[Paper Review] Mapping polygons to the grid with small Hausdorff and Fréchet distance

Quirijn W. Bouts, Irina Kostitsyna+4 more|arXiv (Cornell University)|Jun 21, 2016

Computational Geometry and Mesh Generation17 references4 citations

TL;DR

This paper presents algorithms to map a simple polygon $P$ to a grid polygon $Q$ with small Hausdorff and Fréchet distances, ensuring topological simplicity and geometric fidelity. It proves the existence of grid polygons with constant Hausdorff distance to $P$ and shows that under realistic input assumptions, constant Fréchet distance is also achievable, with practical heuristics improving output quality while preserving distance bounds.

ABSTRACT

We show how to represent a simple polygon $P$ by a grid (pixel-based) polygon $Q$ that is simple and whose Hausdorff or Fréchet distance to $P$ is small. For any simple polygon $P$, a grid polygon exists with constant Hausdorff distance between their boundaries and their interiors. Moreover, we show that with a realistic input assumption we can also realize constant Fréchet distance between the boundaries. We present algorithms accompanying these constructions, heuristics to improve their output while keeping the distance bounds, and experiments to assess the output.

Motivation & Objective

• To address the challenge of representing continuous polygonal shapes as grid-based polygons while preserving geometric similarity.
• To ensure the resulting grid polygon remains simple (non-self-intersecting) and maintains small geometric distance to the original polygon.
• To achieve constant Hausdorff and, under realistic assumptions, constant Fréchet distance between the original and grid polygons.
• To develop practical heuristics that improve output quality without violating distance bounds.
• To evaluate the method on diverse shapes, including country outlines, buildings, and animals, for applications in cartography and puzzle generation.

Proposed method

• Constructs a grid polygon $Q$ by projecting the vertices and edges of a simple polygon $P$ onto the integer grid using a rounding strategy that preserves topological structure.
• Employs a novel construction method to ensure the Hausdorff distance between $P$ and $Q$ is bounded by a constant, independent of polygon complexity.
• Introduces a heuristic-based postprocessing pipeline involving vertex additions, removals, and shifts to refine $Q$ while maintaining bounded Hausdorff and Fréchet distances.
• Uses the symmetric difference as a heuristic metric to guide the selection of optimal $Q_4$ configurations during refinement.
• Applies a realistic input assumption—bounded aspect ratio and minimum edge length—to achieve constant Fréchet distance between boundaries.
• Implements and evaluates algorithms on a diverse dataset, including real-world maps and animal outlines, to assess geometric fidelity and topological correctness.

Experimental results

Research questions

• RQ1 Can a simple polygon be mapped to a grid polygon such that the Hausdorff distance between their boundaries and interiors is bounded by a constant?
• RQ2 Under what geometric assumptions can the Fréchet distance between the boundaries of a polygon and its grid representation be kept constant?
• RQ3 How can postprocessing heuristics improve the visual and geometric quality of the grid polygon without increasing the Hausdorff or Fréchet distance?
• RQ4 What is the trade-off between geometric fidelity (measured by symmetric difference) and topological correctness in grid polygonization?
• RQ5 To what extent can the proposed algorithms preserve shape characteristics in real-world applications such as nonogram generation and cartographic schematization?

Key findings

• The Hausdorff distance between the original polygon $P$ and the constructed grid polygon $Q$ is bounded by a constant, regardless of polygon complexity.
• Under a realistic input assumption (minimum edge length and bounded aspect ratio), the Fréchet distance between the boundaries of $P$ and $Q$ is also bounded by a constant.
• Postprocessing heuristics—additions, removals, and shifts—reduce the normalized symmetric difference by up to 54% on average, with improvements up to 116% on specific inputs like Vietnam.
• The symmetric difference heuristic for selecting $Q_4$ reduces the normalized symmetric difference by up to 17% on average, with significant gains on complex shapes like buildings and animals.
• On average, the symmetric difference of the grid polygon is 270% higher than the optimal (but non-grid) solution, but this is reduced to 197% with Fréchet-based refinement, indicating strong practical performance.
• The method successfully generates topologically simple grid polygons for diverse inputs, including country outlines, building footprints, and animal silhouettes, with minimal geometric distortion.