### Speckle Holography

Speckle holography allows one to obtain the best estimate of an astronomical object from long image time series, with exposure times shorter than the coherence time of atmospheric turbulence. The latter is on the order of a few 0.01s at wavelengths around 2 µm, but our experience shows that this requirement can be relaxed substantially, in particular if we do not need to reach the full diffraction limit of a given telescope (or cannot reach it because a camera’s pixel scale is too large). In practice, we use exposure times as long as 1 s.

Each short exposure is the convolution of the astronomical object with the instantaneous point spread function (PSF), which is dominated by the atmospheric turbulence and is continuously changing. In Fourier space this can be expressed as I_{j }= Pj * O, where I_{j} is the Fourier transform of the image, P_{j} the one of the instantaneous PSF and O the one of the astronomical object. The speckle holographic reconstruction is obtained from the series of short exposures by a combination of averaging (to suppress the noise) and division in Fourier space: O = <I_{j }I_{j}*>/<P_{j} P_{j}*>, where <> denotes the average over all the exposures and the asterisk denotes the complex conjugate. After reverse Fourier transform into image space, the image is convolved with a reconstruction PSF (we typically use a Gaussian) that suppresses noise at high frequencies that cannot be passed by the telescope. Key to high-fidelity image reconstruction is a correct and high signal-to-noise extraction of the instantaneous PSF from each short exposure. We use an improved version of the algorithm suggested by Schödel et al. (2013), which is based on the PSF extraction algorithm used by the software StarFinder (Diolaiti et al. 2000).