🤖 AI Summary
This work addresses the challenge of modeling diffeomorphisms for shape search under geometric symmetry constraints—specifically periodicity, rotation/reflection equivariance, and other invariances. We propose Flow Symmetrization: a method that constructs symmetry-enforced deformation flows via manifold-constrained ODEs, enabling differentiable embedding of explicit geometric constraints—such as isohedral tiling classes and topological adaptation to non-Euclidean identification spaces—under Lie group actions. To our knowledge, this is the first framework unifying tiling theory and differential-geometric optimization within a differentiable diffeomorphism setting. Experiments demonstrate precise contour reconstruction in Euclidean Escher-style tiling tasks and high-fidelity density estimation on non-Euclidean identification spaces—including the torus, sphere, and Klein bottle—outperforming unconstrained baselines by significant margins.
📝 Abstract
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.
