Learning Deformations with Parallel Transport

Many vision problems, such as object recognition and image synthesis, are greatly impacted by deformation of objects. In this paper, we develop a deformation model based on Lie algebraic analysis. This work aims to provide a generative model that explicitly decouples deformation from appearance, which is fundamentally different from the prior work that focuses on deformation-resilient features or metrics. Specifically, the deformation group for each object can be characterized by a set of Lie algebraic basis. Such basis for different objects are related via parallel transport. Exploiting the parallel transport relations, we formulate an optimization problem, and derive an algorithm that jointly estimates the deformation basis for a class of objects, given a set of images resulted from the action of the deformations. We test the proposed model empirically on both character recognition and face synthesis.