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In paper I of this series (Yang et al. 2017, ApJ), we proposed an analyticalmethod of blind separation ({\bf ABS}) to extract the cosmic magnificationsignal in galaxy number distribution and reconstruct the weak lensing powerspectrum. Here we report a new version of the ABS method, with significantlyimproved performance. This version is characterized by a shift parameter$\mathcal{S}$, with the special case of $\mathcal{S}=0$ corresponding to theoriginal ABS method. We have tested this new version, compared to the previousone, and confirmed its supreme performance in all investigated situations.Therefore it supersedes the previous version. The proof of concept studiespresented in this paper demonstrate that it may enable surveys such as LSST andSKA to reconstruct the lensing power spectrum at $z\simeq 1$ with $1\%$accuracy. We will test with more realistic simulations to verify itsapplicability in real data.

Weak lensing power spectrum reconstruction by counting galaxies.-- II: Improving the ABS method with the shift parameter