Go with the flow: Optical flow-based transport operators for image manifolds

Image articulation manifolds (IAMs) play a central conceptual role in a host of computer vision and image understanding problems. The core premise is that we can view a collection of images, each of which is indexed by a small number of degrees of freedom (3D camera pose, motion/deformation, etc.), as a low-dimensional nonlinear manifold. In order to perform parameter estimation and navigation on an IAM, we require a {\em transport operator} that traverses the manifold from image to image. The two current approaches to manifold transport suffer from major shortcomings that have limited the practical impact of manifold methods. First, algebraic methods require that the IAM possess an unrealistic algebraic structure. Second, locally linear methods based on a tangent plane approximation cannot cope with the non-differentiability of IAMs containing images with sharp edges. In this paper, we demonstrate that the {\em optical flow} between pairs of images on an IAM is a valid transport operator with a number of attractive properties. In particular, we establish that the optical flow forms a low-dimensional smooth manifold. Several experiments involving novel-view synthesis, geometric clustering, and manifold charting validate that the optical flow manifold approach both offers performance significantly superior to current approaches and is practical for real-world applications.