  Weak gravitational lensing provides a unique method to directly measure the
distribution of mass in the universe. Because the distortions induced by
lensing in the shape of background galaxies are small, the measurement of weak
lensing requires high precision. Here, we present a new method for obtaining
reliable weak shear measurements. It is based on the Shapelet basis function
formalism of Refregier (2001), in which galaxy images are decomposed into
several shape components, each providing independent estimates of the local
shear. The formalism affords an efficient modelling and deconvolution of the
Point Spread Function. Using the remarkable properties of Shapelets under
distortions, we construct a simple, minimum variance estimator for the shear.
We describe how we implement the method in practice, and test the method using
realistic simulated images. We find our method to be stable and reliable for
conditions analogous to ground-based surveys. Compared to earlier methods, our
method has the advantages of being accurate, linear, mathematically
well-defined, and optimally sensitive, since it uses the full shape information
available for each galaxy.
