  High-resolution imaging of ultracold atoms typically requires custom high
numerical aperture (NA) optics, as is the case for quantum gas microscopy.
These high NA objectives involve many optical elements each of which
contributes to loss and light scattering, making them unsuitable for quantum
back-action limited "weak" measurements. We employ a low cost high NA aspheric
lens as an objective for a practical and economical-although aberrated-high
resolution microscope to image ${^{87}\mathrm{Rb}}$ Bose-Einstein condensates.
Here, we present a novel methodology for digitally eliminating the resulting
aberrations that is applicable to a wide range of imaging strategies and
requires no additional hardware. We recover nearly the full NA of our
objective, thereby demonstrating a simple and powerful digital aberration
correction method for achieving optimal microscopy of quantum objects. This
reconstruction relies on a high quality measure of our imaging system's
even-order aberrations from density-density correlations measured with
differing degrees of defocus. We demonstrate our aberration compensation
technique using phase contrast imaging, a dispersive imaging technique directly
applicable to quantum back-action limited measurements. Furthermore, we show
that our digital correction technique reduces the contribution of photon shot
noise to density-density correlation measurements which would otherwise
contaminate the desired quantum projection noise signal in weak measurements.
