We develop and test a method for measuring the gravitational lensing-induced distortion of faint background galaxies. We first describe how we locate the galaxies and measure two-component "polarization" or ellipticity statistic e_α whose expectation value should be proportional to the gravitational shear γ_α. We then show that an anisotropic instrumental point-spread function (PSF) perturbs the polarization by δe_α = P_(αβ)~sP_β, where p_α is a measure of the PSF anisotropy and P_(αβ)~s is the "linearized smear polarizability tensor." By estimating P_(αβ)~s for each object we can determine p_α from the foreground stars and apply a correction - P_(αβ)~sP_β to the galaxies. We test this procedure using deep high-resolution images from the Hubble Space Telescope (HST) which are smeared with an anisotropic PSF and then have noise added to simulate ground-based observations. We find that the procedure works very well. A similar analysis yields a linear shear polarizability tensor P_(αβ)~γ which describes the response to a gravitational shear. This calibrates the polarization-shear relation, but only for galaxies which are well resolved. To empirically calibrate the effect of seeing on the smaller galaxies we artificially stretch HST images to simulate lensing and then degrade them as before. These experiments provide a rigorous and exacting test of the method under realistic conditions. They show that it is possible to remove the effect of instrumental PSF anisotropy and that the method provides an efficient and quantitative measurement of the gravitational shear.
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