Learning Smooth Pattern Transformation Manifolds

Manifold models provide low-dimensional representations that are useful for processing and analyzing data in a transformation-invariant way. In this paper, we study the problem of learning smooth pattern transformation manifolds from image sets that are observations of geometrically transformed signals. In order to construct a manifold, we build a representative pattern whose transformations accurately fit various input images. The pattern is formed by selecting a good common sparse approximation of the images with parametric and smooth atoms. We examine two aspects of the manifold building problem, where we first target an accurate transformation-invariant approximation of the input images, and then extend this solution for their classification. For the approximation problem, we propose a greedy method that constructs a representative pattern by selecting analytic atoms from a continuous dictionary manifold. We present a DC (Difference-of-Convex) optimization scheme which is applicable for a wide range of transformation and dictionary models, and demonstrate its application to transformation manifolds generated by the rotation, translation and anisotropic scaling of a reference pattern. Then, we generalize this approach to a setting with multiple transformation manifolds, where each manifold represents a different class of signals. We present an iterative multiple manifold building algorithm such that the classification accuracy is promoted in the joint selection of atoms. Experimental results suggest that the proposed methods yield high accuracy in the approximation and classification of data in comparison with some reference methods, while achieving invariance to geometric transformations due to the transformation manifold model.