# Fourier Ptychographic Reconstruction Method of Self-Training Physical Model
## 2 Principle of FPM
### Fourier Ptychographic Microscopy Imaging Process  
In the FPM system, the LED array with multi-angle illumination is used as the illumination light of the system. This section analyzes the transfer process of the sample information of the FPM system under more general illumination, as shown in Figure 2. Suppose the incident light from the \(E^{m}(x_{0},y_{0})\) LED:  
\[E^{m}(x_{0},y_{0})=\exp[i2\pi(\mu^{m}x_{0}+v^{m}y_{0})] \tag{1}\]  
Figure 1: Fourier Ptychographic Microscopy Device Diagram. (**a**) Real FPM System; (**b**) FPM System Diagram. where \((\mu,v)\) denotes that the illumination wave vector is determined by the illumination angle of the incident light; \((x_{0},y_{0})\) refers to the LED light source position. \[\mu^{m}=\frac{n\sin\varphi_{x}^{m}}{\lambda},v^{m}=\frac{n\sin\varphi_{y}^{m}}{\lambda} \tag{2}\]  
where \(n\) is the refractive index of the medium in the illumination path. For the LED array structure in Figure 2, the illumination angle of each LED is calculated as follows:  
\[ \begin{array}{c}\sin\varphi_{x}^{m}=\frac{x_{0}^{center}-x^{m}}{\sqrt{\left(x _{0}^{center}-x^{m}\right)^{2}+\left(y_{0}^{center}-y^{m}\right)^{2}+L^{2}}},\\ \sin\varphi_{y}^{m}=\frac{y_{0}^{center}-y^{m}}{\sqrt{\left(x_{0}^{ center}-x^{m}\right)^{2}+\left(y_{0}^{center}-y^{m}\right)^{2}+L^{2}}}\end{array}  \tag{3}\]  
In Equation (3), \((x_{0}^{center},y_{0}^{center})\) represents the center coordinates of the unanalyzed area on the object surface, \((x^{m},y^{m})\) represents the coordinates of the \(m\)th LED, and \(L\) represents the distance from the LED array to the sample. By combining with Formula (2), the spectral information transmitted to the focal plane behind the objective lens can be obtained as follows:  
\[ \begin{array}{c}U_{0}^{m}\big{(}f_{x},f_{y},0\big{)}P\big{(}f_{x},f_{y} \big{)}=\mathcal{F}\{O_{0}(x_{0},y_{0},0)E^{m}(x_{0},y_{0})\}P\big{(}f_{x},f_{ y}\big{)}\\ =U_{0}\big{(}f_{x}-\mu^{m},f_{y}-v^{m},0\big{)}P\big{(}f_{x},f_{y}\big{)}\\ =U_{0}\big{(}f_{x},f_{y},0\big{)}P\big{(}f_{x}+\mu^{m},f_{y}+v^{m}\big{)}\end{array}  \tag{4}\]  
where \(\mathcal{F}\) represents Fourier transform. Finally, the expression of the low-resolution image acquired by the FPM system can be obtained:  
\[I^{m}(x_{i},y_{i})=||\mathcal{F}^{-1}\{U_{0}\big{(}f_{x},f_{y},0\big{)}P\big{(} f_{x}+\mu^{m},f_{y}+v^{m}\big{)}\}||^{2} \tag{5}\]  
It can be seen that in the case of oblique incident illumination, the spectrum of the sample has shifted as a whole, which can be equivalent to the interception of the pupil function of the system at various locations of the sample spectrum, and the acquisition of information exceeding the original passband is realized. At another angle, the illumination of different LEDs in the FPM system can be equivalent to the pupil function (sub-aperture) at different positions on the sample spectrum. A series of low-resolution images with different information are taken by the camera in turn and iterated in the frequency domain to update the corresponding spectrum information in different sub-apertures, expand the sample spectrum range, and recover the high-frequency complex amplitude information limited by the original objective resolution, which is also the basis for FPM technology to realize the images of super-resolution reconstruction. Since the zero-frequency information related to the background light is located in the center of the spectrum, we divide the captured image into a bright field image and a dark field image. The way to distinguish these is to detect whether the spectrum passing through the system contains zero-frequency information. It is expressed by the formula:  
\[NA_{ill}=n\sqrt{\sin^{2}\varphi_{x}+\sin^{2}\varphi_{y}} \tag{6}\]  
Figure 2: Imaging schematic of Fourier ptychographic microscopy system. Among them, \(NA_{ill}\) is the lighting \(NA\). If \(NA_{ill}\leq NA_{obj}\), the image is a bright field image; if \(NA_{ill}>NA_{obj}\), the image is a dark field image. Figure 3 shows the imaging diagram of bright field and dark field images.