## 3 Core Technologies and Algorithms
### D PSF Evaluation  
The fluorescent normalization intensity of one fluorescent nanoparticle on the object plane is [18]:  
\[f_{particle}(x_{o},y_{o})=\left\{\begin{array}{l}\sqrt{1-(x_{o}^{2}+y_{o}^{2 })/(D_{particle}/2)^{2}},\ (\sqrt{x_{o}^{2}+y_{o}^{2}}\leq D_{particle}/2)\\ 0,\ (\text{\emph{otherwise}})\end{array}\right. \tag{3}\]  
where \(x_{o}\) and \(y_{o}\) are the coordinate values of the object plane along the X and Y axes, respectively; \(D_{particle}\) is the diameter of the fluorescent nanoparticle. When only considering geometrical optic imaging, the intensity distribution of the fluorescent nanoparticle image is \(f_{particle}\) (\(-x_{i}/M\), \(-y_{i}/M\)) after microscopic magnification, where \(M\) is the microscopic magnification and \(x_{i}\) and \(y_{i}\) are the coordinate values of the imaging plane.  
The normalization intensity distribution of the PSF on an object plane can be considered a function of the Gaussian function shape [19; 20]:  
\[I(x_{i},y_{i})=\exp(-\frac{{x_{i}}^{2}+{y_{i}}^{2}}{2\tau_{I}^{2}}) \tag{4}\]  
where \(\tau_{I}\) is standard deviation (\(\tau_{I}\) is the desired value) and the PSF's FWHM is determined by \(\tau_{I}\), \(FWHM_{PSF}=2\tau_{I}\sqrt{2 ln2}\). The normalized microscopic diffraction imaging of one fluorescent nanoparticle is:  
\[F_{imaging}(x_{i},y_{i})=f_{particle}(-x_{i}/M,-y_{i}/M)*I(x_{i},y_{i}) \tag{5}\]  
where \(F_{imaging}\) (\(x_{i},y_{i}\)) is the normalized intensity distribution of the nanoparticle's image and the \(F_{imaging}\) (\(x_{i}\), \(y_{i}\)) central position can be obtained with the Gaussian function. In this way, it is possible to build the convolution coordinates of Equation (5) to use as the central position for the original point.  
In order to obtain \(I\) (\(x_{i},y_{i}\)) so that the standard deviation \(\tau_{I}\) of \(I\) (\(x_{i},y_{i}\)) can be obtained, Equation (5) can be used to solve a minimization problem:  
\[Minimize:\|f_{particle}(-x_{i}/M,-y_{i}/M)*I(x_{i},y_{i})-F_{imaging}(x_{i},y_{i })\| \tag{6}\]  
Equation (6) can be solved by an iteration algorithm. Choosing a fine initial iteration value plays a very important role in whether or not an ultimate optimum solution is obtained. Due to the FWHM of \(F_{imaging}\) (\(x_{i}\), \(y_{i}\)), \(FWHM_{imaging}\) is slightly greater than \(FWHM_{PSF}\); particularly, when \(D_{particle}\) is small, the former is approximately equal to the latter. Therefore, the standard deviation \(\tau_{I}\) of the initial iteration value can be taken as \(FWHM_{imaging}/(2\sqrt{2 ln2})\).  
\(\tau_{IZ}\) is the \(\tau_{I}\) value corresponding to any object plane, and at the focal depth \((\tau_{IZ})^{2}\), the variation approximately acts linearly [19]. Here, we used cubic polynomial \((\tau_{IZ})^{2}=a|Z_{P}|^{3}+b|Z_{P}|^{2}+c|Z_{P}|+(\tau_{lo})^{2}\) to fit the \((\tau_{IZ})^{2}\) variation, where \(Z_{P}\) is any position relative to the focal object plane (original position) along the Z-axis; \(\tau_{lo}\) is the \(\tau_{I}\) value corresponding to the focal object plane; and \(a\), \(b\), and \(c\) are fitting coefficients. The 3D PSF can be expressed as:\[I_{Z}(x_{i},y_{i})=\exp(-\frac{x_{i}^{2}+y_{i}^{2}}{2\tau_{IZ}^{2}}) \tag{7}\]  
Obtaining \(\tau_{IZ}\) on five perpendicular object planes (the focal object plane and any four perpendicular object planes) can initially provide the coefficients \(a\), \(b\), and \(c\). Therefore, the initial iteration value \(\tau_{IZint}\) of Equation (6) can be taken as \(\tau_{IZ}=\sqrt{a|Z_{p}|^{3}+b|Z_{p}|^{2}+c|Z_{p}|+\left(\tau_{Io}\right)^{2}}\) for the process of obtaining \(\tau_{IZ}\) on other perpendicular object planes. All obtained \(\tau_{IZ}\) values and their corresponding \(Z_{p}\) values are used to determine the new fitting coefficients, \(a\), \(b\), and \(c\). This operation is repeated for the perpendicular object planes as much as possible, ultimately confirming the coefficients \(a\), \(b\), and \(c\).