Comparison of Distance Measures for Planar Curves

Abstract

The Hausdorff distance is a very natural and straightforward distance measure for comparing geometric shapes like curves or other compact sets. Unfortunately, it is not an appropriate distance measure in some cases. For this reason, the Fréchet distance has been investigated for measuring the resemblance of geometric shapes which avoids the drawbacks of the Hausdorff distance. Unfortunately, it is much harder to compute. Here we investigate under which conditions the two distance measures approximately coincide, i.e., the pathological cases for the Hausdorff distance cannot occur. We show that for closed convex curves both distance measures are the same. Furthermore, they are within a constant factor of each other for so-called κ-straight curves, i.e., curves where the arc length between any two points on the curve is at most a constant κ times their Euclidean distance. Therefore, algorithms for computing the Hausdorff distance can be used in these cases to get exact or approximate computations of the Fréchet distance, as well.

Shortest Path Problems on a Polyhedral Surface

“…The combinatorial type of a shortest path map edge E is a vertex-edge pair (v, e) such that E has one endpoint at a vertex v and has its other endpoint on an edge e. As the source point s varies continuously along ab, the position of E's endpoint on e is parameterized homographically by g(s) = A+Bs C+Ds for constants A, B, C, D. We also define an edgelet α as a line segment such that the shortest path map for every source point s ∈ α has the same combinatorial structure. 6 Theorem 12. The link-based shortest path map SPM(ab, cd) can have Ω(M 4 ) complexity and can be constructed on N in O(M 7 ) space and either O(M 7 ) expected time or O(M 6 λ 6 (M )) deterministic time.…”

“…The Hausdorff distance [5,6] is a similarity metric commonly used to compare sets of points or sets of higher-dimensional objects such as line segments or triangles. Since the Hausdorff distance relies heavily on nearest neighbor distance calculations, it is often computed with a Voronoi diagram.…”

Detecting Commuting Patterns by Clustering Subtrajectories

“…Since it takes the continuity of the shapes into account it is generally regarded as being a more appropriate distance measure than the Hausdorff distance for curves [5,6,41]. For polygonal curves, the discrete Fréchet distance is a natural variant of the Fréchet distance.…”

Four Soviets Walk the Dog: Improved Bounds for Computing the Fréchet Distance

“…They argued that the Fréchet distance is better suited as a similarity measure, and they described an O(n 2 log n) time algorithm to compute it on a real RAM or pointer machine. 1 Since Alt and Godau's seminal paper, there has been a wealth of research in various directions, such as extensions to higher dimensions [7,23,26,28,33,46], approximation algorithms [9,10,37], the geodesic and the homotopic Fréchet distance [29,34,38,48], and much more [2,22,25,35,36,51,54,55]. Most known approximation algorithms make further assumptions on the curves, and only an O(n 2 )-time approximation algorithm is known for arbitrary polygonal curves [24].…”