The directed Hausdorff distance between imprecise point sets

逼近有向 Hausdorff 距离

Approximating the Directed Hausdorff Distance

本文针对倍增度量中的点集，提出独立预处理与线性时间查询方法，以计算有向 Hausdorff 距离的 $(1+\varepsilon)$ 近似，并支持一次性求所有 $k$ 的有向 $k$-偏距离。

Minimum $L_\infty$ Hausdorff Distance of Point Sets Under Translation: Generalizing Klee's Measure Problem

We present a (combinatorial) algorithm with running time close to $O(n^d)$ for computing the minimum directed $L_\infty$ Hausdorff distance between two sets of $n$ points under translations in any constant dimension $d$. This substantially improves the best previous time bound near $O(n^{5d/4})$ by Chew, Dor, Efrat, and Kedem from more than twenty years ago. Our solution is obtained by a new generalization of Chan's algorithm [FOCS'13] for Klee's measure problem. To complement this algorithmic result, we also prove a nearly matching conditional lower bound close to $Ω(n^d)$ for combinatorial a…

Embedding Ray Intersection Graphs and Global Curve Simplification

We prove that circle graphs (intersection graphs of circle chords) can be embedded as intersection graphs of rays in the plane with polynomial-size bit complexity. We use this embedding to show that the global curve simplification problem for the directed Hausdorff distance is NP-hard. In this problem, we are given a polygonal curve $P$ and the goal is to find a second polygonal curve $P'$ such that the directed Hausdorff distance from $P'$ to $P$ is at most a given constant, and the complexity of $P'$ is as small as possible.

在平移下计算 $L_\infty$ 赫尔德距离：维度、对称性和离散性的相互作用

Computing $L_\infty$ Hausdorff Distances Under Translations: The Interplay of Dimensionality, Symmetry and Discreteness

研究在平移下计算 $L_\infty$ 赫尔德距离的复杂性，揭示维度、对称性和离散性之间的相互作用。

On Optimal Polyline Simplification using the Hausdorff and Fréchet Distance

We revisit the classical polygonal line simplification problem and study it using the Hausdorff distance and Fréchet distance. Interestingly, no previous authors studied line simplification under these measures in its pure form, namely: for a given $\varepsilon$ > 0, choose a minimum size subsequence of the vertices of the input such that the Hausdorff or Fréchet distance between the input and output polylines is at most $\varepsilon$. We analyze how the well-known Douglas-Peucker and Imai-Iri simplification algorithms perform compared to the optimum possible, also in the situation where the a…

Stability of multidimensional persistent homology with respect to domain perturbations

Motivated by the problem of dealing with incomplete or imprecise acquisition of data in computer vision and computer graphics, we extend results concerning the stability of persistent homology with respect to function perturbations to results concerning the stability with respect to domain perturbations. Domain perturbations can be measured in a number of different ways. An important method to compare domains is the Hausdorff distance. We show that by encoding sets using the distance function, the multidimensional matching distance between rank invariants of persistent homology groups is alway…