Nash Meets Wertheimer: Using Good Continuation in Jigsaw Puzzles

Abstract

Jigsaw puzzle solving is a challenging task for computer vision since it requires high-level spatial and semantic reasoning. To solve the problem, existing approaches invariably use color and/or shape information but in many real-world scenarios, such as in archaeological fresco reconstruction, this kind of clues is often unreliable due to severe physical and pictorial deterioration of the individual fragments. This makes state-of-the-art approaches entirely unusable in practice. On the other hand, in such cases, simple geometrical patterns such as lines or curves offer a powerful yet unexplored clue. In an attempt to fill in this gap, in this paper we introduce a new challenging version of the puzzle solving problem in which one deliberately ignores conventional color and shape features and relies solely on the presence of linear geometrical patterns. The reconstruction process is then only driven by one of the most fundamental principles of Gestalt perceptual organization, namely Wertheimer's {\em law of good continuation}. In order to tackle this problem, we formulate the puzzle solving problem as the problem of finding a Nash equilibrium of a (noncooperative) multiplayer game and use classical multi-population replicator dynamics to solve it. The proposed approach is general and allows us to deal with pieces of arbitrary shape, size and orientation. We evaluate our approach on both synthetic and real-world data and compare it with state-of-the-art algorithms. The results show the intrinsic complexity of our purely line-based puzzle problem as well as the relative effectiveness of our game-theoretic formulation.