  We present GLIMPSE - Gravitational Lensing Inversion and MaPping with Sparse
Estimators - a new algorithm to generate density reconstructions in three
dimensions from photometric weak lensing measurements. This is an extension of
earlier work in one dimension aimed at applying compressive sensing theory to
the inversion of gravitational lensing measurements to recover 3D density maps.
Using the assumption that the density can be represented sparsely in our chosen
basis - 2D transverse wavelets and 1D line of sight dirac functions - we show
that clusters of galaxies can be identified and accurately localised and
characterised using this method. Throughout, we use simulated data consistent
with the quality currently attainable in large surveys. We present a thorough
statistical analysis of the errors and biases in both the redshifts of detected
structures and their amplitudes. The GLIMPSE method is able to produce
reconstructions at significantly higher resolution than the input data; in this
paper we show reconstructions with 6x finer redshift resolution than the shear
data. Considering cluster simulations with 0.05 <= z <= 0.75 and 3e13/h Msun <=
Mvir <= 1e15/h Msun, we show that the redshift extent of detected peaks is
typically 1-2 pixels, or Dz <~ 0.07, and that we are able to recover an
unbiased estimator of the redshift of a detected cluster by considering many
realisations of the noise. We also recover an accurate estimator of the mass,
that is largely unbiased when the redshift is known, and whose bias is
constrained to <~ 5% in the majority of our simulations when the estimated
redshift is taken to be the true redshift. This shows a substantial improvement
over earlier 3D inversion methods, which showed redshift smearing with a
typical standard deviation of 0.2-0.3, a significant damping of the amplitude
of the peaks detected, and a bias in the detected redshift.
