We introduce an analytical model for Fourier ptychography (FP). The model represents recorded intensity images as linear equations. Here, the sample’s self-correlating components are the unknown variables, while the optical system’s point-spread function’s corresponding components serve as coefficients. This framework facilitates the direct computation of high-resolution complex images from the sample. A standout feature of our model is its ability to verify the uniqueness of the reconstructed image, a critical attribute for numerous quantitative applications. Our linear model offers clear insights into the impact of various experimental factors on accurate reconstruction, such as scanning step, random scanning mode, spatial resolution, and noise. While these factors have been previously acknowledged, their precise roles have remained nebulous. We have also developed an efficient computational method tailored for our model, adept at managing large matrices, thus enhancing the translation of low to high-resolution images. This research elucidates the foundational mathematics behind FP’s efficacy, underscoring its potential for optical measurements and metrology where mathematical uniqueness is paramount.