will be much larger than the odd moments (and, indeed, upon random rotations, the latter will necessarily average to zero) we can dramatically simplify equation (\[eq:chi2\]). First, we define the susceptibility of each odd moment as: $$\Delta f_{n' m',ij}=\hat{S}^{(2)}_{ij}f_{nm}$$ where $f_{nm}$ represents all of the “even” coefficients, and $n'm'$ represents all of the odd coefficients. Thus, we wish to solve for the relation: $$\sum_{n'm'}(f_{n'm'}-\gamma_{i,j}\Delta f_{n' m',ij})^2=min.$$ where the first term is taken directly from measurement. Taking the derivatives and rearranging, we find: $$\sum_{n'm'} f_{n'm'}\Delta f_{n'm',ij}=\gamma_{i,j}\sum_{n'm'}(\Delta
f_{n'm',ij}\Delta f_{n'm',i'j'})$$ which can readily be inverted to solve for $\gamma_{i,j}$.

In practice, however, there are a number of issues which must be considered. First, if the PSF or pixel scale are relatively large compared to the minimum resolution scale of an image then many of the high-order moments returned by shapelets decomposition will, in fact, not have any information. Thus, the above inversion will yield a systematic underestimate of the true image flexion. Above, we describe a truncation which minimizes this effect.

While the flexion inversion is, at its core, linear algebra, it involves an enormous number of terms. We have thus provided an inversion code for shapelets estimates of flexion along with examples 