How do I create a hidden image only visible when shaking your head?

Here’s a way (with no blurring really, but you can add some if you want) I tested and result is not too far from what you want.
Change your guy’s face (special credit to Shepard Fairey ;-) to black and white with high contrast (should basically be a “bitmap” image, meaning either black or white, no grey shades).
Apply transparency (10% in my example for a quite obvious result) then place the striped pattern over it. Note I created it in Illustrator using Blend object feature.

That’s basically it… Enjoy!

Edit: Huh I didn’t read carefully the other answers before posting. This is exactly what Emilie suggested. :-/ Credit goes to her! This is the illustrated version of her answer.

Result: