Maximum a posteriori estimation of wave-front slopes using a Shack–Hartmann wave-front sensor

Current methods for estimating the wave-front slope at the pupil of a telescope by using a Shack–Hartmann wave-front sensor (SH–WFS) are based on a simple centroid calculation of the irradiance distributions (spots) recorded in each subaperture. The centroid calculation does not utilize knowledge concerning the correlation properties of the slopes over the subapertures or the amount of light collected by the SH–WFS. We present the derivation of a maximum a posteriori (MAP) estimation of the irradiance centroids by incorporating statistical knowledge of the wave-front tilts. Information concerning the light level in each subaperture and the relative spot size is also employed by the estimator. The MAP centroid estimator is found to be unbiased, and the mean square error performance is upper bounded by that exhibited by the classical centroid technique. This error performance is demonstrated by using Kolmogorov wave-front slope statistics for various light levels.