  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.
