Is SIFT scale invariant?

This note is devoted to a mathematicalexploration of whether Lowe's Scale-Invariant Feature Transform (SIFT)21, a very successful image matching method, issimilarity invariant as claimed. It is proved that the method is scale invariant only if the initial image blurs are exactly guessed. Yet, even a large error on the initial blur is quickly attenuated by this multiscale method, when the scale of analysis increases. In consequence, itsscale invariance is almost perfect. The mathematical arguments aregiven under the assumption that the Gaussian smoothing performed by SIFT givesan aliasing free sampling of the image evolution. The validity of this main assumption isconfirmed by a rigorous experimental procedure, and by a mathematicalproof. These results explain why SIFT outperforms all other imagefeature extraction methods when it comes to scale invariance.