Follow the instructions in the mounting a filesystem post, then:
mkdir -p $HOME/raspbian/2012-10-28-wheezy-raspbian && cd $HOME/raspbian/2012-10-28-wheezy-raspbian
Copy files as root (the root image contains device special files that will cause problems for a regular user):
sudo cp -r /mnt/* .
Own the file hierarchy (my username is ‘user’ and I’m in the ‘user’ group):
sudo chown -R user:user *