  Weak lensing observations measure the shear field and hence the gradient of
the dimensionless surface density $\kappa$. We present several new algorithms
to recover $\kappa$ {}from shear estimates on a finite region and compare how
they perform with realistically noisy data. The reconstruction methods studied
here are divided into 2 classes: direct reconstruction and regularized
inversion. Direct reconstruction techniques express $\kappa$ as a 2D integral
of the shear field. This yields an estimator for $\kappa$ as a discrete sum
over background galaxy ellipticities which is straightforward to implement, and
allows a simple estimate of the noise. We study 3 types of direct
reconstruction methods: 1) $\kappa$-estimators that measure the surface density
at any given target point relative to the mean value in a reference region 2) a
method that explicitly attempts to minimize the rotational part of $\nabla
\kappa$ that is due to noise and 3) a novel, exact Fourier-space inverse
gradient operator. We also develop two `regularized maximum likelihood'
methods, one of which employs the conventional discrete Laplacian operator as a
regularizer and the other uses regularization of all components in Fourier
space. We compare the performance of all the estimators by means of simulations
and noise power analysis. A general feature of these unbiased methods is an
enhancement of the low frequency power which, for some of the methods, can be
quite severe. We find the best performance is provided by the maximum
likelihood method with Fourier space regularization although some of the other
methods perform almost as well.
