Approximating the Fréchet distance for realistic curves in near linear time
Proceedings of the twenty-sixth annual symposium on Computational geometry
Fréchet distance of surfaces: some simple hard cases
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part II
The frechet distance revisited and extended
Proceedings of the twenty-seventh annual symposium on Computational geometry
Computing the Fréchet distance between folded polygons
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Partial matching between surfaces using fréchet distance
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
The fréchet distance revisited and extended
ACM Transactions on Algorithms (TALG)
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A suitable measure for the similarity of shapes represented by parameterized curves or surfaces is the Fréchet distance. Whereas efficient algorithms are known for computing the Fréchet 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 decreasing sequence of rationals converging to the Fréchet distance. It follows that the decision problem, whether the Fréchet distance of two given surfaces lies below a specified value, is recursively enumerable. Furthermore, we show that a relaxed version of the Fréchet distance, the weak Fréchet distance can be computed in polynomial time. For this, we give a computable characterization of the weak Fréchet distance in a geometric data structure called the Free Space Diagram.