I don't have a definitive answer for you, but I do have some food for thought. Seeing your picture reminds me of times I've been working on computer video and there's a part of the video memory that got corrupted.
Everything you see on the screen is represented in a big chunk of memory where memory locations hold information on pixel values such as color and brightness. If you should accidentally try to use a bit of that memory for computer operations, you end up with screen data that looks quite similar to what you're seeing on your screen.
Now that same condition can also come about when you have a bit of bad video memory; a part of memory that doesn't correctly hold the values that are programmed into it; you'll see a similar sort of corruption on the screen.
So if it's a software issue your best bet is to wipe the whole phone and restore it back to its original state. That will eliminate any corrupted storage or a bad app that could be causing the problem.
If that doesn't help then you've got two possibilities remaining; although it doesn't look like a high probability, it is still possible it's a screen issue. So if it was my phone, my next step would be to replace the display. Because the last possibility is that it's a memory problem which requires microsoldering expertise and board-level debugging that none of us home DIY-ers are equipped to tackle.
If you get to that point, your best bet is to seek out a repair shop that is experienced in board repairs and see if they can give you a diagnosis and possibly an estimate of cost to repair.
Good luck; let us know what you find and how it turns out!