In this paper, we develop an approach to exploiting kernel methods with manifold-valued data. In many computer vision problems, the data can be naturally represented as points on a Riemannian manifold. Due to the non-Euclidean geometry of Riemannian manifolds, usual Euclidean computer vision and machine learning algorithms yield inferior results on such data. In this paper, we define Gaussian radial basis function (RBF)-based positive definite kernels on manifolds that permit us to embed a given manifold with a corresponding metric in a high dimensional reproducing kernel Hilbert space. These kernels make it possible to utilize algorithms developed for linear spaces on nonlinear manifold-valued data. Since the Gaussian RBF defined with any given metric is not always positive definite, we present a unified framework for analyzing the positive definiteness of the Gaussian RBF on a generic metric space. We then use the proposed framework to identify positive definite kernels on two specific manifolds commonly encountered in computer vision: the Riemannian manifold of symmetric positive definite matrices and the Grassmann manifold, i.e., the Riemannian manifold of linear subspaces of a Euclidean space. We show that many popular algorithms designed for Euclidean spaces, such as support vector machines, discriminant analysis and principal component analysis can be generalized to Riemannian manifolds with the help of such positive definite Gaussian kernels.
In this paper, we focus on the Riemannian manifold of SPD matrices and on the Grassmann manifold. SPD matrices find a variety of applications in computer vision @cite_19 . For instance, covariance region descriptors are used in object detection @cite_28 , texture classification @cite_35 @cite_2 , object tracking, action recognition and human recognition @cite_13 @cite_57 . Diffusion Tensor Imaging (DTI) was one of the pioneering fields for the development of non-linear algorithms on SPD matrices @cite_14 @cite_41 . In optical flow estimation and motion segmentation, structure tensors are often employed to encode important image features, such as texture and motion @cite_46 @cite_30 . Structure tensors have also been used in single image segmentation @cite_22 .
@cite_30 @cite_35 @cite_14 @cite_22 @cite_28 @cite_41 @cite_57 @cite_19 @cite_2 @cite_46 @cite_13
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Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices –especially of high-dimensional ones– comes at a high cost that limits the applicability of existing techniques. In this paper, we introduce algorithms able to handle high-dimensional SPD matrices by constructing a lower-dimensional SPD manifold. To this end, we propose to model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. This lets us formulate dimensionality reduction as the problem of finding a projection that yields a low-dimensional manifold either with maximum discriminative power in the supervised scenario, or with maximum variance of the data in the unsupervised one. We show that learning can be expressed as an optimization problem on a Grassmann manifold and discuss fast solutions for special cases. Our evaluation on several classification tasks evidences that our approach leads to a significant accuracy gain over state-of-the-art methods. Representing images and videos with Symmetric Positive Definite (SPD) matrices and considering the Riemannian geometry of the resulting space has proven beneficial for many recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices –especially of high-dimensional ones– comes at a high cost that limits the applicability of existing techniques. In this paper we introduce an approach that lets us handle high-dimensional SPD matrices by constructing a lower-dimensional, more discriminative SPD manifold. To this end, we model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. In particular, we search for a projection that yields a low-dimensional manifold with maximum discriminative power encoded via an affinity-weighted similarity measure based on metrics on the manifold. Learning can then be expressed as an optimization problem on a Grassmann manifold. Our evaluation on several classification tasks shows that our approach leads to a significant accuracy gain over state-of-the-art methods. We present a new algorithm to detect pedestrian in still images utilizing covariance matrices as object descriptors. Since the descriptors do not form a vector space, well known machine learning techniques are not well suited to learn the classifiers. The space of d-dimensional nonsingular covariance matrices can be represented as a connected Riemannian manifold. The main contribution of the paper is a novel approach for classifying points lying on a connected Riemannian manifold using the geometry of the space. The algorithm is tested on INRIA and DaimlerChrysler pedestrian datasets where superior detection rates are observed over the previous approaches. We propose a novel discriminative learning approach to image set classification by modeling the image set with its natural second-order statistic, i.e. covariance matrix. Since nonsingular covariance matrices, a.k.a. symmetric positive definite (SPD) matrices, lie on a Riemannian manifold, classical learning algorithms cannot be directly utilized to classify points on the manifold. By exploring an efficient metric for the SPD matrices, i.e., Log-Euclidean Distance (LED), we derive a kernel function that explicitly maps the covariance matrix from the Riemannian manifold to a Euclidean space. With this explicit mapping, any learning method devoted to vector space can be exploited in either its linear or kernel formulation. Linear Discriminant Analysis (LDA) and Partial Least Squares (PLS) are considered in this paper for their feasibility for our specific problem. We further investigate the conventional linear subspace based set modeling technique and cast it in a unified framework with our covariance matrix based modeling. The proposed method is evaluated on two tasks: face recognition and object categorization. Extensive experimental results show not only the superiority of our method over state-of-the-art ones in both accuracy and efficiency, but also its stability to two real challenges: noisy set data and varying set size.