While the paper talks about the Legendre Transform, the actual mathematical object that they end up using is a support function transform, which though similar is not exactly what Legendre had in mind.

While this paper was a substantial breakthrough, it is still nowhere near the solution to the problem of recovering the surface of a mirror.

In the years since the publication of this paper, there have been two more pieces of significant work on recovering specular geometry.

The first is the work by Silvio Savarese which we discussed earlier in the class. Besides this there has been work on using rotating an object under a light bean and observing the highlights so created (the exact reference escapes me at the moment, its cited in silvio's papers).