  We present new methods for mapping the curl-free (E-mode) and divergence-free
(B-mode) components of spin 2 signals using spin directional wavelets. Our
methods are equally applicable to measurements of the polarisation of the
cosmic microwave background (CMB) and the shear of galaxy shapes due to weak
gravitational lensing. We derive pseudo and pure wavelet estimators, where E-B
mixing arising due to incomplete sky coverage is suppressed in wavelet space
using scale- and orientation-dependent masking and weighting schemes. In the
case of the pure estimator, ambiguous modes (which have vanishing curl and
divergence simultaneously on the incomplete sky) are also cancelled. On
simulations, we demonstrate the improvement (i.e., reduction in leakage)
provided by our wavelet space estimators over standard harmonic space
approaches. Our new methods can be directly interfaced in a coherent and
computationally-efficient manner with component separation or feature
extraction techniques that also exploit wavelets.
