normal deprojection operator: $${\bf A}=
\left(
\begin{array}{cc}
1-\kappa-\gamma_1 & -\gamma_2 \\
-\gamma_2 & 1-\kappa+\gamma_1
\end{array}
\right)\ ,
\label{eq:Adef}$$ and thus, the second term on right in equation (\[eq:transform\]) represents the flexion signal. ${\bf D}$ may be written as: $$\begin{aligned}
D_{ij1}&=&\left(
\begin{array}{cc}
-2\gamma_{1,1}-\gamma_{2,2} & \ \ -\gamma_{2,1} \\
-\gamma_{2,1} & \ \ -\gamma_{2,2}
\end{array}
\right), \\ \nonumber
D_{ij2}&=& \left(
\begin{array}{cc}
-\gamma_{2,1} & \ \ -\gamma_{2,2}\\
-\gamma_{2,2} & \ \ 2\gamma_{1,2}-\gamma_{2,1}
\end{array}
\right).\end{aligned}$$

These distortions create asymmetries in a lensed image – a skewness and a bending, depending on the values of individual coefficients. Irwin & Shmakova (2005;2006) describe a similar lensing analysis technique in which the elements of ${\bf D}$ are referred to as “Catenoids” and “Displacements.”

BGRT describe an inversion whereby one can estimate the individual components, and thus measure two “flexions”: $${\cal F}\equiv \partial^\ast \gamma$$ $${\cal G}=\partial \gamma$$ where $\partial$ is the complex derivative operator: $$\partial = \partial_1+i\partial_2\ .$$ Figure \[fg:shapes\] is reproduced from BGRT and shows the effect of a first or second flexion on a circular source.

An object with first flexion, ${\cal F}$, appears skewed, while an object with second flexion, ${\cal G}$, appears arced, especially if the image has an induced shear as well. The 