Orientation Preserving Diffeomorphisms

Orientation-preserving diffeomorphisms, smooth and invertible transformations preserving object orientation, are central to shape analysis and image registration, aiming to find optimal mappings between shapes or images. Current research focuses on developing efficient algorithms and neural network architectures, such as those based on Euler-Poincaré equations or compositions of elementary diffeomorphisms, to compute these mappings, often within optimal control or Riemannian geometry frameworks. These advancements enable improved accuracy and speed in applications like medical image registration, robot motion planning, and shape analysis, offering more robust and computationally feasible solutions to complex geometric problems.