Learning Manifolds in the Wild
Chinmay D Hegde, Aswin C. Sankaranarayanan, Richard G. Baraniuk · 2012
Despite the promise of low-dimensional manifold models for image processing, computer vision, and machine learning tasks, their utility has been hamstrung in practice by two fundamental challenges. First, practical image manifolds are non-isometric to their underlying parameter space, while the state-of-the-art manifold modeling and learning frameworks assume isometry. Second, practical image manifolds are strongly perturbed by nuisance parameters such as illumination variations, occlusions, and clutter. In this paper, we develop new theory and practical algorithms for manifold modeling, learning, and processing that directly address these challenges. To address the isometry challenge, we show that the Earth Mover’s Distance (EMD) is a more natural metric for inter-image distances than the standard Euclidean distance and use it to establish the isometry of manifolds generated by translations and rotations of a reference image. To the best of our knowledge, this is the first rigorous result on manifold isometry for generic grayscale image familes. To address the nuisance parameter challenge, we advocate an image representation based on local keypoint features and use it to define a new keypoint articulation manifold (KAM). We employ the KAM framework on a number of real-world image datasets acquired “in the wild” to demonstrate its improved performance over state-of-the-art manifold modeling approaches. A particularly compelling application of our approach is the automatic organization of large, unstructured collections of photographs gathered from the internet.