Curvilinear distortion for mapping texture on quad sphere

Question

I'm using the formula I've found here to map procedural texture on to a quad sphere.

For example I'm working with a 3d panoramic cube map that I've generated with a simplex noise.

Each face of the cube will be seamless. That's not the point.

My problem is that the corners of each face will have the typical distortion after mapping it onto the quad sphere.

Therefore I want to "manipulate" or prepare the texture ( in this case, the heightmap ) of each face individually with a curvilinear pincushion distortion.

1. Am I assuming right that the curvilinear pincushion distortion will compensate the distortion effect on the quad sphere ?
2. Can someone point me to a formula of the curvilinear distortion or the corresponding calculation of the formula I've linked ?

A: original texture. B: what my texture should look like after preparing.

note that B is only a sketch !

Answer 1

1. No, Pincushion distortion is not actually the inverse of Barrel distortion. (Proof below)

2. This paper seems to be shooting for exactly what you want: A Real-time FPGA Implementation of a Barrel Distortion Correction Algorithm with Bilinear Interpolation (by Gribbon, Johnston, & Bailey) (Formulas inside)

2 Algorithms

Barrel distortion occurs when the magnification of the lens decreases with axial distance causing each image point to move radially towards the centre of the image. This results in the characteristic “barrel” shape. The barrel distortion model [2] is:

\$r_u=r_d(1+kr^2_d)\$        (1)

where \$r_u\$ and \$r_d\$ are the distance from the centre of distortion in the undistorted and distorted images respectively, as shown in Figure 1, and \$k\$ is the distortion parameter, which is specific to the lens.

Figure 1: Illustration of barrel distortion model

Proof as promised: By contradiction for simplicity.

Extracting the relevant fact about Barrel (and/or Pincushion) distortion:

(2): \$R_{skew}=f_k(R_{flat})\$

We construct an equation we can solve for our correction constant "j" in terms of the distortion constant "k":

(3): \$R_{flat} = f_j(f_k(R_{flat}))\$

(4): \$R_{{flat}}=R_{{flat}}(R_{{flat}}^2k+1) R_{{flat}}^2j(R_{{flat}}^2k+1)^2+1)\$

Which we can then re-arrange to isolate "j":

(5): \$j=\frac{-k}{R_{{flat}}^6 k^3+3 R_{{flat}}^4 k^2+3 R_{{flat}}^2 k+1}\$

Which depends on the independent variable, and thus is not a constant. Therefore a Pincushion or Barrel distortion does not compensate its complement distortion.