How does REGIONPROPS define "normalized second central moment"?

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Matt J 2013-12-13

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https://ww2.mathworks.cn/matlabcentral/answers/109708-how-does-regionprops-define-normalized-second-central-moment

评论： Shinuo Weng 2018-4-26

From the documentation (R2013a) on REGIONPROPS

'MajorAxisLength' — Scalar specifying the length (in pixels) of the major axis of the ellipsethat has the same normalized second central moments as the region.

However according to the definition of normalized central moments given here, the normalized second central moment is always 1 with identity. So, clearly REGIONPROPS uses a different definition. What is the definition that REGIONPROPS uses?

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Shinuo Weng 2018-4-26

the normalized second central moment is the second central moment divided by the area.

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Answer 1

Image Analyst 2013-12-13

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https://ww2.mathworks.cn/matlabcentral/answers/109708-how-does-regionprops-define-normalized-second-central-moment#answer_118326

I'm not seeing how it says that the normalized second central moment is always 1.

I'm not sure what they use, but if I were going to do it, I'd use these formulas: https://en.wikipedia.org/wiki/Image_moment#Central_moments. Note that f(x,y) always equals 1 for a binary blob.

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Matt J 2013-12-13

I'm not seeing how it says that the normalized second central moment is always 1.

The expression they give E((x-mu)^n/sigma^n is identically 1 for n=2. Also, there is a table at the link some lines above showing that second standardized moments (synonymous they say with normalized moments) are 1.

but if I were going to do it, I'd use these formulas:

Yes, that's what I'd have expected, too, but I don't see how those moments are "normalized".

Cris Luengo 2017-2-8

This is an old question, but just for reference value:

In that link to Wikipedia page on image moments, look at mu', not mu. mu' is normalized by the "object weight", the 0th order moment. Those are used to compute axes lengths (4 * sqrt(lambda)).

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