2 Image formation in adaptive optics

Within the isoplanatic patch, the AO corrected image is a convolution of the true object ($o({\bf r})$) with a residual PSF ($h({\bf r})$) characterizing the turbulence, the telescope and the AO bench. The detected intensity distribution $d({\bf r})$ of the corrected image is given by:  

$$d({\bf r}) = o({\bf r})\star h({\bf r}) + n({\bf r})$$ (1)

$n({\bf r})$ is an additive zero mean random process that accounts for the noise affecting the image (photon and detector noise). The AO correction is always partial and does not provide a full compensation for the atmospheric turbulence. The correction performance depends on the observing conditions: turbulence strength characterized by the Fried parameter (r₀) (Fried 1965), magnitude and angular size of the source used for wavefront sensing. It also depends on the AO system itself: number of corrected modes, servo-loop bandwidth etc. (Rousset et al. 1990; Rigaut 1991; Conan 1994). The partially corrected image can be fully characterized by the PSF $h({\bf r})$.

A posteriori deconvolution is therefore required to estimate the object $o({\bf r})$ from the data $d({\bf r})$.