Not yet reviewed by Pith; the record is open.

Computational imaging methods empower modern microscopy with the ability of producing high-resolution, large field-of-view, aberration-free images. One of the dominant computational label-free imaging methods, Fourier ptychographic microscopy (FPM), effectively increases the spatial-bandwidth product of conventional microscopy by using multiple tilted illuminations to achieve high-throughput imaging. However, its iterative reconstruction method is prone to parameter selection, can be computationally expensive and tends to fail under excessive aberrations. Recently, spatial Kramers-Kronig methods show it is possible to analytically reconstruct complex field but lacks the ability of correcting aberrations or providing extended resolution enhancement. Here, we present a closed-form method, termed APIC, which weds the strengths of both methods. A new analytical phase retrieval framework is established in APIC, which demonstrates, for the first time, the feasibility of analytically reconstructing the complex field associated with darkfield measurements. In addition, APIC can analytically retrieve complex aberrations of an imaging system with no additional hardware. By avoiding iterative algorithms, APIC requires no human designed convergence metric and always obtains a closed-form complex field solution. The faithfulness and correctness of APIC's reconstruction are guaranteed due to its analytical nature. We experimentally demonstrate that APIC gives correct reconstruction result while FPM fails to do so when constrained to the same number of measurements. Meanwhile, APIC achieves 2.8 times faster computation using image tile size of 256 (length-wise). We also demonstrate APIC is unprecedentedly robust against aberrations compared to FPM - APIC is capable of addressing aberration whose maximal phase difference exceeds 3.8${\pi}$ when using a NA 0.25 objective in experiment.