  This paper investigates the challenge of learning image manifolds,
specifically pose manifolds, of 3D objects using limited training data. It
proposes a DNN approach to manifold learning and for predicting images of
objects for novel, continuous 3D rotations. The approach uses two distinct
concepts: (1) Geometric Style-GAN (Geom-SGAN), which maps images to
low-dimensional latent representations and maintains the (first-order) manifold
geometry. That is, it seeks to preserve the pairwise distances between base
points and their tangent spaces, and (2) uses Euler's elastica to smoothly
interpolate between directed points (points + tangent directions) in the
low-dimensional latent space. When mapped back to the larger image space, the
resulting interpolations resemble videos of rotating objects. Extensive
experiments establish the superiority of this framework in learning paths on
rotation manifolds, both visually and quantitatively, relative to
state-of-the-art GANs and VAEs.
