The importance of wild video based image set recognition is becoming monotonically increasing. However, the contents of these collected videos are often complicated, and how to efficiently perform set modeling and feature extraction is a big challenge for set-based classification algorithms. In recent years, some proposed image set classification methods have made a considerable advance by modeling the original image set with covariance matrix, linear subspace, or Gaussian distribution. As a matter of fact, most of them just adopt a single geometric model to describe each given image set, which may lose some other useful information for classification. To tackle this problem, we propose a novel algorithm to model each image set from a multi-geometric perspective. Specifically, the covariance matrix, linear subspace, and Gaussian distribution are applied for set representation simultaneously. In order to fuse these multiple heterogeneous Riemannian manifoldvalued features, the well-equipped Riemannian kernel functions are first utilized to map them into high dimensional Hilbert spaces. Then, a multi-kernel metric learning framework is devised to embed the learned hybrid kernels into a lower dimensional common subspace for classification. We conduct experiments on four widely used datasets corresponding to four different classification tasks: video-based face recognition, set-based object categorization, video-based emotion recognition, and dynamic scene classification, to evaluate the classification performance of the proposed algorithm. Extensive experimental results justify its superiority over the state-of-the-art.
Manifold Dimensionality Reduction Based Image Set Classification: To circumvent the above problem, some algorithms that jointly perform linear mapping and metric learning directly on the original Riemannian manifold have been suggested recently @cite_50 @cite_19 @cite_53 , and therefore a discriminative lower-dimensional one can be yielded. Harandi . @cite_50 produce a lower-dimensional SPD manifold with an orthogonal mapping obtained by devising a discriminative metric learning framework with respect to the original high-dimensional data. To simplify the computational complexity, Huang . @cite_53 put forward a novel Log-Euclidean metric learning algorithm to form a desirable SPD manifold by directly embedding the tangent space of original SPD manifold into a lower-dimensional one. Similarly, Huang . @cite_19 try to learn a lower-dimensional and more discriminative Grassmannian-valued feature representations for the original high dimensional Grassmann manifold under a devised projection metric learning framework. Thanks to the advantage of fully considering the manifold geometry, the above algorithms show good classification performance. Yet, they also have an inherent design flaw, that is the mapping which is defined and learned on the non-linear Riemannian geometry is linear, which seems to be unreasonable.
@cite_19 @cite_53 @cite_50
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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. 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. In video based face recognition, great success has been made by representing videos as linear subspaces, which typically lie in a special type of non-Euclidean space known as Grassmann manifold. To leverage the kernel-based methods developed for Euclidean space, several recent methods have been proposed to embed the Grassmann manifold into a high dimensional Hilbert space by exploiting the well established Project Metric, which can approximate the Riemannian geometry of Grassmann manifold. Nevertheless, they inevitably introduce the drawbacks from traditional kernel-based methods such as implicit map and high computational cost to the Grassmann manifold. To overcome such limitations, we propose a novel method to learn the Projection Metric directly on Grassmann manifold rather than in Hilbert space. From the perspective of manifold learning, our method can be regarded as performing a geometry-aware dimensionality reduction from the original Grassmann manifold to a lower-dimensional, more discriminative Grassmann manifold where more favorable classification can be achieved. Experiments on several real-world video face datasets demonstrate that the proposed method yields competitive performance compared with the state-of-the-art algorithms. The manifold of Symmetric Positive Definite (SPD) matrices has been successfully used for data representation in image set classification. By endowing the SPD manifold with Log-Euclidean Metric, existing methods typically work on vector-forms of SPD matrix logarithms. This however not only inevitably distorts the geometrical structure of the space of SPD matrix logarithms but also brings low efficiency especially when the dimensionality of SPD matrix is high. To overcome this limitation, we propose a novel metric learning approach to work directly on logarithms of SPD matrices. Specifically, our method aims to learn a tangent map that can directly transform the matrix logarithms from the original tangent space to a new tangent space of more discriminability. Under the tangent map framework, the novel metric learning can then be formulated as an optimization problem of seeking a Mahalanobis-like matrix, which can take the advantage of traditional metric learning techniques. Extensive evaluations on several image set classification tasks demonstrate the effectiveness of our proposed metric learning method.