on the (unlensed) quadrupole moments of the light distribution. This is due to a relatively subtle effect not present in shear analysis. Since the flexion signal is asymmetric, the center of brightness in the image plane will no longer necessarily correspond to the center of brightness in the source plane, and since the shapelet decomposition is performed around the center of light, we need to correct for this.

Most important, though, is the fact that second order lensing terms yield transfer of power between indices with $|\Delta n|+|\Delta m|=$ 1 or 3. To second order, then, a lensed image can be expressed as: $$f'=(1+\kappa \hat{K}+\gamma_j \hat{S}_j
+\hat{S}_{ij}^{(2)}\gamma_{i,j})f \ .$$

Flexion analysis assumes (as does shear analysis) that the intrinsic flexion is random, and thus all “odd” (defined as n+m) moments are expected to be zero. Thus, from a set of shapelet coefficients, a best estimate for the flexion signal may be found via $\chi^2$ minimization, where: $$\chi^2\equiv\left[\mu_{n_1m_1}-f_{n_1m_1}+(\gamma_i \hat{S}_i^{(1)}+\gamma_{i,j}S_{ij}^{(2)})
  \overline{f}_{n'_1m'_1}\right]V^{-1}_{n_1m_1 n_2
  m_2}
 \left[(\mu_{n_2m_2}-f_{n_2m_2}+(\gamma_i \hat{S}_i^{(1)}+\gamma_{i,j}S_{ij}^{(2)})
  \overline{f}_{n'_2m'_2}\right],
\label{eq:chi2}$$ $V_{n_1m_1 n_2m_2}$ is the covariance matrix of the shapelet coefficients, and $\mu_{nm}$ is the “unlensed” estimate of a shapelet coefficient. For odd modes, this is zero. For even modes, the relative effect of shear is 