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I am trying to implement a computer vision algorithm, but I'm having a problem with some notation used in a article. They define an image as a set of RGB colors with an index $\mathbb{z} = (z_1, z_2, ..., z_n)$.

They then use this formula: $\langle \| z_p - z_q \|^2 \rangle$, with $\langle \cdot \rangle$ defined as the expectation operator over the whole image.

My question is, how can a color be passed to an expectation operator? As far as I understand it, an expectation operator needs a random variable. Somehow, I think I'm interpreting this wrong. Can anyone shed some light on the meaning of the $\langle \cdot \rangle$ operator?

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2 Answers 2

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It is possible that they mean empirical expectation, ie they don't really know the exact distribution of the pixels, but they define for instance $p_i=\frac{\#(z_j==i)}{N}$, where $N$ is the total number of pixels and $j\in[1,N]$, which yields the relative number of pixels valued $i$ in the entire image. Now you can define the empirical expectation as $\hat{E}[f(x)] = \sum_i p_i f(x)$ .

I do not know if this is the case here, but it occurs sometimes.

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I have found the answer:

It seems like the meaning of $\langle \| z_p - z_q \|^2 \rangle$ is the color variance measured of the whole image. I dont see their logic, but it seems they meant it like that.

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