## 2 Principle of FPM  
The FPM algorithm can be divided into a forward downsampling algorithm and an inverse reconstruction algorithm. The forward downsampling algorithm downscales high-resolution amplitude and phase maps into low-resolution amplitude images corresponding to specific illumination conditions, typically used to simulate the low-resolution intensity images captured by a CCD camera in an FPM system, thus generating simulation datasets. This algorithm employs two high-resolution images to simulate the amplitude \(A(x,y)\) and phase \(P(x,y)\), thereby modeling the optical response \(O(x,y)\) of the sample under FPM system illumination:  
\[O(x,y)=A(x,y)e^{i\cdot P(x,y)}=o(x,y)e^{i\left(k_{x}\cdot x+k_{y}\cdot y\right)}, \tag{1}\]  
where \(o(x,y)\) represents the sample transmission function, and \(k_{x},k_{y}\) denotes the plane wave vectors along the x-axis and y-axis. The wave vector \(k_{i}\) corresponding to the \(i\)-th LED can be expressed as follows:  
\[k_{i}=\left(k_{xi},k_{yi}\right)=\left(\frac{\sin\theta_{xi}}{\lambda},\frac{ \sin\theta_{yi}}{\lambda}\right), \tag{2}\]  
where \(\left(\theta_{xi},\theta_{yi}\right)\) represents the illumination angle of the i-th LED, and \(\lambda\) denotes the wavelength. When the i-th LED illuminates the sample with the wave vector \(k_{i}\), it effectively shifts the sample spectrum \(O_{i}(k)\) to be centered around \(k_{xi},k_{yi}\), expressed as  
\[O_{i}(k_{x}-k_{xi},k_{y}-k_{yi})=F\Big{\{}o(x,y)e^{i\left(k_{xi}\cdot x+k_{yi} y\right)}\Big{\}}, \tag{3}\]  
where \(F\{\cdot\}\) denotes the Fourier transform. As light passes through the objective lens, the field of view undergoes low-pass filtering via the pupil function. Here, we define the initial pupil function \(P_{0}\big{(}k_{x},k_{y}\big{)}\) as the contrast transfer function of an aberration-free system's objective lens:  
\[P_{0}\big{(}k_{x},k_{y}\big{)}=CTF\big{(}k_{x},k_{y}\big{)}=\left\{ \begin{array} []{l}1,{k_{x}}^{2}+{k_{y}}^{2}\leq(NA\bullet\frac{2\pi}{\lambda})^{2}\\ 0,{k_{x}}^{2}+{k_{y}}^{2}>(NA\bullet\frac{2\pi}{\lambda})^{2}\end{array} \right., \tag{4}\]  
where \(NA\) represents the numerical aperture of the objective lens. Thus, the intensity image \(I_{i}(x,y)\) recorded by the CCD corresponding to the i-th LED is expressed as  
\[I_{i}(x,y)=|g_{i}(x,y)|^{2}=\Big{|}F^{-1}\big{\{}\Phi_{i}\big{(}k_{x},k_{y} \big{)}\,\big{\}}\Big{|}^{2}=\Big{|}F^{-1}\big{\{}P_{i}\big{(}k_{x},k_{y} \big{)}O_{i}\big{(}k_{x}-k_{xi},k_{y}-k_{yi}\big{)}\,\big{\}}\Big{|}^{2}. \tag{5}\]  
The inverse reconstruction algorithm in FPM achieves phase recovery and aperture synthesis super-resolution by iteratively updating the sample transmission function \(O_{i}(x,y)\) between the spatial and frequency domains, simultaneously shifting in the frequency domain and introducing amplitude information in the spatial domain. This process can be expressed as  
\[\Phi_{i}{}^{\prime}\big{(}k_{x},k_{y}\big{)}=F\bigg{\{}\sqrt{I_{ci}(x,y)} \frac{g_{i}(x,y)}{|g_{i}(x,y)|}\bigg{\}}, \tag{6}\]  
\[O_{i+1}\big{(}k_{x}-k_{xi},k_{y}-k_{yi}\big{)}=O_{i}\big{(}k_{x}-k_{xi},k_{y} -k_{yi}\big{)}+\alpha\frac{P_{i}^{*}\big{(}k_{x},k_{y}\big{)}}{\big{|}P_{i} \big{(}k_{x},k_{y}\big{)}\big{|}^{2}_{\max}}\big{[}\[PERSON]_{i}{}^{\prime}\big{(} k_{x},k_{y}\big{)}-\[PERSON]_{i}\big{(}k_{x},k_{y}\big{)}\big{]}, \tag{7}\]\[P_{i+1}\big{(}k_{x},k_{y}\big{)}=P_{i}\big{(}k_{x},k_{y}\big{)}+\beta\frac{O_{i}^{* }\big{(}k_{x}-k_{xi},k_{y}-k_{yi}\big{)}}{\big{|}O_{i}\big{(}k_{x}-k_{xi},k_{y}- k_{yi}\big{)}\big{|}_{\max}^{2}}\big{[}\Phi_{i}^{\prime}\big{(}k_{x},k_{y}\big{)}- \Phi_{i}\big{(}k_{x},k_{y}\big{)}\big{]}, \tag{8}\]  
where \(O_{i+1}\big{(}k_{x}-k_{xi},k_{y}-k_{yi}\big{)}\) and \(P_{i+1}\big{(}k_{x},k_{y}\big{)}\) are optimized based on the gradient descent method [34], and \(\alpha\) and \(\beta\) are update step sizes that need to be predefined. Repeating Equations (5)-(8), the complex field corresponding to the spectra of all \(i\) illuminations is updated sequentially, completing one iteration cycle.