see also a nice review by Bartelmann & Schneider, 2001) developed perhaps the most comprehensive approach to dealing with the second moments (the ellipticity) with noisy observing, and with a (potentially anisotropic) PSF.

Our approach is similar. We have only worked with a Gaussian window thus far, but the approach is generalizable for any circularly symmetric weighting. We thus define a window function: $$W({\bf
  \theta})=\frac{1}{2\pi\sigma_{W}}\exp\left(-\frac{\theta_1^2+\theta_2^2}{2\sigma_W^2}\right)$$ where the origin is taken to be the center of light, and the integral is normalized to unity. Further, we define the weighted moments as, for example: $$\hat{Q}_{11}=\frac{1}{\hat{F}}\int (\theta_1-\overline{\theta_1})^2 f({\bf
  \theta}) W({\bf \theta}) d^2{\bf \theta}\ .$$ We can thus redefine all HOLICs and moments similarly. We have found through experimentation (see below) that for a sky noise limited source, a reasonable value of $\sigma_W$ is 1.5 times the half-light radius.

If we were to simply replace all elements in ${\cal M}$, $\zeta$, and $\delta$ from equation (\[eq:Fsolve\]) with their weighted counterparts we would not get an unbiased estimate of the flexion. There are two corrections. One has to do with the fact that centroid shift will differ from the unweighted case to the weighted case. Consider an extreme scenario in which the window width is 