Exploring geometrical structures in image and video data for classification : from subspaces to matrix manifolds (PhD)

Abstract

One of the important tasks in computer vision and machine learning is data classification, which needs data to be represented as feature vectors that are compact,informative as well as discriminative. This thesis makes an attempt to generate such representations using data geometry. The geometry being referred to here is either subspace or manifold geometry.Two types of data are being considered; those which can be modeled as points in a union of subspaces and those which lie on a matrix manifold. For the former, kernel based and autoencoder based transformation learning approaches are described. The methods are based on the fact that real data models usually deviate by a large extent from an ideal union of subspace structure; thereby resulting in non-linearities. These non-linearities are handled by the two above-mentioned approaches. In manifold geometry, distance measures existing over the manifolds are useful for measuring similarity between two points. Same class data points tend to be nearby points in manifolds while the points from different classes would be far away. This thesis uses distance functions over matrix manifolds to come up with distance based positive definite kernels defined on a manifold space. For all the kernel based approaches, a discriminative representation corresponding to each data point is obtained by diagonalizing the kernel-gram matrix while for the autoencoder based approach, representation is directly obtained as the encoder output.These representations are the outcome of the assumed underlying geometrical models. In order to test the performance of the models designed for image classification task,images are considered as points in a union of subspaces. Similarly, image sets and videos are modeled as points on matrix manifolds for their respective classification tasks. While affine Grassmann manifold (AGM) geometry is considered for image sets, two geometrical models are proposed for videos - product Grassmann manifold (PGM) and product manifold of symmetric positive semi-definite (SPSD) matrices.The obtained results show the significance of these geometries in data classification.