ELTHON: An Escher-like Tile design method using Hierarchical Optimization

Abstract

Escher-like tiling attempts to design a tile whose copies cover a plane with no overlaps and no gaps. Deforming a given image shape into a tileable shape while maintaining the original shape as far as possible is a difficult problem. A tileable shape similar to a given image can be generated by an analytical optimization method (AOM) that requires no iterative calculations. However, the generated shape is often non-tileable due to edge self-intersections; moreover, as the method is sensitive to small changes of the input image shape, the input shape must be adjusted manually by much trial and error. To avoid this problem, this paper proposes an Escher-Like Tile design method using Hierarchical OptimizatioN (ELTHON), which divides the tile-design problem and resolves the subproblems by two different methods. The given problem (such as the goal figure shape) is modified by an upper-layer optimizer, and feasible solutions (such as the tileable shapes) are found by a lower-layer optimizer based on AOM. Because the upper layer employs metaheuristics such as a genetic algorithm, which support flexible objective functions and constraints, different types of figure similarity metrics and constraints can be selected to avoid the generation of non-tileable shapes. Experimental results showed that ELTHON designed tiles without violating the constraints, whereas 58.9% of the solutions found by AOM violated the constraints. Additionary, in 16 of 32 figures tested, ELTHON produced tileable shapes with higher similarity to given images compared with AOM, whereas AOM outperformed ELTHON in only two figures.