regardless of the focal plane mask size adopted.

Designing a PIAACMC for a non circular aperture
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We consider in this work PIAACMC designs that perform a lossless PIAA apodization of the pupil to produce a generalized prolate function for the aperture. We note that other apodization functions could be adopted, and could potentially lead to superior performance, but this is not explored in this paper. In the unobstructed circular pupil case [@2010ApJS..190..220G], designing the PIAACMC is relatively simple, as PIAA apodization using a radial remapping function preserves the circular aperture shape. The prolate function can thus be first computed, and then realized with a radial PIAA apodization.

Designing a PIAACMC for complex shaped apertures is considerably more challenging because the PIAA apodization modifies the aperture shape, which itself changes the generalized prolate function. In addition to this circular problem, if the aperture is not circularly symetric, the generalized prolate is also not symmetric, and the required remapping function therefore cannot be written as a radial function. While PIAA optics can be designed for any radial remapping [@2003AA...404..379G], an arbitrarily chosen 2D remapping function can almost never be realized with a set of two PIAA optics.

To overcome the two 