  Upcoming ground-based cosmic microwave background experiments will provide
CMB maps with high sensitivity and resolution that can be used for high
fidelity lensing reconstruction. However, the sky coverage will be incomplete
and the noise highly anisotropic, so optimized estimators are required to
extract the most information from the maps. We focus on quadratic-estimator
based lensing reconstruction methods that are fast to implement, and compare
new more-optimally filtered estimators with various estimators that have
previously been used in the literature. Input CMB maps can be optimally
inverse-signal-plus-noise filtered using conjugate gradient (or other)
techniques to account for the noise anisotropy. However, lensing
reconstructions from these filtered input maps have an anisotropic response to
the lensing signal and are difficult to interpret directly. We describe a
second-stage filtering of the lensing maps and analytic response model that can
be used to construct lensing power spectrum estimates that account for the
anisotropic response and noise inhomogeneity in an approximately optimal way
while remaining fast to compute. We compare results for simulations of upcoming
Simons Observatory and CMB Stage-4 experiments to show the robustness of the
more optimal lensing reconstruction pipeline and quantify the improvement
compared to less optimal estimators. We find a substantial improvement in
reconstructed lensing power variance between optimal anisotropic and isotropic
filtering of CMB maps, and up to 30% improvement in variance by using the
additional filtering step on the reconstruction potential map. Our approximate
analytic response model is unbiased to within a small percent-level additional
Monte Carlo correction.
