Structured Discrete Shape Approximation: Theoretical Complexity and Practical Algorithm

TL;DR

This paper investigates the complexity of approximating 2D shapes with discrete systems, proving NP-hardness, and introduces an algorithmic framework that effectively produces good shape approximations despite the problem's computational difficulty.

Contribution

The paper presents a novel algorithmic approach combining shape sampling and cardinality-minimization for discrete shape approximation, and demonstrates its effectiveness on Zometool-based shape contours.

Findings

Deciding feasibility of shape approximation is NP-hard.

The proposed algorithm produces good approximations efficiently.

Successful application to Zometool shape contour approximation.

Abstract

We consider the problem of approximating a two-dimensional shape contour (or curve segment) using discrete assembly systems, which allow to build geometric structures based on limited sets of node and edge types subject to edge length and orientation restrictions. We show that already deciding feasibility of such approximation problems is NP-hard, and remains intractable even for very simple setups. We then devise an algorithmic framework that combines shape sampling with exact cardinality-minimization to obtain good approximations using few components. As a particular application and showcase example, we discuss approximating shape contours using the classical Zometool construction kit and provide promising computational results, demonstrating that our algorithm is capable of obtaining good shape representations within reasonable time, in spite of the problem's general intractability. We conclude the paper with an outlook on possible extensions of the developed methodology, in particular regarding 3D shape approximation tasks.