# American Institute of Mathematical Sciences

February  2011, 5(1): 115-136. doi: 10.3934/ipi.2011.5.115

## Is SIFT scale invariant?

 1 CMLA, ENS Cachan, 61 avenue du Président Wilson, 94235 Cachan Cedex, France 2 CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France

Received  October 2010 Revised  November 2010 Published  February 2011

This note is devoted to a mathematical exploration of whether Lowe's Scale-Invariant Feature Transform (SIFT)[21], a very successful image matching method, is similarity 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, its scale invariance is almost perfect. The mathematical arguments are given under the assumption that the Gaussian smoothing performed by SIFT gives an aliasing free sampling of the image evolution. The validity of this main assumption is confirmed by a rigorous experimental procedure, and by a mathematical proof. These results explain why SIFT outperforms all other image feature extraction methods when it comes to scale invariance.
Citation: Jean-Michel Morel, Guoshen Yu. Is SIFT scale invariant?. Inverse Problems & Imaging, 2011, 5 (1) : 115-136. doi: 10.3934/ipi.2011.5.115
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