I would like to understand the fisheye effect for images from the "vector field"-point of view.
Suppose we are given a bounded connected set $B\subset[0,1]^2$ (pre-fisheye) and we distort this set via a homeomorphism say $\phi$, i.e. $\phi(B)\subset[0,1]^2$ (post-fisheye).
My question: How would we model this function $\phi$?
My thoughts: Say this distortion is circular (as in most pictures with this effect) and towards the boundary of $[0,1]^2$ points should get "cramped up". Meanwhile close to some centerpoint $c\in ]0,1[^2$ point should get spread apart. So I think modelling such a function should involve such a center point and a degree of "cramping" that increases towards the boundary. However, I do not know how to model it precisely. In Photoshop there most certainly is a function for this, hence modelling it shouldn't be too hard, but I am struggling....
Follow Up Question: Suppose more generally that we have not necessarily a circular distortion, but any type of distortion induced by some closed injective curve $r:[0,1]\to[0,1]^2$. How do we model $\phi$ then? And what about negative fisheye-distortions?
thanks in advance for any help.