This is at heart a mathematical problem, but is best motivated in physical terms. I'll introduce a very special case and move on to the general case later.
Special case An object, taken for convenience to be a cube of side $a$, is placed at distance $d$ in front of a convex spherical mirror of radius $R$. What is the shape (more precisely, the equation of the shape) of the reflected image?
Assume that light travels in perfect rays and we don't use the paraxial approximation (or any approximation,for that matter). This has now become a purely mathematical problem, which may not resemble physical image formation.
Here's the heart of the problem. For point objects, the image is given by the circle catacaustic , but AFAIK, there is no easy way to extend this to solid bodies. I would expect that exact equations for the image would be known (either in closed form or as a series) only for a select set of objects. Can you offer more insight?
General case Now consider a general case where the mirror surface and the object surface can both be parametrized with polynomial functions of coordinates $u,v$. Is there a general approach to this problem? Note that this is for mathematical tractability only ; it has no remote resemblance to real life.