Which approach is better, shapelets or HOLICs? From a signal perspective, the shapelets technique is better. It is designed to provide optimal weighting and return optimal signal-noise. Moreover, as described above, inversion of the PSF is a straightforward and well-designed process. In the absence of noise, the two techniques produce very similar results.

On the other hand, the HOLICs technique has several practical advantages, especially for large surveys. For one, the HOLICs code is typically much faster than shapelets. For an N pixel image, the HOLICs technique is an ${\cal O}(N)$ calculation, whereas the shapelets is ${\cal O}(N^2)$. Additionally, some values of $\beta$ produce very bad reconstructions, and hence, minimization of $\chi^2$ can be time-consuming and may not converge to a minimum.

As a simple test, we created images with brightness profiles of: $$I(r)\propto \exp[-(r/r_0)^n]$$ and though we found similar results for a reasonable range of exponents, the results presented below are for $n=1.5$. We have used a constant source ellipticity, typical of those observed in the field, $\varepsilon=0.2$, and had measurement errors which were dominated by sky brightness. In each case, we had no intrinsic arciness or skewness (that is, the flexion of the unlensed objects were zero), since 