Pictorial and apictorial polygonal jigsaw puzzles from arbitrary number of crossing cuts

TL;DR

This paper introduces a new class of convex polygonal jigsaw puzzles generated by arbitrary straight cuts, analyzing their theoretical properties and proposing a dynamical system-based solution approach.

Contribution

It formalizes a novel puzzle generation model inspired by the Lazy caterer sequence and develops an automatic solving method using a spring-mass dynamical system.

Findings

Puzzles are fully solvable automatically.

Theoretical analysis of puzzle properties and challenges.

Proposed solution effectively handles geometrical noise.

Abstract

Jigsaw puzzle solving, the problem of constructing a coherent whole from a set of non-overlapping unordered visual fragments, is fundamental to numerous applications, and yet most of the literature of the last two decades has focused thus far on less realistic puzzles whose pieces are identical squares. Here, we formalize a new type of jigsaw puzzle where the pieces are general convex polygons generated by cutting through a global polygonal shape with an arbitrary number of straight cuts, a generation model inspired by the celebrated Lazy caterer sequence. We analyze the theoretical properties of such puzzles, including the inherent challenges in solving them once pieces are contaminated with geometrical noise. To cope with such difficulties and obtain tractable solutions, we abstract the problem as a multi-body spring-mass dynamical system endowed with hierarchical loop constraints and a layered reconstruction process. We define evaluation metrics and present experimental results on both apictorial and pictorial puzzles to show that they are solvable completely automatically.