I've always wanted a set up like this, but im not sure how the filtration works??
The air bubbling up the tube creates an uplift force and causes the water to also flow up and out. So water moves through the tube from one side of the filter foam to the other. Gravity forces the water to then flow back through the foam. The surface area of the foam provides tons of surface area and results in a very slow flow rate through the foam. Both of those aspects are great for biological filtration - lots of surface area for bacteria to colonize and slow filtration rate to allow effective reduction of ammonia and nitrite.
Because of the slow flow rate it's not good for mechanical filtration. I keep these tanks bare bottom and only have small fish - easy to siphon out the waste that builds up on the bottom and the poops aren't large enough to turn the filter into a mess. Since minimal waste builds up, not to mention the large surface area, the filter material doesn't clog or require frequent cleaning - I gently rinse them out once every 9-12 months.
I was in the local JoAnn Fabric's store here in town a while back, and they carry very large pieces of foam there, they have white and green. Anything from 1in to I believe 5in thick. Im wondering if this foam would basically work the same for filtration. I sure looks the same as the blue foam pads here? 
The material I used is very stiff - cutting it slightly larger than the aquarium dimensions allows it to hold itself in place, compressed against the sides of the tank. Anything you use should be open celled - open cells allow water to flow through it. Lots of foams are either closed cell (should be obvious) or a sloppy mixture (harder to tell - try blowing through it to tell how open it is). A rigid material also maintains the cell structure better without collapsing. Lastly you'll want something that hasn't been treated in any manner - anti-fungal, anti-bacterial, etc. I buy my filter floss at JoAnns and am always careful to get the 100% pure, non-allergenic, non-treated batting.
So it might work - you'll probably have to take a closer look and I'd suggest trying it with some fish that won't leave you heartbroken should the experiment not work.