a set of four radial spiders at 90 deg), and the APLC, which could be designed to take into account central obstruction and was found to be quite robust to spiders at the $10^{-5}$ contrast level. The coronagraphs concepts for which ground-based designs compatible with central obscuration have been proposed (shaped aperture, APLC, band-limited Lyot coronagraph) are unfortunately not able to offer IWA less than $\approx 2 \lambda/D$, and also do not enable high contrast (approximately $10^{9}$) coronagraphy on centrally obscured or segmented apertures.

The work presented in this paper is aimed at demonstrating that high performance coronagraphy is possible in monochromatic light on centrally obscured and/or segmented pupils for both ground-based and space-based telescopes. The Apodized Pupil Complex Mask Lyot Coronagraph (APCMLC) and Phase-Induced Amplitude Apodization complex mask coronagraph (PIAACMC) concepts, previously described for circular unobstructed apertures in [@2010ApJS..190..220G], are here adapted to arbitrarily shaped apertures. Section \[sec:APCMLC\] describes how the APCMLC can be adapted to non-circular apertures, and a step by step process to design a APCMLC for any aperture shape is proposed and examples are shown. In Section \[sec:PIAACMC\], the PIAACMC is shown to offer performance superior to the APCMLC, and its design for centrally obscured and 