first flexion has an $m=1$ rotational symmetry, and thus behaves like a vector. In particular, it is a direct tracer of the gradient of the convergence: $${\cal F}_1 {\bf i}+{\cal F}_2 {\bf j}=\nabla \kappa$$ where ${\cal F}_1$ is the real component and ${\cal F}_2$ is the imaginary part (as with the second flexion and, as the standard convention, with shear).

The second flexion has an $m=3$ rotational symmetry, though unlike the first flexion, it has no simple physical interpretation like that of the first flexion. It is, however, roughly proportional to the local derivative of the magnitude of the shear. A more complete discussion of flexion formalism can be found in BGRT.

Shapelets Decomposition {#sec:shapelets}
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Review of Shapelets
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Measurement of flexion ultimately requires very accurate knowledge of the distribution of light in an image. The shapelets (Refregier 2003; Refregier & Bacon 2003) method of image reconstruction decomposes an image into 2D Hermite polynomial bases: $$f({\bf \theta})=\sum_{n,m} {\cal B}_{nm}({\bf \theta}) f_{nm}\ .$$ This technique has a number of very natural advantages. In the absence of a PSF, all shapelet coefficients will have equal noise. Moreover, the basis set is quite localized (Hermite polynomials have a Gaussian smoothing filter), and 