arbitrarily small and in which the unlensed image was circularly symmetric with a peak at the center. In that case, the centroid will essentially remain at the center (peak brightness) even if the unweighted moments shift.

Thus, compared to the unweighted moments, the centroid will shift: $$\Delta \overline\theta_l=\frac{\sum_{ijk}D_{ijk}\hat{Q}_{ijkl}}{\sigma_W^2}$$ where we have used the explicit fact that for a Gaussian: $$\frac{dW({\bf \theta})}{d\theta_i}=-\frac{\theta_i}{\sigma_W^2}W({\bf
    \theta})\ ,$$

The other correction has to do with the fact that though lensing preserves surface brightness, it does not preserve total flux. This is normally related by the Jacobian of the coordinate transformation. However, when considering a window function, we need consider that transformation explicitly: $$W(\beta)d^2\beta=\left|\frac{\partial \beta}{\partial
  \theta}\right|W(\beta)d^2\theta$$ as used by Okura et al. (2006), and where we have simply multiplied both sides by the factor $W(\beta)$. In this context, ${\bf \beta}$ refers to the image coordinate in the source plane. Ignoring the terms proportional to shear (which cannot be directly addressed by this method at any rate), we have the approximate relation: $$W(\beta)\simeq W(\theta)+\frac{1}{2}D_{ijk}\theta_i\theta_j
\frac{dW}{d\theta_k}$$ or, as we have already asserted: $$W(\beta)\simeq
W(\theta)-\frac{1}{2}D_{ijk}\frac{\theta_i\theta_j\theta_k}{\sigma_W^2}
W(\theta)\ .$$ Note that the latter term contains an odd number of position elements, and thus, coupling to the generating equations for $\zeta$ and 