  Weak lensing provides a direct way of mapping the density distribution in the
universe. To reconstruct the density field from the shear catalog, an important
step is to build the shear field from the shear catalog, which can be quite
nontrivial due to the inhomogeneity of the background galaxy distribution and
the shape noise. We propose the PDF-Folding method as a statistically optimal
way of reconstructing the shear field. It is an extention of the PDF-SYM
method, which is previously designed for optimizing the stacked shear signal as
well as the shear-shear correlation for the Fourier\_Quad shear estimators.
PDF-Folding does not require smoothing kernels as in traditional methods,
therefore it suffers less information loss on small scales, and avoids possible
biases due to the spatial variation of shear on the scale of the kernel. We
show with analytic reasoning as well as numerical examples that the new method
can reach the optimal signal-to-noise ratio on the reconstructed shear map
under general observing conditions, i.e., with inhomogeneous background
densities or masks. We also show the performance of the new method on real data
around foreground galaxy clusters.
