Are digital adaptive optics possible? Adaptive optics generally apply slight changes to mirrors to account for atmospheric turbulence. These generally require either a mirror to be broken up into smaller mirrors, each with their own actuator, or for a single large mirror to be slightly deformed by many different actuators.

Is it possible to do much of this digitally? 

Couldn't software slightly attenuate or amplify each pixel at each time-slice collected from a passive mirror? These changes would depend on the distortions measured by a guide star or laser (this digital system would still require a guiding light)? I suspect this would probably be less accurate than true adaptive optics, but it would also be far less expensive (as you wouldn't need so many actuators).

Has this approach been studied and dropped? Is it ridiculous and not even theoretically possible? I'm curious if anything has researched this idea.
The problem with your approach is that the deformable mirror changes the phase of the light across the mirror, where the light is not focussed.  The light at the sensor array is focussed, and what you get is the intensity which is, roughly, the Fourier Transform of the phase front at the mirror.   By the time you have intensity, the phase information has been lost.  

edit for clarity:

The sensor array measures the intensity, at which point the phase info is irretrievable.  If you remove the sensor array and re-measure past the focal plane, yes, you can get phase info -- see "Plenoptic cameras."  

Now, there are other techniques -- you might be interested in searching for papers on "Lucky Imaging" , which basically takes as many images as possible and throws away the distorted ones.