For cosmic shear to become an accurate cosmological probe, systematic errors in the shear measurement method must be unambiguously identified and corrected for. We demonstrate that cosmic shear can be measured accurately in Fourier space in the presence of background noise, source Poisson noise, and finite pixel size, without assumptions on the morphologies of galaxy and PSF. Our noise treatment can be generalized for images made of multiple exposures through MultiDrizzle. This is demonstrated with the COSMOS/ACS data. Our shear measurement method can achieve sub-percent accuracy even for images of signal-to-noise ratio less than 5 in general, making it the most promising technique for cosmic shear measurement in the ongoing and upcoming large scale galaxy surveys.