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Revisiting the Fréchet distance between piecewise smooth curves

Jacobus Conradi, Anne Driemel, Benedikt Kolbe

TL;DR

This work extends the Fréchet distance computation from polygonal curves to piecewise smooth curves in fixed dimension by introducing a combinatorial free-space framework. It provides a $O(mn)$-time decision algorithm for algebraically bounded pieces and, via parametric search, computes the distance in $O(mn\\log(mn))$ time, with a $O(cn/\\varepsilon)$-time $ (1+\\varepsilon)$-approximation for $c$-packed curves. A key innovation is a refined decomposition of the free-space diagram based on the boundary $B_{\delta}$, enabling a purely combinatorial reconstruction via a graph $G_{\delta}$ and monotone reachability, avoiding curvature-based calculations. The paper also develops a curve simplification method that preserves packing properties, enabling efficient, near-linear-time approximations through the notion of $\\epsilon$-relative free-space complexity. Overall, the results generalize efficient Fréchet-distance computation to smooth curves and higher dimensions, with practical impact for graphics, robotics, and motion-analysis pipelines.

Abstract

Since its introduction to computational geometry by Alt and Godau in 1992, the Fréchet distance has been a mainstay of algorithmic research on curve similarity computations. The focus of the research has been on comparing polygonal curves, with the notable exception of an algorithm for the decision problem for planar piecewise smooth curves due to Rote (2007). We present an algorithm for the decision problem for piecewise smooth curves that is both conceptually simpler and naturally extends to the first algorithm for the problem for piecewise smooth curves in $\mathbb{R}^d$. We assume that the algorithm is given two continuous curves, each consisting of a sequence of $m$, resp.\ $n$, smooth pieces, where each piece belongs to a sufficiently well-behaved class of curves, such as the set of algebraic curves of bounded degree. We introduce a decomposition of the free space diagram into a controlled number of pieces that can be used to solve the decision problem similarly to the polygonal case, in $O(mn)$ time, leading to a computation of the Fréchet distance that runs in $O(mn\log(mn))$ time. Furthermore, we study approximation algorithms for piecewise smooth curves that are also $c$-packed for some fixed value $c$. We adapt the existing framework for $(1+ε)$-approximations and show that an approximate decision can be computed in $O(cn/ε)$ time for any $ε> 0$.

Revisiting the Fréchet distance between piecewise smooth curves

TL;DR

This work extends the Fréchet distance computation from polygonal curves to piecewise smooth curves in fixed dimension by introducing a combinatorial free-space framework. It provides a -time decision algorithm for algebraically bounded pieces and, via parametric search, computes the distance in time, with a -time -approximation for -packed curves. A key innovation is a refined decomposition of the free-space diagram based on the boundary , enabling a purely combinatorial reconstruction via a graph and monotone reachability, avoiding curvature-based calculations. The paper also develops a curve simplification method that preserves packing properties, enabling efficient, near-linear-time approximations through the notion of -relative free-space complexity. Overall, the results generalize efficient Fréchet-distance computation to smooth curves and higher dimensions, with practical impact for graphics, robotics, and motion-analysis pipelines.

Abstract

Since its introduction to computational geometry by Alt and Godau in 1992, the Fréchet distance has been a mainstay of algorithmic research on curve similarity computations. The focus of the research has been on comparing polygonal curves, with the notable exception of an algorithm for the decision problem for planar piecewise smooth curves due to Rote (2007). We present an algorithm for the decision problem for piecewise smooth curves that is both conceptually simpler and naturally extends to the first algorithm for the problem for piecewise smooth curves in . We assume that the algorithm is given two continuous curves, each consisting of a sequence of , resp.\ , smooth pieces, where each piece belongs to a sufficiently well-behaved class of curves, such as the set of algebraic curves of bounded degree. We introduce a decomposition of the free space diagram into a controlled number of pieces that can be used to solve the decision problem similarly to the polygonal case, in time, leading to a computation of the Fréchet distance that runs in time. Furthermore, we study approximation algorithms for piecewise smooth curves that are also -packed for some fixed value . We adapt the existing framework for -approximations and show that an approximate decision can be computed in time for any .
Paper Structure (15 sections, 16 theorems, 7 equations, 8 figures)

This paper contains 15 sections, 16 theorems, 7 equations, 8 figures.

Table of Contents

  1. Introduction and motivation
  2. Problem definition
  3. Results
  4. Comparison to previous work
  5. Computational model
  6. A combinatorial description of the free space diagram
  7. Overview of the algorithm
  8. Refining the free space diagram
  9. Monotone paths in the free space diagram and the decision problem
  10. The structure of the boundary $B_{\delta}$ of the free space
  11. The minimization problem of computing the Fréchet distance
  12. Simplification of piecewise smooth curves
  13. Computing the simplification $\mathop{\mathrm{simp}}\nolimits(\gamma)$ of a piecewise smooth curve $\gamma$ with respect to $\mu\in\mathbb{R}$.
  14. Preserving $c$-packedness
  15. The relative free space complexity of simplified smooth curves

Key Result

Lemma 1

The boundary $B_{\delta}$ of the free space in the interior of each subcell in $\{S\}$ is a union of smooth arcs that are monotone in both coordinates of $\mathbb{R}^2$ and are disjoint except possibly at the boundary of a subcell.

Figures (8)

  • Figure 1: Two smooth curves in $\mathbb{R}^3$ and a contour plot of the associated distance function in the joint parametric space of the curves.
  • Figure 2: The decomposition of a cell of the free space diagram into subcells arising from the horizontal and vertical lines at extremities of the cyan forbidden region in the coordinate directions.
  • Figure 5: Two different sets of slope information and their combinatorial structures in a subcell.
  • Figure 6: Finding slope information by evaluating the distance at points.
  • Figure 7: Computing the reachable intervals (in orange) on the top and right edges of a cell.
  • ...and 3 more figures

Theorems & Definitions (23)

  • Lemma 1
  • Lemma 2
  • Remark 3
  • Definition 4
  • Proposition 5
  • Remark 6
  • Example 7
  • Remark 8
  • Proposition 9
  • Remark 10
  • ...and 13 more