Speckle imaging signal-to-noise performance as a function of frame integration time

In the past 10 years astronomical image reconstruction based on a variety of speckle techniques have become popular means of enhancing and improving the resolution of turbulence degraded images. These techniques are based on the Fourier processing of a large number of turbulence degraded \snapshots" or frames of the irradiance in the system's image plane. The number of snapshots needed is largely a function of the signal-to-noise ratio (SNR) of the Fourier components of a single snapshot. Estimating these Fourier components is hampered by the inherent noise induced by the photon detection process and the random eeect of the turbulent atmosphere. In most cases the SNR is much less than unity and many frames are averaged to improve the overall SNR. It is well established that if the frames are uncorrelated then the SNR improves over the single frame SNR by a factor of p m where m is the number of frames averaged. This fact implies that the smallest exposure time possible is desirable in order to maximize m for a given observation time. On the other hand, due to nite photon ux levels and read noise eeects it can be shown that the exposure time should be increased at the cost of reducing m. This paper describes and presents results from a detailed SNR analysis of estimating the modulus of an object's Fourier spectrum from turbulence distorted images. Unlike previous analyses, the work presented in this paper takes into account the proper temporal correlation properties of the atmosphere, the spatial frequency being estimated, as well as the inter-frame correlations that degrade the SNR improvement factor from the ideal case of p m.