There are several non-parametric methods to reconstruct dark matter in 2D which can be broadly split into two categories: convergence techniques  and potential techniques . In the former one measures the projected surface mass density (or convergence) directly by applying a convolution to the measured shear under the assumption that . Potential techniques perform a minimization and are better suited to the cluster regime and can also incorporate strong lensing information . In the majority of methods, choices need to be made about smoothing scales to optimize signal-to-noise whilst preserving reconstruction resolution. Using a wavelet method circumvents this choice [860, 497] but makes the resulting significance of the reconstruction difficult to measure.
The lensing distortion depends on the total projected surface mass density along the line of sight and a geometrical factor that increases with source distance. This redshift dependence can be used to recover the full 3D gravitational potential of the matter density as described in [455, 72] and applied to the COMBO-17 survey in  and the COSMOS survey in . This work has been extended in  to reconstruct the full 3D mass density field and applied to the STAGES survey in .
All 3D mass reconstruction methods require the use of a prior based on the expected mean growth of matter density fluctuations. Without the inclusion of such a prior,  have shown that one is unable to reasonably constrain the radial matter distribution, even for densely sampled space-based quality lensing data. Therefore 3D maps cannot be directly used to infer cosmological parameters.
The driving motivation behind the development of 3D reconstruction techniques was to enable an unbiased 3D comparison of mass and light. Dark haloes for example would only be detected in this manner. However the detailed analysis of noise and the radial PSF in the 3D lensing reconstructions presented for the first time in  show how inherently noisy the process is. Given the limitations of the method to resolve only the most massive structures in 3D the future direction of the application of this method for the Euclid Wide survey should be to reconstruct large scale structures in the 3D density field. Using more heavily spatially smoothed data we can expect higher quality 3D resolution reconstructions as on degree scales the significance of modes in a 3D mass density reconstruction are increased . Adding additional information from flexion may also improve mass reconstruction, although using flexion information alone is much less sensitive than shear .
Living Rev. Relativity 16, (2013), 6
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