The DNG Specification (page 87) describes a simple algorithm for converting a "fisheye" photo into a regular photo (just a few lines of basic geometry).
The transformation is configured with six numbers: $ k_0, k_1, k_2, k_3, c_x, c_y $, Allows to present identity (ie not to change the photo at all)? To get identity, the must $ f $ should always be one, so the bracket $ (k_0 theta + k_1 theta ^ 3 + k_2 theta ^ 5 + k_3 theta ^ 7) $ should always be $ r $,
In other words:
$ w (r) = k_0 theta + k_1 theta ^ 3 + k_2 theta ^ 5 + k_3 theta ^ 7 $, from where $ theta = arctan (r) $
What should be $ k $s, so that $ w $ is the identity for $ r in[0,1]$?