$\delta$ produces contributions of $6^{th}$ moments in ${\cal M}$: $$\Delta
\hat{Q}_{ijk}=-\frac{1}{2}D_{lmn}\frac{\hat{Q}_{ijklmn}}{\sigma_W^2}\ ,$$ which, in turn, must be corrected for.

We may thus say that: $$\hat{\cal M}={\cal M}(\hat{Q}_{ij},...)+\Delta{\cal M}\ ,$$ where the latter expression can also be found in Appendix A.

PSF Correction in HOLICs {#sec:psfcorrection}
------------------------

As with our discussion of shapelets, above, we must also consider PSF deconvolution in our HOLICs pipeline. We define the PSF function in equation (\[eq:PSFdef\]), and all unweighted moments of the PSF are denoted by $P_{ij}$, etc. In principle, because of the higher signal-noise of the PSF, the unweighted moments are easier to estimate than the moments of the detected image. While we argued, above, that the induced flexion from a PSF is likely to be small, it is still the case, as with shear, that the PSF will reduce the measured flexion. Let’s first consider the case in which we were able to measure the unweighted moments of both the PSF and the observed image. It is straightforward to show that: $$Q_{ij}=Q_{ij}^{(0)}+P_{ij}\ .
\label{eq:quadrel}$$

Thus may be computed via the relation: $$Q_{ij}=\frac{1}{F}\int \theta_i \theta_j f^{(0)}(\theta')P(\theta-\theta')
d^2\theta d^2\theta'$$ Making the substitution, $$\theta''=\theta-\theta'$$ yields $$Q_{ij}=\frac{1}{F}\int (\theta_i'\theta_j'+\theta_i''\theta_j''+
\theta_i'\theta_j''+ \theta_i''\theta_j')f^{(0)}(\theta')P(\theta'')
d^2\theta'd^2\theta''$$ It is straightforward to show that 