## 2 Materials and Methods  
### Theory  
In order to analytically describe how the real counts can be recovered from the image counts, some basic assumptions have been made. Specifically, the method proposed here takes into account only how the system PSF affects the PVE under ideal conditions of uniform radioactive source surrounded by uniform background.  
From Equation (7) and Equation (8) it is possible to obtain the value of the source counts concentration s (see Equation (9)), as a function of the background counts concentration b, Cimage and H1D(l,o):  
\[s=\frac{1}{H_{1D}(l,\sigma)}C_{image}-b\frac{1-H_{1D}(l,\sigma)}{H_{1D}(l, \sigma)} \tag{9}\]  
Equation (9) shows that it is possible to quantify s from the knowledge of:  
1. the function H1D(l,o), which is an analytic function of measurable parameters: the dimension of the source and the \(\sigma\) of the image system's PSF;
2. Cimage directly measurable on the image space;
3. \(b\) that, in the practice, can be evaluated considering a background region of interest (ROI) away from the source (i.e., away with respect to the FWHM of the system's PSF) on the image space. It is important to notice that:  
1. \(0\leq H_{1D}(l,\sigma)\leq 1\);
2. \(\frac{1}{H_{1D}(l,\sigma)}\) is the factor which corrects the effect of loss of counts on the image space inside the ROI (_spill out_);
3. \(\frac{1-H_{1D}(l,\sigma)}{H_{1D}(l,\sigma)}\) is the factor that corrects the effect of increase of counts on the image space inside the ROI due to the background (_spill in_).  
The proposed approach can now be generalized in a 3D ideal scenario (Equation (10)), i.e., by considering a 3D source surrounded by a homogeneous background, which can be described as:  
\[S_{3D}=\left(x,l_{x};y,\ l_{y};z,l_{z}\right)=\left\{\begin{array}{cc}s& \forall(x,y,z)\in\left[\frac{-l_{x}}{2};\frac{l_{y}}{2}\right];\ \frac{\left[-l_{y}}{2};\frac{l_{y}}{2}\right];\left[\frac{-l_{z}}{2};\frac{l_ {z}}{2}\right]\\ b&\text{outside}\end{array}\right. \tag{10}\]  
Since the PSF of the 3-D (three-dimensional) imaging system can be derived as:  
\[\begin{array}{c}PSF_{3D}\left(x,\ \sigma_{x};y,\sigma_{y};z,\ \sigma_{z} \right)=\frac{1}{\left(\sigma_{x},\sigma_{y}\sigma_{z}\right)\sqrt{8\pi}}e^{- \frac{x^{2}}{\sigma_{x}^{2}}-\frac{y^{2}}{\sigma_{y}^{2}}-\frac{x^{2}}{\sigma _{z}^{2}}}=\\ PSF_{1D}(x,\ \sigma_{x})PSF_{1D}\big{(}x,\ \sigma_{y}\big{)}PSF_{1D}(x,\ \sigma_{z}) \end{array} \tag{11}\]  
it is easy to verify that:  
\[\begin{array}{c}I_{3D}\big{(}x,l_{x},\sigma_{x};y,l_{y},\sigma_{y};z,l_{z},\sigma_{z}\big{)}=\frac{1}{\left(\sigma_{x},\sigma_{y}\sigma_{z}\right)\sqrt {8\pi}}\int_{-\frac{l_{x}}{2}}^{\frac{l_{y}}{2}}\int_{-\frac{l_{z}}{2}}^{\frac {l_{z}}{2}}e^{-\frac{\left(x-x^{\prime}\right)^{2}}{\sigma_{x}^{2}}-\frac{ \left(y-y^{\prime}\right)^{2}}{\sigma_{y}^{2}}-\frac{\left(z-x^{\prime}\right) ^{2}}{\sigma_{z}^{2}}}dx^{\prime}dy^{\prime}\ dz^{\prime}=\\ I_{1D}(x,l_{x},\sigma_{x})\times I_{1D}\big{(}y,l_{y},\sigma_{y}\big{)}\times I _{1D}(z,l_{z},\sigma_{z})\end{array} \tag{12}\]  
and  
\[\begin{array}{c}H_{3D}\big{(}x,\ \sigma_{x};y,\ \sigma_{y};z,\ \sigma_{z} \big{)}=\frac{\int_{-\frac{l_{x}}{2}}^{\frac{l_{y}}{2}}\int_{-\frac{l_{y}}{2}}^ {\frac{l_{y}}{2}}\int_{-\frac{l_{z}}{2}}^{\frac{l_{z}}{2}}\int_{3D(x,\ \sigma_{x}\beta y,\ \sigma_{y}\beta z,\ \sigma_{z})\text{d} utydz}}{\int_{-\infty}^{\frac{l_{y}}{2}}\int_{-\infty}^{ \frac{l_{y}}{2}}\int_{-\infty}^{\frac{l_{y}}{2}}\int_{3D(x,\ \alpha z,\beta y,\ \sigma_{y}\beta z,\ \sigma_{z})\text{d} utydz}}=\\ H_{1D}(l_{x},\sigma_{x})H_{1D}(l_{y},\sigma)H_{1D}(l_{z},\sigma_{z})\end{array} \tag{13}\]  
From Equations (12) and (13), it is possible to derive Equation (14), which describes the final 3D generalization of Equation (7) simply as:  
\[s=\frac{1}{H_{1D}(l_{x},\sigma_{x})H_{1D}(l_{y},\sigma_{y})H_{1D}(l_{z}, \sigma_{z})}C_{image}-b\frac{1-H_{1D}(l_{x},\sigma_{x})H_{1D}(l_{y},\sigma_{y} )H_{1D}(l_{z},\sigma_{z})}{H_{1D}(l_{x},\sigma_{x})H_{1D}(l_{y},\sigma_{y})H_ {1D}(l_{z},\sigma_{z})} \tag{14}\]