Riemannian Manifold Clustering and Dimensionality Reduction for Vision-Based Analysis

Abstract

Segmentation is one fundamental aspect of vision-based motion analysis, thus it has been extensively studied. Its goal is to group the data into clusters based upon image properties such as intensity, color, texture, or motion. Most existing segmentation algorithms proceed by associating a feature vector to each pixel in the image or video and then segmenting the data by clustering these feature vectors. This process can be phrased as a manifold learning and clustering problem, where the objective is to learn a low-dimensional representation of the underlying data structure and to segment the data points into different groups. Over the past few years, various techniques have been developed for learning a low-dimensional representation of a nonlinear manifold embedded in a high-dimensional space. Unfortunately, most of these techniques are limited to the analysis of a single connected nonlinear manifold. In addition, all these manifold learning algorithms assume that the feature vectors are embedded in a Euclidean space and make use of (at least locally) the Euclidean metric or a variation of it to perform dimensionality reduction. While this may be appropriate in some cases, there are several computer vision problems where it is more natural to consider features that live in a Riemannian space. To address these problems, algorithms for performing simultaneous nonlinear dimensionality reduction and clustering of data sampled from multiple submanifolds of a Riemannian manifold have been recently proposed. In this book chapter, we give a summary of these newly developed algorithms as described in Goh and Vidal (Conference on Computer Vision and Pattern Recognition, 2007 and 2008; European Conference on Machine Learning, 2008; and European Conference on Computer Vision, 2008) and demonstrate their applications to vision-based analysis.