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The technique of maximizing sharpness metrics has been used to estimate and compensate for aberrations with adaptive optics, to correct phase errors in synthetic-aperture radar, and to restore images. The largest class of sharpness metrics is the sum over a nonlinear point transformation of the image intensity. How the second derivative of the point nonlinearity varies with image intensity determines the effects of various metrics on the imagery. Some metrics emphasize making shadows darker, and other emphasize making bright points brighter. One can determine the image content needed to pick the best metric by computing the statistics of the image autocorrelation or of the Fourier magnitude, either of which is independent of the phase error. Computationally efficient, closed-form expressions for the gradient make possible efficient search algorithms to maximize sharpness.
Fienup et al. (2003) studied this question.