Shapes and shears, stars and smears: Optimal measurements for weak lensing

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 and then use this geometric interpretation to analyze 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 point-spread function (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 expressed as Laguerre expansions, a two-dimensional generalization of the Edgeworth expansion, making the PSF correction and shape measurement relatively straightforward and computationally efficient. We also discuss sources of noise-induced bias in weak-lensing measurements selection biases, and centroid biases arising from noise recti cation and provide a solution for these and previously identified biases.