=A_{x}^{+}\bigcup A_{y}^{+}\bigcup A_{x}^{-}\bigcup A_{y}^{-}={{A}_{1}},{{A}_{2}},...,{{A}_{n}}
{EQUATION}::
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:
::{EQUATION(size="75")}$
X=\frac{W_{x}^{+}-W_{x}^{-}}{E}
{EQUATION}::
::{EQUATION(size="75")}$
Y=\frac{W_{y}^{+}-W_{y}^{-}}{E}
{EQUATION}::
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{SUP()}+{SUP}{SUB()}x{SUB}, W{SUP()}+{SUP}{SUB()}y{SUB} and W{SUP()}-{SUP}{SUB()}x{SUB}, W{SUP()}-{SUP}{SUB()}y{SUB} carry not only position information, but also energy information. The position estimated by W{SUP()}+{SUP}{SUB()}x{SUB}, W{SUP()}+{SUP}{SUB()}y{SUB} and W{SUP()}-{SUP}{SUB()}x{SUB}, W{SUP()}-{SUP}{SUB()}y{SUB} 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{SUB()}{HTML()}γ{HTML}{SUB} ≤ 600 keV {HTML()}γ{HTML} 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).