Deconstructing Approximate Offsets

TL;DR

This paper introduces an efficient algorithm to determine if a polygonal shape can be approximated as a Minkowski sum with a disk, enabling simplified shape representations with applications in shape analysis and processing.

Contribution

It provides an exact O(n log n)-time decision algorithm for the offset-deconstruction problem, including a practical implementation and specialized convex shape solutions.

Findings

Exact decision algorithm runs in O(n log n) time for general polygons.

Convex shapes allow a simplified O(n) decision and solution computation.

The implementation handles uncertainties and computes approximate solutions when applicable.

Abstract

We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(n log n)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using CGAL, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter \delta; its running time additionally depends on \delta. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one.