When it involves the other senses.
Do you hear the bandit step on a branch, or smell them because they haven't bathed? You're still rolling perception for initiative (unless the circumstance allows you to roll something else, as normal).
There are also Special Senses such as Tremorsense.
When one creature might detect another, the GM almost always uses the most precise sense available. (Vague Senses)
If the opponent cannot be seen, then it falls back onto these other senses. These rolls would not gain the benefit of the Eyes of the Eagle.
(All links Archives of Nethys)