details in an image. Structure on the individual pixel scale may simply represent noise.

From a practical perspective, our goal is to optimize selection of $\beta$, and the maximum coefficient index, $n_{max}$. Refregier (2003) suggests the following parameters: $$\begin{aligned}
  \beta\simeq\sqrt{\theta_{min}\theta_{max}}\nonumber\\
  n_{max}\simeq\frac{\theta_{max}}{\theta_{min}} -1\ ,\end{aligned}$$ where $\theta_{min}$ and $\theta_{max}$ represent the minimum and maximum scales of image structure, respectively.

R. Massey (private communication) has found that rather than performing overlap integrals to solve for the shapelet coefficients (as was done, for example, in the analysis of Goldberg & Bacon 2005), the ideal approach is to do a $\chi^2$ minimization of the reconstructed image with the original postage stamp. This may seem complicated, and it is. Fortunately, a shapelets package is available in IDL at the shapelets webpage.[^2]

For our sample of co-added, background-subtracted, HST ACS images of Abell 1689, we find that $\theta_{min} = 0.4$ pixels and $\theta_{max}
= 1.8\sqrt{a^2+b^2}$ give good shapelet reconstructions, where $a$ and $b$ are the semi-major and semi-minor axes of the galaxy as measured by SExtractor. However, it is important to note that these parameters are somewhat dependent on the noise level in the images.

For a sky-limited sample, we have found that the optimal choice of 