  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.
