We present a novel computational puzzle solver for square-piece image jigsaw puzzles with no prior information such as piece orientation or anchor pieces. By “piece” we mean a square $d$d x $d$d block of pixels, where we investigate pieces as small as 7 × 7 pixels. To reconstruct such challenging puzzles, we propose to find maximum geometric consensus between pieces, specifically hierarchical piece loops. The proposed algorithm seeks out loops of four pieces and aggregates the smaller loops into higher order “loops of loops” in a bottom-up fashion. In contrast to previous puzzle solvers which aim to maximize compatibility measures between all pairs of pieces and thus depend heavily on the pairwise compatibility measures used, our approach reduces the dependency on the pairwise compatibility measures which become increasingly uninformative for small scales and instead exploits geometric agreement among pieces. Our contribution also includes an improved pairwise compatibility measure which exploits directional derivative information along adjoining boundaries of the pieces. We verify the proposed algorithm as well as its individual components with mathematical analysis and reconstruction experiments.

中文翻译：

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通过分层循环约束解决方形拼图

我们提出了一种新颖的计算谜题求解器，用于解决方片图像拼图难题，而没有先验信息（例如，块方向或锚定块）。“块”是指正方形的$ d $ dx $ d $ d像素块，在这里我们调查小至7×7像素的块。为了重建此类具有挑战性的难题，我们建议在零件之间找到最大的几何一致性，特别是在分层零件回路之间。所提出的算法找出四个部分的循环，并以自下而上的方式将较小的循环聚合为更高阶的“循环”。与以前的难题解算器旨在最大程度地提高所有成对部件之间的兼容性度量并因此严重依赖于所使用的成对兼容性度量相比，我们的方法减少了对成对兼容性度量的依赖，成对兼容度量对于小规模而言变得越来越无用，而是利用片段之间的几何一致性。我们的贡献还包括改进的成对兼容性度量，该度量利用了沿片段相邻边界的方向派生信息。我们通过数学分析和重建实验来验证所提出的算法及其各个组成部分。