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 6 \times finer redshift resolution than the shear data .
Considering cluster simulations with 0.05 \leq z _ { cl } \leq 0.75 and 3 \times 10 ^ { 13 } h ^ { -1 } M _ { \odot } \leq M _ { vir } \leq 10 ^ { 15 } h ^ { -1 } M _ { \odot } , we show that the redshift extent of detected peaks is typically 1 - 2 pixels , or \Delta z \lesssim 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 \lesssim 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 \sigma \sim 0.2 - 0.3 , a significant damping of the amplitude of the peaks detected , and a bias in the detected redshift .