How to remap a fisheye image?

I want to flatten a series of fisheye images by remapping them to a rectinlinear projection.

To achieve this, I need to be able to remap the pixels of the image using fisheye correction formulas for the x-, and y-coordinates. How can I achieve this? I have found this question and this question but I wonder how to use this for fisheye correction.

So far I've tried to use ImageTransformation for this, but I can't get the function to work properly.

  f[pt_] := With[{s = {.5, .5}},
  Module[{rd, polarcoor, ru, newcoor},
   rd = Norm[pt - s]^2/Norm[s];
   polarcoor = 
    CoordinateTransform[{"Cartesian" -> "Polar", 2}, (pt - s)];
   ru = 1*Tan[2*ArcSin[((polarcoor[[1]])/(2*1))]];
   CoordinateTransform["Polar" -> "Cartesian", {ru, polarcoor[[2]]}]
   ]
  ]
ImageTransformation[image,f]

This should first translate the image coordinates to polar coordinates, then calculate the new r (ru=r undistorted), and than translate these back to cartesian coordinates. The ru = 1*Tan[2*ArcSin[((polarcoor[[1]])/(2*1))]] is based on the above links, with a random value chosen as f. It fi

I get an error message, saying the function doesn't map. Anyone have an idea how to fix this and any further suggestions on how to improve the code?

Update 2 22-10-2013 Changed the code to:

image = Import["http://i.stack.imgur.com/JDX9f.jpg"];
r[pt_] := Module[{rd, ru, polarcoor, a},
  rd = Norm[pt];
  ru = *Transformation formula* 
  a = ArcTan @@ (pt);

  ru {Cos[a], Sin[a]}
  ]

Using ru = Sqrt[rd]; ImageTransformation[image, r, DataRange -> {{-1, 1}, {-1, 1}}] gives

Using ru = ArcTan[rd]; ImageTransformation[image, r, DataRange -> {{-2, 2}, {-2, 2}}] gives

Both look like a step in the right direction, with straightened lines, but I got them by trial-and-error so I don't know the correctness.