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?

Example fisheye image

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 enter image description here

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

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.

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