Diffeomorphisms play a crucial role while searching for shapes with fixed topological properties, allowing for smooth deformation of template shapes. Several approaches use diffeomorphism for shape search. However, these approaches employ only unconstrained diffeomorphisms. In this work, we develop Flow Symmetrization - a method to represent a parametric family of constrained diffeomorphisms that contain additional symmetry constraints such as periodicity, rotation equivariance, and transflection equivariance. Our representation is differentiable in nature, making it suitable for gradient-based optimization approaches for shape search. As these symmetry constraints naturally arise in tiling classes, our method is ideal for representing tile shapes belonging to any tiling class. To demonstrate the efficacy of our method, we design two frameworks for addressing the challenging problems of Escherization and Density Estimation. The first framework is dedicated to the Escherization problem, where we parameterize tile shapes belonging to different isohedral classes. Given a target shape, the template tile is deformed using gradient-based optimization to resemble the target shape. The second framework focuses on density estimation in identification spaces. By leveraging the inherent link between tiling theory and identification topology, we design constrained diffeomorphisms for the plane that result in unconstrained diffeomorphisms on the identification spaces. Specifically, we perform density estimation on identification spaces such as torus, sphere, Klein bottle, and projective plane. Through results and experiments, we demonstrate that our method obtains impressive results for Escherization on the Euclidean plane and density estimation on non-Euclidean identification spaces.

翻译：微分同胚在具有固定拓扑性质的形状搜索中发挥着关键作用，可实现模板形状的光滑形变。已有多种方法利用微分同胚进行形状搜索，但这些方法仅使用无约束微分同胚。本研究提出流对称化方法——一种能够表示包含周期对称性、旋转等变性及反射等变性等额外对称约束的约束微分同胚参数族的方法。该表示方法具有可微性，适用于基于梯度优化的形状搜索方法。由于这些对称约束自然地出现在镶嵌类型中，我们的方法特别适合表示属于任意镶嵌类别中的瓷砖形状。为验证方法有效性，我们设计了两套框架分别解决埃舍尔镶嵌与密度估计两大难题。第一套框架专攻埃舍尔镶嵌问题，通过参数化不同等面类型的瓷砖形状，利用基于梯度的优化方法将模板瓷砖形变为目标形状。第二套框架聚焦于识别空间中的密度估计，通过利用镶嵌理论与识别拓扑之间的内在联系，设计出平面上的约束微分同胚，使其在识别空间上转化为无约束微分同胚。具体而言，我们在环面、球面、克莱因瓶和射影平面等识别空间上进行密度估计。实验结果证明，本方法在欧氏平面的埃舍尔镶嵌与非欧识别空间的密度估计任务中均取得了优异效果。