Pupil functions are compact and modifiable descriptions of the three-dimensional (3D) imaging properties of wide-field optical systems. The pupil function of a microscope can be computationally estimated from the measured point spread function (PSF) using phase retrieval algorithms. The compaction of a 3D PSF into a 2D pupil function suppresses artefacts and measurement noise without resorting to rotational averaging. We show here that such 'phase-retrieved' pupil functions can reproduce features in the optical path, both near the sample and in the microscope. Unlike the PSF, the pupil function can be easily modified to include known aberrations, such as those induced by index-mismatched mounting media, simply by multiplying the pupil function by a calculated aberration function. PSFs calculated from such a modified pupil function closely match the corresponding measured PSFs collected under the aberrated imaging conditions. When used for image deconvolution of simulated objects, these phase-retrieved, calculated PSFs perform similarly to directly measured PSFs.