How do I reconstruct a covered-over part of a regular pattern
You can try using the Hough transform to detect the disk and the visible ellipses.  

Also, you may want to do a search for "texture generation".

Most importantly, keep in mind that just because a person can do something effortlessly, it doesn't mean that it is easy to program a computer to do it. Virtually any vision task is very easy for people, but are extremely hard for computers.