Pictorial and Apictorial Polygonal Jigsaw Puzzles from Arbitrary Number of Crossing Cuts
Summary: arXiv:2008.07644v3 Announce Type: replace-cross
Jigsaw puzzle solving is a fundamental problem that involves constructing a coherent whole from a set of non-overlapping unordered visual fragments. Despite its significance in various applications, existing literature predominantly focuses on less realistic puzzles, particularly those composed of identical square pieces. In this article, we present a novel approach that formalizes a new type of jigsaw puzzle, where the pieces are general convex polygons created by cutting through a global polygonal shape using an arbitrary number of straight cuts. This innovative generation model is inspired by the well-known Lazy Caterer sequence.
Theoretical Properties of Polygonal Jigsaw Puzzles
We delve into the theoretical aspects of these polygonal jigsaw puzzles, exploring several inherent challenges that arise during the solving process, particularly when pieces are affected by geometrical noise. Such noise can complicate the reconstruction of the original shape, making it difficult for algorithms to accurately identify and assemble the pieces.
Modeling the Puzzle Solving Process
To address these challenges and obtain tractable solutions, we abstract the jigsaw puzzle problem as a multi-body spring-mass dynamical system. This model includes:
- Hierarchical loop constraints: These constraints help manage the interactions between different pieces, ensuring that they adhere to certain geometric relationships.
- Layered reconstruction process: This process allows for a systematic approach to reassemble the puzzle, facilitating incremental progress as pieces are combined.
Evaluation Metrics and Experimental Results
In our research, we define specific evaluation metrics to assess the performance and effectiveness of our approach. These metrics not only measure the accuracy of the assembled puzzles but also evaluate the efficiency of the solving process. We conduct experiments on both apictorial and pictorial puzzles, demonstrating that our method allows for complete automated solvability.
Conclusion
In conclusion, the introduction of polygonal jigsaw puzzles generated through arbitrary cuts presents a significant advancement in the domain of jigsaw puzzle solving. By applying a multi-body spring-mass dynamical system framework, we can effectively tackle the complexities associated with geometrical noise and demonstrate that both apictorial and pictorial puzzles can be solved automatically. This research opens new avenues for future studies in computational geometry and artificial intelligence, offering a more realistic and challenging perspective on puzzle-solving algorithms.
For further details, please refer to the complete paper available on arXiv: arXiv:2008.07644v3.