I'm not familiar with the machine you are picturing but a filter is not a problem so long as it's not (step 1) creating a mixture and then (2) separating it. So the coffee filters that are placed on top of the cup and water is poured through them should be fine.
As Rabbi David Sperling writes here
This is based on the ruling of the Shulchan Aruch (Orach Haim, 319,9
and the Mishna Brurah 33 ibid) which allows pouring water into a sieve
that has grape dregs in it, in order that the water flow though the
dregs (absorbing their taste) and out through the sieve. The logic
here is that the water is poured into the sieve was not mixed, and it
comes out not mixed. Based on this it is permitted to pour water over
tea leaves (making sure not to break the laws of cooking on Shabbat -
aiyn bishul achar afiya and kil shlishi) that are in a strainer, as
the tea just flows through the strainer, absorbing the tea taste, and
out the other side (see Shmirat Shabbat 3, 64 [Hebrew, new edition],
and the Chazon Ish, Orach Haim, 53).
This is exactly the use of the
coffee filter that sits on the top of a cup. The filter cup has coffee
in it, and a paper filter on its base. The water is poured into this
filter cup and then drips through into the cup below after having
absorbed the taste of the coffee beans.