Is it possible to restore scrambled/damaged text? Given an ease with which a human can read a text scrambled in a special way, like the following passage popular a while ago


  Aoccdrnig to a rscheearch at Cmabrigde Uinervtisy, it deosn't mttaer
  in waht oredr the ltteers in a wrod are, the olny iprmoatnt tihng is
  taht the frist and lsat ltteers be at the rghit pclae. The rset can be
  a toatl mses and you can sitll raed it wouthit porbelm. Tihs is
  bcuseae the huamn mnid deos not raed ervey lteter by istlef, but the
  wrod as a wlohe.


could you train a computer to restore the original? Are there any studies/known instruments that can achieve this?

I am talking about mildly scrambled/damaged text, like the above, not a totally random permutations or random omission of letters.
You can rephrase your question as a statistical problem, where you for each word or n-gram try to predict the most likely correctly spelled version of each word or n-gram based on your knowledge of the distributions of these across a corpus of text in the target language.

This is how spell-checkers work.

Peter Norvig (who wrote Artificial Intelligence: A Moderne Approach with Stuart Russell) has an article on the this with a short, easy-to-read code example and a description of the statistics behind it on his web page: How to write a Spelling Corrector.