  Structured illumination microscopy (SIM) improves resolution by
down-modulating high-frequency information of an object to fit within the
passband of the optical system. Generally, the reconstruction process requires
prior knowledge of the illumination patterns, which implies a well-calibrated
and aberration-free system. Here, we propose a new \textit{algorithmic
self-calibration} strategy for SIM that does not need to know the exact
patterns {\it a priori}, but only their covariance. The algorithm, termed
PE-SIMS, includes a Pattern-Estimation (PE) step requiring the uniformity of
the sum of the illumination patterns and a SIM reconstruction procedure using a
Statistical prior (SIMS). Additionally, we perform a pixel reassignment process
(SIMS-PR) to enhance the reconstruction quality. We achieve 2$\times$ better
resolution than a conventional widefield microscope, while remaining
insensitive to aberration-induced pattern distortion and robust against
parameter tuning.
