on the flexion webpage.

HOLICs Analysis {#sec:holics}
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Higher Order Moments
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Okura et al. (2006) recently related flexion directly to the 3rd moments of observed images. This is a significant extension of flexion, and very much along the lines of Goldberg & Natarajan’s (2002) original work which talked about “arciness” in terms of the measured octopole moments. Throughout our discussion, we will use the notation: $$Q_{ij}=\frac{1}{F}\int (\theta_i-\overline{\theta_i}) (\theta_j-\overline{\theta_j}) f({\bf
  \theta}) d^2{\bf \theta}$$ to refer, in this case, to the unweighted quadrupole moments, with all higher moments being defined by exact analogy. In this context, $F$ refers to the unweighted integrated flux.

They define the complex terms: $$\zeta\equiv\frac{(Q_{111}+Q_{122})+i(Q_{112}+Q_{222})}{\xi}$$ and $$\delta\equiv\frac{(Q_{111}-3Q_{122})+i(3Q_{112}-Q_{222})}{\xi}$$ where $$\xi\equiv Q_{1111}+2Q_{1122}+Q_{2222}\ .$$ These terms are collectively referred to as HOLICs.

If a galaxy is otherwise perfectly circular (i.e. no ellipticity), and in the absence of noise, then the HOLICs may be directly related to estimators of the flexion (subject to an unknown bias of $1-\kappa$). Namely: $${\cal F}\simeq \frac{4 \zeta\xi}{9\xi-6(Q_{11}^2+Q_{22}^2)}
\label{eq:skewness}$$ $${\cal G}\simeq \frac{4 \delta}{3}
\label{eq:arciness}$$ where the latter term in the denominator of ${\cal F}$ does not appear in the Okura et al. analysis. Bacon and Goldberg (2005) show that a flexion induces a shift in the 