Is SIFT scale invariant?

Pages: 115 - 136, Volume 5, Issue 1, February 2011

doi:10.3934/ipi.2011.5.115       Abstract        References        Full Text (3458.6K)       Related Articles

Jean-Michel Morel - CMLA, ENS Cachan, 61 avenue du Président Wilson, 94235 Cachan Cedex, France (email)
Guoshen Yu - CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France (email)

Abstract: 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.

Keywords:  SIFT, scale invariance, Shannon interpolation, Gaussian blur, sampling theory, aliasing.
Mathematics Subject Classification:  Primary: 68T10, 68T40; Secondary: 68T45, 93C85.

Received: October 2010;      Revised: November 2010;      Published: February 2011.

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