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We propose a method to compute scale invariant features in omnidirectional images. We present a formulation based on Riemannian geometry for the definition of differential operators on non-Euclidian manifolds that correspond to the particular form of mirrors in omnidirectional imaging. These operators lead to a scale-space analysis that preserve the geometry of the visual information in omnidirectional images. We eventually build novel scale- invariant omniSIFT features inspired by the planar SIFT framework. We apply our generic solution to omnidirectional images captured with parabolic mirrors. Simple descriptors that uses omniSIFT characteristics offer promising performance in the case of image rotation or translation where visual features can be preserved due to the proper handling of the implicit image geometry.