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We present Spin-SILC, a new foreground component separation method thataccurately extracts the cosmic microwave background (CMB) polarisation $E$ and$B$ modes from raw multifrequency Stokes $Q$ and $U$ measurements of themicrowave sky. Spin-SILC is an internal linear combination method that usesspin wavelets to analyse the spin-2 polarisation signal $P = Q + iU$. Thewavelets are additionally directional (non-axisymmetric). This allows differentmorphologies of signals to be separated and therefore the cleaning algorithm islocalised using an additional domain of information. The advantage of spinwavelets over standard scalar wavelets is to simultaneously andself-consistently probe scales and directions in the polarisation signal $P = Q+ iU$ and in the underlying $E$ and $B$ modes, therefore providing the abilityto perform component separation and $E$-$B$ decomposition concurrently for thefirst time. We test Spin-SILC on full-mission Planck simulations and data andshow the capacity to correctly recover the underlying cosmological $E$ and $B$modes. We also demonstrate a strong consistency of our CMB maps with thosederived from existing component separation methods. Spin-SILC can be combinedwith the pseudo- and pure $E$-$B$ spin wavelet estimators presented in acompanion paper to reliably extract the cosmological signal in the presence ofcomplicated sky cuts and noise. Therefore, it will provide acomputationally-efficient method to accurately extract the CMB $E$ and $B$modes for future polarisation experiments.

Spin-SILC: CMB polarisation component separation with spin wavelets