Can we Compute the Similarity Between Surfaces?

TL;DR

This paper investigates the computability of the Fréchet distance for surfaces, establishing its upper semi-computability and providing a polynomial-time algorithm for the weak Fréchet distance using a geometric data structure.

Contribution

It proves the upper semi-computability of the Fréchet distance for triangulated surfaces and introduces a polynomial-time algorithm for the weak Fréchet distance via the free space diagram.

Findings

Fréchet distance for surfaces is upper semi-computable.

Decision problem for Fréchet distance is recursively enumerable.

Weak Fréchet distance can be computed in polynomial time.

Abstract

A suitable measure for the similarity of shapes represented by parameterized curves or surfaces is the Fr\'echet distance. Whereas efficient 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. Here, using a discrete approximation we show that it is {\em upper semi-computable}, i.e., there is a non-halting Turing machine which produces a monotone decreasing sequence of rationals converging to the result. It follows that the decision problem, whether the Fr\'echet distance of two given surfaces lies below some specified value, is recursively enumerable. Furthermore, we show that a relaxed version of the problem, the computation of the {\em weak Fr\'echet distance} can be solved in polynomial time. For this, we give a computable characterization of the weak Fr\'echet distance in a geometric data structure called the {\em free space diagram}.