Abstract

Dimensionality reduction is an essential aspect of visual pro- cessing. Traditionally, linear dimensionality reduction techniques such as principle components analysis have been used to find low dimensional linear subspaces in visual data. However, sub-manifolds in natural data are rarely linear, and consequently many recent techniques have been de- veloped for discovering non-linear manifolds. Prominent among these are Local Linear Embedding and Isomap. Unfortunately, such techniques cur- rently use a naive appearance model that judges image similarity based solely on Euclidean distance. In visual data, Euclidean distances rarely correspond to a meaningful perceptual difierence between nearby images. In this paper, we attempt to improve the quality of manifold inference techniques for visual data by modeling local neighborhoods in terms of natural transformations between images—for example, by allowing im- age operations that extend simple difierences and linear combinations. We introduce the idea of modeling local tangent spaces of the manifold in terms of these richer transformations. Given a local tangent space representation, we then embed data in a lower dimensional coordinate system while preserving reconstruction weights. This leads to improved manifold discovery in natural image sets.