  We present the theoretical and analytical bases of optimal techniques to
measure weak gravitational shear from images of galaxies. We first characterize
the geometric space of shears and ellipticity, then use this geometric
interpretation to analyse images. The steps of this analysis include:
measurement of object shapes on images, combining measurements of a given
galaxy on different images, estimating the underlying shear from an ensemble of
galaxy shapes, and compensating for the systematic effects of image distortion,
bias from PSF asymmetries, and `"dilution" of the signal by the seeing. These
methods minimize the ellipticity measurement noise, provide calculable shear
uncertainty estimates, and allow removal of systematic contamination by PSF
effects to arbitrary precision. Galaxy images and PSFs are decomposed into a
family of orthogonal 2d Gaussian-based functions, making the PSF correction and
shape measurement relatively straightforward and computationally efficient. We
also discuss sources of noise-induced bias in weak lensing measurements and
provide a solution for these and previously identified biases.
