In § \[sec:holics\], we discuss a new, conceptually simpler, form of flexion analysis developed by Okura et al. (2006), which uses moments, rather than basis functions to measure flexion. They call their technique Higher Order Lensing Image’s Characteristics, or HOLICs. We refine the HOLICs approach somewhat, and develop a KSB (Kaiser, Squires, & Broadhurst, 1995)-type approach using a Gaussian filter to perform an inversion, as well as describe a technique for PSF deconvolution. In § \[sec:simulate\], we discuss comparisons of the two techniques using simulated lenses and simulated PSFs. In § \[sec:measurement\], we compare shapelets and HOLICs inversions on HST images. Finally, in § \[sec:discuss\], we discuss the implications of this study.

In Appendix A, we also present the explicit HOLICs inversion matrix, so the reader can write his/her own code. He/she need not do so, however, as all codes discussed herein are available from the flexion webpage.[^1]

Flexion
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What is flexion? Conceptually, flexion represents local variability in the shear field which expresses itself as second-order distortions in the coordinate transformation between unlensed and lensed images: $$\beta_i\simeq A_{ij}\theta_j +\frac{1}{2}D_{ijk} \theta_j \theta_k,
\label{eq:transform}$$ with $$D_{ijk}=\partial_k A_{ij} \ ,$$ where $\partial_k$ is shorthand for $\partial/\partial x_k$. Here, ${\bf A}$ is the 