  We have developed a method for measuring higher-order weak lensing
distortions of faint background galaxies, namely the weak gravitational
flexion, by fully extending the Kaiser, Squires & Broadhurst method to include
higher-order lensing image characteristics (HOLICs) introduced by Okura,
Umetsu, & Futamase. We take into account explicitly the weight function in
calculations of noisy shape moments and the effect of higher-order PSF
anisotropy, as well as isotropic PSF smearing. Our HOLICs formalism allows
accurate measurements of flexion from practical observational data in the
presence of non-circular, anisotropic PSF. We test our method using mock
observations of simulated galaxy images and actual, ground-based
  Subaru observations of the massive galaxy cluster A1689 ($z=0.183$). From the
high-precision measurements of spin-1 first flexion, we obtain a
high-resolution mass map in the central region of A1689. The reconstructed mass
map shows a bimodal feature in the central $4'\times 4'$ region of the cluster.
The major, pronounced peak is associated with the brightest cluster galaxy and
central cluster members, while the secondary mass peak is associated with a
local concentration of bright galaxies. The refined, high-resolution mass map
of A1689 demonstrates the power of the generalized weak lensing analysis
techniques for quantitative and accurate measurements of the weak gravitational
lensing signal.
