Can we Compute the Similarity Between Surfaces?

A suitable measure for the similarity of shapes represented by parameterized curves or surfaces is the Fr echet distance. Whereas e cient algorithms are known for computing the Fr echet distance of polygonal curves, the same problem for triangulated surfaces is NP-hard. Furthermore, it remained open whether it is computable at all. Using a discrete approximation we show that it is upper semi-computable, i.e., there is a non-halting Turing machine which produces a monotone decreasing sequence of rationals converging to the Fr echet distance. It follows that the decision problem, whether the Fr echet distance of two given surfaces lies below a speci ed value, is recursively enumerable. Furthermore, we show that a relaxed version of the Fr echet distance, the weak Fr echet distance can be computed in polynomial time. For this, we give a computable characterization of the weak Fr echet distance in a geometric data structure called the free space diagram.