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I am reading a paper on SIFT by Lowe.

In it, gradient magnitudes and orientations are calculated using these formulae:

enter image description here $$$$ I don't know why m(x,y) is calculated above. I heard that gradient magnitude is usually calculated like the following.

$$\sqrt{f_x ^2 + f_y^2} $$

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  • $\begingroup$ what is f_x and f_y in your case? they are gradients, right? try to compare it with terms in SIFT. $\endgroup$ Commented Jul 13, 2013 at 6:41

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Horizontal and vertical gradients are computed by taking neighbor pixel differences:

$$g_{x}=L(x+1,y)-L(x-1,y)\\g_{y}=L(x,y+1)-L(x,y-1)$$

Gradient magnitude is computed the same way as in your formula:

$$m(x,y)=\sqrt{g_x^2+g_y^{2}}$$

Replacing $g_{x}$ and $g_{y}$ with above will give you the original formula.

Gradients are usually computed by forward or central difference formula, but I think in this case the above is still accurate enough since the image $L$ is smoothed.

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  • $\begingroup$ I still have a little doubt, but thank you very much :) $\endgroup$
    – jakeoung
    Commented Jul 15, 2013 at 5:31
  • $\begingroup$ Okay. You can consult the Image gradient wiki article. There is some brief explanation and similar formulas (same meaning). $\endgroup$
    – Libor
    Commented Jul 15, 2013 at 5:36

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