Fig. 2: Reconstruction pipeline for APIC.

By changing the illuminating angle, we effectively shift the CTF to different positions in the spatial frequency domain, and samples different regions of sample’s spectrum. For measurements under NA-matching angle illumination, we first use Kramers–Kronig relation to recover the corresponding spectrums. The phase differences of two spectrums with overlaps in their sampled spectrum are used to extract the imaging system’s aberration. Then, the image-wise reconstructed spectrums are corrected for aberration and get stitched, which forms our prior knowledge in the reconstruction process involving darkfield measurements. To extend the spectrum using darkfield measurement, the known spectrum in the ith measurement is used to isolate cross-correlation from other autocorrelation terms. By solving a linear equation involving the isolated cross-correlation, the unknown spectrum can be analytically obtained. We then use the newly reconstructed spectrum to extend the recovered spectrum. The extended spectrum then serves as the prior for the (i+1)th measurement.