Marginal estimation of aberrations and image restoration by use of phase diversity

We propose a novel method called marginal estimator for estimating the aberrations and the object from phase-diversity data. The conventional estimator found in the literature concerning the technique first proposed by Gonsalves has its basis in a joint estimation of the aberrated phase and the observed object. By means of simulations, we study the behavior of the conventional estimator, which is interpretable as a joint maximum a posteriori approach, and we show in particular that it has undesirable asymptotic properties and does not permit an optimal joint estimation of the object and the aberrated phase. We propose a novel marginal estimator of the sole phase by maximum a posteriori. It is obtained by integrating the observed object out of the problem. This reduces drastically the number of unknowns, allows the unsupervised estimation of the regularization parameters, and provides better asymptotic properties. We show that the marginal method is also appropriate for the restoration of the object. This estimator is implemented and its properties are validated by simulations. The performance of the joint method and the marginal one is compared on both simulated and experimental data in the case of Earth observation. For the studied object, the comparison of the quality of the phase restoration shows that the performance of the marginal approach is better under high-noise-level conditions.