  We present a new shear estimator for weak lensing observations which properly
accounts for the effects of a realistic point spread function (PSF). Images of
faint galaxies are subject to gravitational shearing followed by smearing with
the instrumental and/or atmospheric PSF. We construct a `finite resolution
shear operator' which when applied to an observed image has the same effect as
a gravitational shear applied prior to smearing. This operator allows one to
calibrate essentially any shear estimator. We then specialize to the case of
weighted second moment shear estimators. We compute the shear polarizability
which gives the response of an individual galaxy's polarization to a
gravitational shear. We then compute the response of the population of
galaxies, and thereby construct an optimal weighting scheme for combining shear
estimates from galaxies of various shapes, luminosities and sizes. We define a
figure of merit --- an inverse shear variance per unit solid angle --- which
characterizes the quality of image data for shear measurement. The new method
is tested with simulated image data. We discuss the correction for anisotropy
of the PSF and propose a new technique involving measuring shapes from images
which have been convolved with a re-circularizing PSF. We draw attention to a
hitherto ignored noise related bias and show how this can be analyzed and
corrected for. The analysis here draws heavily on the properties of real PSF's
and we include as an appendix a brief review, highlighting those aspects which
are relevant for weak lensing.

Astrophysics