I just need to know if you think Nolan's batman is Frank Miller's vision on film?

I do, both Nolans are huge fans of Nolan's reinvention and reboot of Batman (the best version in the comics back in the 80s) and Alan Moore's take when Nolan rebooted the series as much as Miller did with the comics and you can tell how much the Nolan brothers borrowed from these sources:

Batman Begins inspired by:
-Batman: Year One
-The Man Who Falls
-Batman: The Long Halloween
-Batman: Haunted Knight (the line "Dr. Crane isn't here right now, but if you'd like to make an appointment..." came from this comic)

The Dark Knight was influenced by:
-The Killing Joke
-Batman: The Long Halloween
-the very first two Joker stories printed in Batman #1 and published in 1940
-Arkham Asylum: A Serious House on Serious Earth

The Dark Knight Rises was influenced by:
-The Dark Knight Returns
-No Man's Land
-Batman: The Cult (probably)
-Kingdom Come (the line "So that's what that feels like..." came from this comic)
-Selina is referred to as "The Cat" (as in her earliest comic appearances)

Nolan's trilogy has a lot of influences from various comics.