Signal-to-noise ratio for astronomical imaging by deconvolution from wave-front sensing

One method for improving the quality of astronomical images measured through a atmospheric turbulence uses simultaneous short-exposure measurements of both an image and the output of a wave-front sensor exposed to an image of the telescope pupil. The wave-front sensor measurements are used to reconstruct an estimate of the instantaneous generalized pupil function of the telescope, which is used to compute an estimate of the instantaneous optical transfer function, which is then used in a deconvolution procedure. This imaging method has been called both deconvolution from wave-front sensor (DWFS) measurements and self-referenced speckle holography. We analyze the signal-to-noise ratio (SNR) behavior of this imaging method in the spatial frequency domain. The analysis includes effects arising from differences in the correlation properties of the incident and the estimated pupil phases and the fact that the object-spectrum estimator is a randomly filtered doubly stochastic Poisson random process. SNR resultsobtained for the DWFS method are compared with the speckle-imaging powerspectrum SNR for equivalent seeing conditions and light levels. It is shown that for unresolved stars the power-spectrum SNR is superior to the DWFS SNR. However, for extended objects the power-spectrum SNR and the DWFS SNR are similar. Since speckle imaging uses a separate Fourier phasereconstruction process not required by the DWFS method, the DWFS method provides an alternative to speckle imaging that uses simple postprocessing at the cost of a wave-front sensor measurement but with no loss of SNR performance for extended objects.