This paper describes an approach to the problem of reconstructing a star which has been blurred by atmospheric turbulence and further corrupted by sensor noise. The Knox-Thompson speckle-imaging technique is used to obtain initial estimates of the amplitude and phase spectra of the object. When the noise level is high, these estimates are quite inaccurate at the higher spatial frequencies, and the associated reconstruction is very poor. We attempted to alleviate the problem by throwing away the noisy high-frequency terms and then using a spectral extrapolation algorithm to reconstruct them from the more accurate low frequencies. Spectral extrapolation was done using both the Gerchberg algorithm and a new algorithm described in the paper. These two algorithms and others are discussed in terms of a convenient eigenfunction expansion, and some experimental results are presented. Although these particular results favor the new algorithm, general conclusions must be drawn with caution.