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Computational imaging methods empower modern microscopy with the ability ofproducing high-resolution, large field-of-view, aberration-free images. One ofthe dominant computational label-free imaging methods, Fourier ptychographicmicroscopy (FPM), effectively increases the spatial-bandwidth product ofconventional microscopy by using multiple tilted illuminations to achievehigh-throughput imaging. However, its iterative reconstruction method is proneto parameter selection, can be computationally expensive and tends to failunder excessive aberrations. Recently, spatial Kramers-Kronig methods show itis possible to analytically reconstruct complex field but lacks the ability ofcorrecting aberrations or providing extended resolution enhancement. Here, wepresent a closed-form method, termed APIC, which weds the strengths of bothmethods. A new analytical phase retrieval framework is established in APIC,which demonstrates, for the first time, the feasibility of analyticallyreconstructing the complex field associated with darkfield measurements. Inaddition, APIC can analytically retrieve complex aberrations of an imagingsystem with no additional hardware. By avoiding iterative algorithms, APICrequires no human designed convergence metric and always obtains a closed-formcomplex field solution. The faithfulness and correctness of APIC'sreconstruction are guaranteed due to its analytical nature. We experimentallydemonstrate that APIC gives correct reconstruction result while FPM fails to doso when constrained to the same number of measurements. Meanwhile, APICachieves 2.8 times faster computation using image tile size of 256(length-wise). We also demonstrate APIC is unprecedentedly robust againstaberrations compared to FPM - APIC is capable of addressing aberration whosemaximal phase difference exceeds 3.8${\pi}$ when using a NA 0.25 objective inexperiment.

High-resolution, large field-of-view label-free imaging via aberration-corrected, closed-form complex field reconstruction