Fig. 1

Flow chart of structured illumination microscopy based on principal component analysis. a The raw 3-step phase-shifting SIM images and the separated spectral components. b The 1-order spectrum after being shifted \(k_{int}\). c Phases of the inverse Fourier transform of b (c1), magnified phase map from the boxed regions in c1 (c2), the ideal phases of c2 (c3) and the unwrapped version of c3 (normalized to \(-\pi\) to \(\pi\)) (c4). The sub-pixel wave vector \({{{\textbf {k}}}_{sub}}\) is reflected in the 2D phase slope of the ideal phases, but in practice, various disturbances produce serious noises. d The phases of the phasor matrix after different operations: the original phases (top first), phases after applying the masking operator (top second), phases after applying PCA (top third), and phases after least-squares fitting (top forth); and the phase distributions along the light blue line (bottom). After using the masking operator and PCA, the irrelevant components in the original phasor matrix are effectively “cleared up”. The cleaned version is close to its ideal form. e Obtained wave vector with sub-pixel accuracy (bottom), merged spectrums in one direction (middle), and merged spectrums of three directions rotating 120 degrees from each other (top). The whole process of PCA-SIM is summarized as: Step 1: obtain the Fourier spectrums of the raw SIM images; Step 2: separate the 0- and \({{\pm }}\)1-order spectrums and shift the \({{\pm }}\)1-order spectrums with integer-pixel displacement; Step 3: use a masking operator to extract the center signals for inverse Fourier transform and obtain the exponential term; Step 4: SVD and extract the principal component; Step 5: fit two principal vectors with the least square method after removing starting error points; Step 6: obtain accurate sub-pixel wave vector; Step 7: obtain the initial phase and modulation depth; Step 8: merge separated spectrums and perform super-resolution reconstruction