Geometric methods on special manifolds for visual recognition

Many computer vision methods assume that the underlying geometry of images is Euclidean. This assumption is generally not valid. Therefore, this dissertation introduces new nonlinear geometric frameworks based upon special manifolds, namely Grasmann and Stiefel manifolds, for visual recognition. The motivation for this thesis is driven by the intrinsic geometry of visual data in which the visual data can be either a still image or video. Visual data are represented as points in appropriately chosen parameter spaces. The idiosyncratic aspects of the data in these spaces are then exploited for pattern classification. Three major research results are presented in this dissertation: face recognition for illumination spaces on Stiefel manifolds, face recognition on Grasmann registration manifolds, and action classification on product manifolds. Previous work has shown that illumination cones are idiosyncratic for face recognition in illumination spaces. However, it has not been addressed how a single image relates to an illumination cone. In this dissertation, a Bayesian model is employed to relight a single image to a set of illuminated variants. The subspace formed by these illuminated variants is characterized on a Stiefel manifold. A new distance measure called Canonical Stiefel Quotient (CSQ) is introduced. CSQ performs two projections on a tangent space of a Stiefel manifold and uses the quotient for classification. The proposed method demonstrates that illumination cones can be synthesized by relighting a single image to a set of images, and the synthesized illumination cones are discriminative for face recognition. Experiments on the CMU-PIE and YaleB data sets reveal that CSQ not only achieves high recognition accuracies for generic faces but also is robust to the choice of training sets. Subspaces can be realized as points on Grasmann manifolds. Motivated by image perturbation and the geometry of Grasmann manifolds, we present a method called Grasmann Registration Manifolds (GRM) for face recognition. First, a tangent space is formed by a set of affine perturbed images where the tangent space admits a vector space structure. Second, the tangent spaces are embedded on a Grasmann manifold and chordal distance is used to compare subspaces. Experiments on the FERET database suggest that the proposed method yields excellent results using both holistic and local features. Specifically, on the FERET Dup2 data set, which is generally considered the most difficult data set on FERET, the proposed method achieves the highest rank one identification rate among all non-trained methods currently in the literature. Human actions compose a series of movements and can be described by a sequence of video frames. Since videos are multidimensional data, data tensors are the natural choice for data representation. In this dissertation, a data tensor is expressed as a point on a product manifold and classification is performed on this product space. First, we factorize a data tensor using a modified High Order Singular Value Decomposition (HOSVD) and recognize each factorized space as a Grasmann manifold. Consequently, a data tensor is mapped to a point on a product manifold and the geodesic distance on the product manifold is computed for tensor classification. The proposed method is geometrically sound and the metric is naturally inherited from the factor manifolds. Experiments on the Cambridge-Gesture and KTH human action data sets show that the proposed method outperforms the current state-of-the-art. The use of special manifolds for visual recognition has just emerged. This dissertation shows that the underlying geometry of space is an important feature for pattern recognition. The proposed geometric frameworks are particularly suitable for high dimensional data, and will lead to many possible future work.