I came across a problem working with image moments [1]. It is stated that
$\eta_{ij} = \frac{\mu_{ji}}{\mu_{00}^{k}}$
where $k = 1 + \frac{i+j}{2}$
is scale invariant.
However, if I try to reproduce this, it does not appear scale invariant at all.
Consider a simple example:
In a binary image, we calculate $\eta_{20}$ of a 2x2 block of 4 pixels:
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$\mu_{20} = 0.5^2 + 0.5^2 + (-0.5)^2 + (-0.5)^2 = 4 \cdot 0.25 = 1$
$\mu_{00} = 4$
$k = 1 + \frac{2+0}{2} = 1+1 = 2$
$\eta_{20} = \frac{1}{4^2} = \frac{1}{16} = 0.0625$
Now, let's scale this block by the factor two:
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$\mu_{20} = 4 \cdot 1.5^2 + 4 \cdot 0.5^2 + 4 \cdot (-0.5)^2 + 4 \cdot (-1.5)^2 = 8 \cdot 2.25 + 8 \cdot 0.25 = 18 + 2 = 20$
$\mu_{00} = 16$
$k = 1 + \frac{2+0}{2} = 1+1 = 2$
$\eta_{20} = \frac{20}{16^2} = \frac{20}{256} = 0.078125$
Why do we have a different result after scaling the object if $\eta_{ij}$ is supposedly scale invariant? Is there any formal proof of the scale invariance of $\eta$?