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This paper introduces a generalization of the Fisher vectors to the Riemannian manifold. The proposed descriptors, called Riemannian Fisher vectors, are defined first, based on the mixture model of Riemannian Gaussian distributions. Next, their expressions are derived and they are applied in the context of texture image classification. The results are compared to those given by the recently proposed algorithms, bag of Riemannian words and R-VLAD. In addition, the most discriminant Riemannian Fisher vectors are identified.

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