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A correction algorithm will probably rely more on the shape of the lens than the shape of an 'ideal' lens.

A single image also won't technically contain enough information to even allow you to reconstruct the image, but with sparse reconstruction you might be able to get it done.

In fact I'm fairly sure people have tried to do exactly that, and phone manufacturers seem pretty willing to throw software at the problem as well, although the latter don't really care much for the method as long as it looks nice.

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