Center of Image - Brightness Distribution

[BOAS]

Hello,

I am reading a review on weak gravitational lensing (https://arxiv.org/pdf/astro-ph/0509252.pdf) and they define the center of an image as follows:

$$\vec \theta_c = \frac{\int d^2 \theta I(\vec \theta) q_I [I(\vec \theta)] \vec \theta}{\int d^2 \theta I(\theta) q_I[I(\vec \theta)]}$$

where $I(\vec \theta)$ is the brightness distribution of an image isolated in the sky and $q_I(I)$ is some weight function.

I am having some trouble seeing that this does indeed define the center of an image and was hoping someone could help me see it.

 

[haruspex]

trouble seeing that this does indeed define the center of an image

It looks like a standard weighted average formula, $\bar x=\frac{\int f(x)x.dx}{\int f(x).dx}$, where f(x) is the weight function. Just the same as finding the centroid of a lamina.
If your question is how they get that weighting function then I am hampered by not knowing how to interpret d²θ.

 

[BOAS]

It looks like a standard weighted average formula, $\bar x=\frac{\int f(x)x.dx}{\int f(x).dx}$, where f(x) is the weight function. Just the same as finding the centroid of a lamina.
If your question is how they get that weighting function then I am hampered by not knowing how to interpret d²θ.

Ah ha! Thank you, I was missing that.

$\vec \theta$ is an angular position on the flat approximation of the sky, I've attached an image to illustrate it. A common example in the literature is to use the heaviside step function as the weight function, which I think just defines a sharp cutoff of the image along an isophote.

 

[haruspex]

Ah ha! Thank you, I was missing that.

$\vec \theta$ is an angular position on the flat approximation of the sky, I've attached an image to illustrate it. A common example in the literature is to use the heaviside step function as the weight function, which I think just defines a sharp cutoff of the image along an isophote.

Ok, but can you shed any light on the d²θ? I would have understood dθ as just the usual integration notation. I suspect it has something to do with the fact that we really want to integrate over the annulus rdrdφ, where r is proportional to θ and φ runs from 0 to 2π, but that would give something like θdθ, not d²θ.