  Sub-percent level accuracy in shear measurement is required by the Stage-IV
weak lensing surveys. One important challenge is about suppressing the shear
bias on source images of low signal-to-noise ratios (SNR$\lesssim10$).
Previously, it has been demonstrated that the shear estimators defined in the
Fourier_Quad (FQ) method can achieve sub-percent accuracy at the very faint end
(SNR$\lesssim5$) through ensemble averaging. Later, it is found that we can
approach the minimum statistical error (the Cramer-Rao Bound) by symmetrizing
the full PDF of the FQ shear estimators (the PDF_SYM approach), instead of
taking ensemble averages. Recently, with a large amount of mock galaxy images,
we are able to identify some small amount of shear biases in the PDF_SYM
approach at the faint end. The multiplicative bias goes up to
$1-2\times10^{-2}$ at SNR $\lesssim 10$, and the anisotropy of the point spread
function (PSF) causes an additive bias that can reach a few times $10^{-4}$. We
find that these biases originate from the noise-source coupling in the galaxy
power spectrum. It turns out that this problem can be largely fixed by adding
additional terms to the FQ shear estimators. The resulting multiplicative and
additive biases can be significantly suppressed to the level of $10^{-3}$ and
$10^{-5}$ respectively. These corrections substantially extend the available
SNR range for accurate shear measurement with the PDF_SYM approach.

Cosmology and Nongalactic Astrophysics