The Escherization problem involves finding a closed figure that tiles the plane that is most similar to a given goal figure. In Koizumi and Sugihara's formulation of the Escherization problem, the tile and goal figures are represented as $n$-point polygons where the similarity between them is measured based on the difference in the positions between the corresponding points. This paper presents alternative similarity measures (distance functions) suitable for this problem. The proposed distance functions focus on the similarity of local structures in several different manners. The designed distance functions are incorporated into a recently developed framework of the exhaustive search of the templates for the Escherization problem. Efficient exhaustive and incomplete search algorithms for the formulated problems are also developed to obtain results within a reasonable computation time. Experimental results showed that the proposed algorithms found satisfactory tile shapes for fairly complicated goal figures in a reasonable computation time.

中文翻译：

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集中于局部结构相似性的广义距离函数的随机化

Escherization问题涉及找到一个闭合图形，该图形将与给定目标图形最相似的平面平铺。在Koizumi和Sugihara提出的Escherization问题中，图块和目标图形表示为$ n $点多边形，其中，它们之间的相似度是根据相应点之间的位置差异来衡量的。本文提出了适合此问题的替代相似性度量（距离函数）。提出的距离函数以几种不同的方式关注局部结构的相似性。设计的距离函数被合并到最近开发的框架中，该框架详尽地搜索了模板化问题。还针对提出的问题开发了有效的穷举和不完整搜索算法，以在合理的计算时间内获得结果。实验结果表明，所提出的算法在合理的计算时间内就可以为相当复杂的目标人物找到令人满意的瓦片形状。