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Diffeomorphisms play a crucial role while searching for shapes with fixedtopological properties, allowing for smooth deformation of template shapes.Several approaches use diffeomorphism for shape search. However, theseapproaches employ only unconstrained diffeomorphisms. In this work, we developFlow Symmetrization - a method to represent a parametric family of constraineddiffeomorphisms that contain additional symmetry constraints such asperiodicity, rotation equivariance, and transflection equivariance. Ourrepresentation is differentiable in nature, making it suitable forgradient-based optimization approaches for shape search. As these symmetryconstraints naturally arise in tiling classes, our method is ideal forrepresenting tile shapes belonging to any tiling class. To demonstrate theefficacy of our method, we design two frameworks for addressing the challengingproblems of Escherization and Density Estimation. The first framework isdedicated to the Escherization problem, where we parameterize tile shapesbelonging to different isohedral classes. Given a target shape, the templatetile is deformed using gradient-based optimization to resemble the targetshape. The second framework focuses on density estimation in identificationspaces. By leveraging the inherent link between tiling theory andidentification topology, we design constrained diffeomorphisms for the planethat result in unconstrained diffeomorphisms on the identification spaces.Specifically, we perform density estimation on identification spaces such astorus, sphere, Klein bottle, and projective plane. Through results andexperiments, we demonstrate that our method obtains impressive results forEscherization on the Euclidean plane and density estimation on non-Euclideanidentification spaces.

Flow Symmetrization for Parameterized Constrained Diffeomorphisms