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 backaction 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 Rb-87 Bose-Einstein condensates. Here, we present a 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 backaction 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.