Using Differentiable Rendering (reconstructing scene setup parameter(s) from an image), possibly, although it would depend on the number of unknown params.

Mitsuba2 had an example of matching caustic patterns by modifying geometry I think...

My gut tells me it would work like a hash function, the diamond -> dot pattern function being trivial, but the dot pattern -> diamond function taking exponentially more effort

There is nothing intrinsically hard about it. Crystallography is the science of reversing this dot pattern.