Calculation geometry – removing the fish eye effect

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 $$r in[0,1]$$?