There's been another report of gravitational lensing in the news. Wikipedia has a good image of an example of gravitational lensing taken by the Hubble Space Telescope. This brings me to wonder that, since they understand the math behind the lensing, shouldn't they be able to reconstitute the original image of the galaxy behind the galaxy that is acting as a lens? That is, take the image and run it through a computer program that reconstructs the image using known parameters of the mass and the position of the lensing galaxy?
That would be possible theoretically. However, in practice, the lensed images are highly non-linearly distorted and due to the limits in resolution, they are often limited to a thickness of a few pixels at best. Due to the non-linear transformations, this would cause the reconstructed original image to be even blurrier.
Having said that, I don't see any real need of reconstructing the original image. You can get most parameters (like a rough estimate of the size and type of the background galaxy) from the distorted image anyway.
The method of doing this is called Deconvolution. In theory, yes, it's possible if you know the exact distortion function, and an exact result image - but, in reality, the image you have is not exact, so the result of deconvolution may not be very useful.