My friend unfortunately I have bad news for you, while your display still works it is permanently damaged. There is no fixing this issue, there is only replacing the screen with a new one.
What has happened is the display has been damaged below the first layer of glass, you cannot see the damage because the front glass is covering it, by removing the front glass you will be able to see the damage.
It must be a minor crack in your LED, small enough to only distort the display and not make it go out completely.
But its only a mater of time before the device gets bumped to hard or dropped and it will finish the small crack and make the display stop working completely.
Its time for a new screen.
Hope this helps get you going in the right direction.
Though the LCD isn't broken on your device, it looks like it is defective, if you have experience taking the S3 apart, and access to an additional S3, you could simply switch motherboards to test it. Honestly, it would be best for you if it is the screen, because the only other possibility is that your main logic board is having issues, which isn't good at all, in that case you would have to get a new phone all together.
