Learning Lie Groups for Invariant Visual Perception

One of the most important problems in visual perception is that of visual invariance: how are objects perceived to be the same despite undergoing transformations such as translations, rotations or scaling? In this paper, we describe a Bayesian method for learning invariances based on Lie group theory. We show that previous approaches based on first-order Taylor series expansions of inputs can be regarded as special cases of the Lie group approach, the latter being capable of handling in principle arbitrarily large transfonnations. Using a matrix-exponential based generative model of images, we derive an unsupervised algorithm for learning Lie group operators from input data containing infinitesimal transfonnations. The on-line unsupervised learning algorithm maximizes the posterior probability of generating the training data. We provide experimental results suggesting that the proposed method can learn Lie group operators for handling reasonably large 1-D translations and 2-D rotations.