x}/\beta)\ ,$$ and $\alpha$, $\beta$ and $\gamma$ are the characteristic scales of $f^{(0)}$, $P$ and $f$, respectively. We may then define a PSF convolution matrix as: $$P_{nmn'm'}\equiv\sum_{n''m''} C_{nmn'm'n''m''}P_{n''m''}\ .$$ If only low order terms in the convolution matrix are included, it may be inverted to perform a deconvolution via: $$f^{(0)}_{nm}=\sum_{n'm'} (P^{-1})_{nmn'm'}f_{n'm'}\ .$$ This provides a good estimate of the low order coefficients, but high order information is lost. An alternative inversion scheme involves fitting the observed galaxy coefficients using a $\chi^2$ minimization scheme. Refregier and Bacon (2003) note that the $\chi^2$ scheme may be more robust numerically, and can take full account of variations in the noise characteristics across an image (although it is strictly only valid in the case of Gaussian noise). It is this scheme that is implemented in the shapelets IDL software.

### Flexion Inversion

If the shapelet coefficients are statistically independent (as they will be in the absence of an explicit PSF deconvolution), formal inversion of the flexion operator is quite straightforward. Under these circumstances, we also have the benefit that the measurement error for each moment is identical (see Refregier 2003 for discussion).

Noting that, in most galaxies, the coefficients corresponding to the $n+m=$even moments 