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In music theory, an octave is an interval in frequency, from a frequency $f$ to frequency $2f$. For example "an octave higher" means "twice the frequency". Expressed as wavelength inversely proportional to frequency, $\lambda \propto \frac{1}{f}$, an octave would be the interval from a $\lambda$ to $\frac{1}{2}\lambda$. In the SIFT paper'...

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Actually, the purpose of all this is to approximate a Laplacian of Gaussian! This computation is part of the corner detection of SIFT. You can find corners by examining extrema of the Laplacian of Gaussians (2nd order derivative). You use Gaussians for denoising, and a Laplacian to find inflection points. However, it is classical to not deal directly with ...

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As a starter, the 2014 IPOL paper Anatomy of the SIFT Method by Ives Rey Otero and Mauricio Delbracio provides a nice description and decryption of the SIFT method, with step-by-step pseudo-code, caveat and additional C code. SIFT was meant to be robust to translation, rotation and scaling/zoom, and also to mild noise/blur, contrast variations. For this, ...

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