  We present a fast iterative FFT-based reconstruction algorithm that allows
for non- parallel redshift-space distortions (RSD). We test our algorithm on
both N-body dark matter simulations and mock distributions of galaxies designed
to replicate galaxy survey conditions. We compare solenoidal and irrotational
components of the redshift distortion and show that an approximation of this
distortion leads to a better estimate of the real-space potential (and
therefore faster convergence) than ignoring the RSD when estimating the
displacement field. Our iterative reconstruction scheme converges in two
iterations for the mock samples corresponding to BOSS CMASS DR11 when we start
with an approximation of the RSD. The scheme takes six iterations when the
initial estimate, measured from the redshift-space overdensity, has no RSD
correction. Slower convergence would be expected for surveys covering a larger
angle on the sky. We show that this FFT based method provides a better estimate
of the real space displacement field than a configuration space method that
uses finite difference routines to compute the potential for the same grid
resolution. Finally we show that a lognormal transform of the overdensity, used
as a proxy for the linear overdensity, is beneficial in estimating the full
displacement field from a dense sample of tracers. However the lognormal
transform of the overdensity does not perform well when estimating the
displacements from sparser simulations with a more realistic galaxy density.
