This paper proposes a fully-automatic, text-guided generative method for producing perfectly-repeating, periodic, tile-able 2D imagery, such as the one seen on floors, mosaics, ceramics, and the work of M.C. Escher. In contrast to square texture images that are seamless when tiled, our method generates non-square tilings which comprise solely of repeating copies of the same object. It achieves this by optimizing both geometry and texture of a 2D mesh, yielding a non-square tile in the shape and appearance of the desired object, with close to no additional background details, that can tile the plane without gaps nor overlaps. We enable optimization of the tile's shape by an unconstrained, differentiable parameterization of the space of all valid tileable meshes for given boundary conditions stemming from a symmetry group. Namely, we construct a differentiable family of linear systems derived from a 2D mesh-mapping technique - Orbifold Tutte Embedding - by considering the mesh's Laplacian matrix as differentiable parameters. We prove that the solution space of these linear systems is exactly all possible valid tiling configurations, thereby providing an end-to-end differentiable representation for the entire space of valid tiles. We render the textured mesh via a differentiable renderer, and leverage a pre-trained image diffusion model to induce a loss on the resulting image, updating the mesh's parameters so as to make its appearance match the text prompt. We show our method is able to produce plausible, appealing results, with non-trivial tiles, for a variety of different periodic tiling patterns.

翻译：本文提出了一种全自动、文本引导的生成方法，用于生成完美重复、周期性、可平铺的二维图像，例如地板、马赛克、陶瓷制品以及M.C.埃舍尔作品中所见的图案。与平铺时无缝衔接的方形纹理图像不同，我们的方法生成非方形平铺图案，这些图案完全由同一物体的重复副本构成。该方法通过优化二维网格的几何形状和纹理来实现这一目标，生成一个在形状和外观上符合目标物体的非方形图块，该图块几乎不包含额外的背景细节，并且能够无间隙、无重叠地平铺整个平面。我们通过一种无约束、可微分的参数化方法，对给定对称群所决定的边界条件下所有有效可平铺网格的空间进行参数化，从而实现了对图块形状的优化。具体而言，我们通过将网格的拉普拉斯矩阵视为可微分参数，构建了一个源自二维网格映射技术——Orbifold Tutte嵌入——的可微分线性方程组族。我们证明了这些线性方程组的解空间恰好是所有可能的有效平铺配置，从而为整个有效图块空间提供了端到端的可微分表示。我们通过可微分渲染器对带纹理的网格进行渲染，并利用预训练的图像扩散模型对生成的图像施加损失，更新网格参数，使其外观与文本提示相匹配。我们展示了该方法能够为多种不同的周期性平铺图案生成合理的、具有吸引力的、包含非平凡图块的结果。