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Reconstruction of gravitational lensing effects in the CMB from current andupcoming surveys is still dominated by temperature anisotropies. Extragalacticforegrounds in temperature maps can induce significant biases in the lensingpower spectrum obtained with the standard quadratic estimators. Techniques suchas masking cannot remove these foregrounds fully, and the residuals can stilllead to large biases if unaccounted for. In this paper, we study the"shear-only" estimator, an example of a class of geometric methods thatsuppress extragalactic foreground contamination while making only minimalassumptions about foreground properties. The shear-only estimator has only beenformulated in the flat-sky limit and so is not easily applied to wide surveys.Here, we derive the full-sky version of the shear-only estimator and itsgeneralisation to an $m=2$ multipole estimator that has improved performancefor lensing reconstruction on smaller scales. The multipole estimator isgenerally not separable, and so is expensive to compute. We explore separableapproximations based on a singular-value decomposition, which allow efficientevaluation of the estimator with real-space methods. Finally, we apply theseestimators to simulations that include extragalactic foregrounds and verifytheir efficacy in suppressing foreground biases.

CMB lensing with shear-only reconstruction on the full sky