Optics and mathematics of dewarping a reflection in a generalized cylinder

This is a slightly modified version of this question, asked to mathematicians, as I suspect more physicists have confronted this particular problem. It is a generalization of a simpler problem posted to physicists.

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I would like to computationally dewarp the image in a generalized cylinder, as illustrated in Scott Fraser’s Reflections:

We can make some simplifying assumptions:

• The viewer (artist) and objects are far from the mirror… much larger than the mirror’s radius of curvature
• The mirror is radially symmetric about a (vertical) axis
• We can see the outer contour ("occluding contour") and hence know the radius as a function of height, $r(h)$.

I’m wondering if anyone is familiar with a paper or book in which this optical problem has been solved. I ask for two reasons: First, I don’t want to have to "re-invent the wheel," and 2) I want to give credit and cite whoever has solved this problem.

Please don’t point out any of the myriad papers on the much simpler problem of a simple cylinder ($r(h) =$ constant)… I’m extremely familiar with cylindrical anamorphic art, and have published in the field.

Of course if in the unlikely event it seems this problem hasn’t been solved, well of course I’d be extremely grateful if someone derives the full equations.