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Several descriptors have been proposed in the past for 3D shape analysis, yet none of them achieves best performance on all shape classes. In this paper we propose a novel method for 3D shape analysis using the covariance matrices of the descriptors rather than the descriptors themselves. Covariance matrices enable efficient fusion of different types of features and modalities. They capture, using the same representation, not only the geometric and the spatial properties of a shape region but also the correlation of these properties within the region. Covariance matrices, however, lie on the manifold of Symmetric Positive Definite (SPD) tensors, a special type of Riemannian manifolds, which makes comparison and clustering of such matrices challenging. In this paper we study covariance matrices in their native space and make use of geodesic distances on the manifold as a dissimilarity measure. We demonstrate the performance of this metric on 3D face matching and recognition tasks. We then generalize the Bag of Features paradigm, originally designed in Euclidean spaces, to the Riemannian manifold of SPD matrices. We propose a new clustering procedure that takes into account the geometry of the Riemannian manifold. We evaluate the performance of the proposed Bag of Covariance Matrices framework on 3D shape matching and retrieval applications and demonstrate its superiority compared to descriptor-based techniques.
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