Solution

When is known, the PSF is known and the numerical procedure is the modification of the Richardson-Lucy iterations (Richardson 1972, Lucy 1974) described by Snyder (1990) and Snyder, Hammoud, and White (1993):

where and

When is unknown, the problem is more difficult. For our initial investigation, we consider the simplified situation for which the flat-field response function is a constant , the PSF is spatially invariant, and the light is monochromatic. For this case, the EM algorithm for updating both the object and the aberration parameters becomes,

and

where

and where

Notice that is independent of the aberration parameters . The optimization problem defined in Eq. 12 can be performed by using standard optimization techniques; however, the maximum need not be found. Instead, if is chosen such that

then the iterations will be a generalized EM algorithm (Dempster et al. 1977) and still have the desirable property

Eq. 15 can be satisfied, for instance, by performing a few iterations of any numerical optimization procedure.