Image Warping – Bulge Effect Algorithm

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See if I understood what you want. Suppose your image coordinates go from 0 to 1.

If you do: 

r = Sqrt[(x - .5)^2 + (y - .5)^2]
a = ArcTan[x - .5, y - .5]
rn = r^2.5/.5

And then remap your pixels according to: 

  x -> rn*Cos[a] + .5 
  y -> rn*Sin[a] + .5

You get:

enter image description here

You may adjust the parameters to get bigger or smaller bulges.

Edit

Let’s see if I understood your comment about warping. The following images are generated using 

rn = r^k {k: 1 ... 2}:

enter image description here

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