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Position determination – Anger principle
We have to determine the precise point of impact of the gamma photon making use of the digital signal generated by the electronics discussed previously. The Anger principle makes this possible, which is basically an energy-weighted position averaging based on the signal and the position of the PMTs. The point of interaction – its X and Y coordinates – can be calculated using the following formula:
{EQUATION()}X=\frac {\sum_i {Px_i \cdot S_i}}{\sum_i S_i}{EQUATION} and {EQUATION()}Y=\frac {\sum_i {Py_i \cdot S_i}}{\sum_i S_i}{EQUATION}
where {EQUATION()}Px_i {EQUATION} and {EQUATION()}Py_i{EQUATION} are the spatial coordinates of the i-th PMT, and {EQUATION()}S_i{EQUATION} is the integrated signal of the i-th PMT, which is proportional to the energy given off by the interacting gamma photon.
::{IMG(src="tiki-download_file.php?fileId=109&display",max="300",desc="Figure 1.: Anger image with linearity phantom, simulation")}{IMG}::
This calculation is basically independent of the energy of the gamma photon, but if the PMT signals are shifted and cut during signal processing in order to remove the low signals, imaging will be highly energy dependent. In case of a Compton scattered photon applying the Anger principle results in a blur at the point of detection. Furthermore, it can also be easily seen that no image can be formed with this method outside of the centre of the outermost PMTs, since there are no coordinates used by the formula. Consequently, the field of view of the camera decreases as compared to the size of the scintillator.
The advantage of the Anger principle is that it allows for fast calculation but the image it provides is highly distorted, so further corrections need to be applied.
Position determination – NLCC
The intrinsic resolution (the full width at half maximum of the point response function) is determined by the noise of the PMTs. Therefore, if we can compensate for the noise with subsequent signal processing, we can achieve better resolution. There are several methods of compensation. One of them includes shifting then cutting the signal of the PMT analogously. The disadvantage of this, as mentioned previously, is the fact that the imaging is energy dependent, due to which image distortions are produced (e.g. Spatreg error, discussed later).
::{img fileId="3337" max="400" desc="Figure 2."}::
We can also use another method, when the signal of the PMT is changed with a nonlinear correction function in a way that the excessively large signals are reduced nonlinearly above a certain threshold (THi), and the so-called bias is subtracted from the signal. The method is based on the fact that the relative standard deviation of the PMT signal is higher when a given PMT is stricken farther from its centre.
::{img fileId="3336" max="400" desc="Figure 3."}::
The advantage of this method is that it provides a more linear intrinsic image with a significantly better resolution. Its disadvantage is that it can only be used with digital cameras because every PMT signal needs to be sampled.
Position determination – statistical calculation
The point of entry of the gamma photon can also be calculated with a statistical method, which is more precise than the Anger principle. The average response ({EQUATION()}M_i{EQUATION}) and standard deviation ({EQUATION()}\sigma_i{EQUATION}). of the PMT signals are determined (in practice by simulations) on the whole surface of the camera along a discrete grid. This way a light response surface is obtained for every PMT, which determines the amplitude of the PMT signal as a function of the position of the light pulse. The surface obtained this way will have a quasi-Gaussian shape, it will differ significantly from the normal distribution at the edge of the field of view only. This surface can be interpolated arbitrarily with splines in later calculations.
::{img src="tiki-download_file.php?fileId=117&display" max="300" desc="Figure 4.: Linearity image, MC simulation, energy without filtering"}::
When an {EQUATION()} S_i {EQUATION} signal vector measured in a scintillation arrives, we search for the (x,y) position where the weighted distance of the signal vector from the average response of the PMT is minimal:
{EQUATION()}\underset {x,y}{min} \sum_i {\sigma(x,y)^{-2}(S_i-M_i(x,y))^2}{EQUATION}
The image formed this way shows high intrinsic linearity and its FOV (Field of View) is larger than what we get when applying the Anger principle. Its disadvantage is that it requires a lot of calculation, since calculations are iterative; however, we can speed the process up by performing the calculations applying the Anger principle and using the result as a preliminary estimate. The calculations can be made energy-independent by calculating the position after the normalization of the PMT signals. This corresponds to projecting onto an equi-energy surface and discriminating there.
When applying this method, the question arises as to what metrics should be used in the D-dimensional space of the PMTs, where D can be as high as 60. In case of the most ideal metrics the full width at half maximum of the average point response is the lowest, the useful field of view is the largest and the standard deviation of position is minimal.
It is important to note here that since the resolution, i.e. the standard deviation of the point response changes locally, the structure of the PMTs appears in the linearized image.
::{IMG(src="tiki-download_file.php?fileId=111&display",max="300",desc="Figure 5.: The standard deviation of the point response draws the structure of the PMT")}{IMG}::