The DNG specification (page 87) describes a simple algorithm for converting a "fish-eye" photo into a regular photo (just several lines of basic geometry).
The transformation is configured using six numbers: $k_0, k_1, k_2, k_3, c_x,c_y$. Does it allow to represent identity (i.e. not changing the photo at all)? In order to get identity, the $f$ should be always one, so the bracket $(k_0\theta + k_1\theta^3+ k_2\theta^5+ k_3\theta^7)$ should be alwqys $r$.
In other words:
$w(r) = k_0\theta + k_1\theta^3+ k_2\theta^5+ k_3\theta^7$, where $\theta=arctan(r)$
What should be $k$s, so that $w$ is the identity for $r\in[0,1]$?