Water filter not filtering ice
The white stuff is probably calcium (99% of the time)... so your water is "hard". When the water is cooled, the calcium aggregates and precipitates (white stuff). A filter will not remove dissolved ions (like calcium). 

A water softener will replace calcium with sodium. That should solve the the white precipitate issue. However, before spending money, have the water tested. 

You can (often) get your water tested for free from a kinetico dealer. However you should know what they say is a biased toward selling a water treatment system, so don't offer them any details of your problem; just ask them to check it. But if they say that you have hard water, then it's probably true.

The other possibility (1% of the time) is dissolved silicates. Again, the kinetico dealer would let you know for free. They would say you need a Reverse Osmosis (RO) system, which is expensive.

Edit Here's a simple home-test that you could do: Add a tablespoon of (regular, distilled) vinegar to a cup of water. Freeze one ice cube tray with regular water (for comparison) and another tray with the water containing a water + vinegar. 

If precipitate does not form in the water+vinegar (or if the precipitate is comparably reduced) then calcium/hardness is very likely of the issue. If the amount of precipitate is about the same, then the issue is more likely silica.