Improving weak lensing reconstructions in 3D using sparsity ´ Jean-Luc Starck Adrienne Leonard, Franc¸ois-Xavier Dupe, Service d’Astrophysique, IRFU, CEA centre de Saclay, Gif-sur-Yvette, France
Nonlinear Approach with Sparsity
Introduction Weak gravitational lensing is a powerful tool, which allows us to map the distribution of dark matter in the Universe. With the advent of large, high-resolution and multi-wavelength surveys, it has recently become possible to use photometric redshift information to reconstruct the matter distribution in three dimensions, rather than a two-dimensional projection. This is no easy task, as the inverse problem is ill posed, the data are noise-dominated, and the lensing efficiency kernel is very broad along the line of sight. State-of-the-art linear methods to recover the density distribution from lensing measurements typically exhibit a line-of-sight bias in the location of detected peaks, and a broad smearing of the density distribution along the line of sight. We present here a non-linear proximal minimization method incorporating a sparse prior, which allows us to recover the underlying density distribution from lensing measurements with minimal bias and smearing, thus allowing for more accurate mapping of the three-dimensional density distribution.
Results
I In order to address the line of sight smearing, damping and bias, we consider the problem in one
I Comparison with linear methods: We consider a cluster at z = 0.25, and the reconstruction is computed at the same resolution as the input data (the number of redshift bins Ninput = Nrec = 20).
dimension; i.e. we are concerned with the inversion of the equation κ = Qδ + ε ,
Transverse Wiener Filter
SVD Filter
Radial Wiener Filter
Sparse approach
and lines of sight are considered independently. I The line of sight lensing operator Q effectively convolves the density with a broad radial kernel. In other words, a discrete density spike is mapped to a lensing signal over a broad range of redshifts.
Weak Gravitational Lensing Lensing efficiency kernel convolving the density distribution.
I Weak lensing measures the tiny elliptical distortions induced in images of background galaxies due to
the gravitational effects of matter concentrations along the line of sight.
Signal produced by a density spike as a function of redshift.
I This implies that the signal is compressible, and will have a sparse representation in an appropriate
dictionary Φ. We aim to construct an estimator that exploits this property as a prior on the reconstruction.
I The Optimisation Problem:
Imposing sparsity while retaining data fidelity implies solving the following minimisation problem:
T minn Φ δ 1 s.t. 21 kκ − Qδk2Σ−1 ≤ , δ ∈ C
I Varying redshifts, super-resolution: We consider clusters at z = 0.2, 0.6, and 1.0. The redshift resolution on the reconstruction is 1.25 × that of the input data (Ninput = 20, Nrec = 25).
δ∈R
Photo credit: courtesy of Bell Labs/Lucent; http://www.nsf.gov/od/lpa/news/press/00/pr0029.htm
I A linear operation maps the measured shear on a given source plane, γ, onto the dimensionless
projected mass density, the convergence κ, on that plane:
I
γ = Pγκκ.
Z
w
dw
0 fK (w
0
0
Φ : The dictionary Σ : The covariance matrix of the noise
)fK (w − w 0) δ[fK (w 0)θ, w 0] , fK (w) a(w 0)
where a is the scale factor of the Universe, w is a comoving distance and fK (w) gives the comoving angular diameter distance, which is a function of the curvature, K , of the Universe. I We can therefore write: κ = Qδ, or, equivalently, γ = PγκQδ. Our goal in 3D lensing is therefore to invert this equation in the presence of noise to recover a map of the density contrast δ.
I
z = 0.2
: the size of the `2 ball constraining the data fidelity C : a closed convex set
z = 0.6
Data: The observed image κ, the dictionary Φ, the number of iterations Niter, proximal steps ω, τ > 0, threshold parameter λ. Begin Initialization: y10 = y20 = δˆ0 = x 0 = 0. For t = 0 to Niter − 2 do 1 1. Initialise auxiliary variables: t1n = y1n + ωΦT δˆn , t2n = y2n + ωΣ− 2 Qδˆn . 2. Splitting Step: 2.1 Sparsity-promoting term: y1n+1 = t1n − ωStλ/ω (t1n /ω) 1 2.2 Data fidelity term: y2n+1 = t2n − ω(Σ− 2 κ + ζ)
√ 1 1
− − n n 2
t2 /ω − Σ 2 κ/ω < N t2 /ω − Σ κ/ω 2 1 √ − n ζ = N(t 2 /ω−Σ 2 κ/ω)
otherwise
t n /ω−Σ− 12 κ/ω
2
N = number of elements in κ 1 3. Projection: x t+1 = PrjC [x n − τ (Φy1n+1 + Q†Σ 2 y2n+1)]. 4. Update estimate: δˆ = 2x n+1 − x n . endfor End Exit: An estimate of the density contrast δˆ along the line of sight.
Linear Inversion Methods I We can express the problem as one of the form:
ε ∼ N (0, σ 2) ,
Discussion
I Example lines of sight:
with d being the measurements, contaminated by gaussian noise ε, and s being the true underlying density. I Simon et al. (2010) propose to invert this using a Saskatoon filter incorporating a Wiener prior (either transverse or radial) as:
I We have presented a simplified 1D approach to reconstruct the density distribution along the line of
sight from weak lensing measurements I Our approach outperforms linear methods in 3 key areas:
ˆ = [α1 + SR†Σ−1R]−1SR†Σ−1d , s
1. Minimal bias is seen in the location of detected peaks 2. Minimal smearing is seen in the reconstruction along the line of sight 3. The recovered amplitude of the reconstruction is ∼ 75% of the true value (or better), compared with < 50% with the state-of-the-art linear methods
where S is the signal covariance, encoding the prior, Σ is the noise covariance, and α is a tuning parameter. I VanderPlas et al. (2011) propose to carry out a singular value decomposition on the matrix e ≡ Σ− 21 R, and filtering by thresholding the matrix of singular values, with no priors placed on the R signal covariance: −1
ˆ = VΛ s
z = 1.0
We use a primal-dual proximal splitting algorithm proposed by Chambolle & Pock (2010):
2
d = Rs + ε,
I
I The Algorithm:
The operator Pγκ represents here a 2D convolution on a plane perpendicular to the line of sight. I A second linear operation, amounting to a convolution along the line of sight, maps the convergence κ onto the matter overdensity δ = ρ/ρ − 1, where ρ is the mean matter density in the universe: 3H02ΩM κ(θ, w) = 2c 2
I
†
−1/2
UΣ
I Our approach allows us to deal with underdetermined problems, therefore can attain
I Chambolle A., Pock T., 2010, A first-order primal-dual algorithm for convex problems with
applications to imaging, JMIV, 40, 120 I Leonard A., Dupe ´ F.-X., Starck J.-L., 2011, A compressed sensing approach to 3D weak lensing,
A&A, in prep. I Simon P., Taylor A. N., Hartlap J., 2009, Unfolding the matter distribution using three-dimensional
weak gravitational lensing, MNRAS, 399, 48 I VanderPlas J. T., Connolly A. J., Jain B., Jarvis M., 2011, Three-dimensional reconstruction of the
density field: An SVD approach to weak-lensing tomography, ApJ, 727, 118
super-resolution in the reconstruction. I We demonstrate performance in reconstructing clusters up to a redshift of z = 1.
d,
I False detections, arising from overfitting, can appear at high redshift when using the sparse
e = UΛV† represents the SVD decomposition. where R I These methods suffer from the same three fundamental problems: 1. There is a broad smearing of the reconstructed density along the line of sight. 2. There is a bias in the radial location of detected structures; they are shifted relative to their true positions in redshift space. 3. The amplitude of the reconstruction is damped, sometimes heavily, with respect to the true underlying density distribution.
3D Lensing Reconstructions with Sparsity
References
approach. However, these false detections are localised to individual lines of sight, and do not form coherent structures I The algorithm proposed is robust, and entirely general. An extension of the method to a full 3D
treatment involves changing only the dictionary Φ. Noisy convergence data (points with error bars) with fits obtained using our algorithm for various values of .
Reconstructions of the density contrast corresponding to the fits in the left panel
I A full 3D treatment will allow for a more sparse representation of the data and, by seeking coherent
structures in 3D should remove many pixel-scale false detections and artefacts.
http://jstarck.free.fr/sparseastro.html
Acknowledgements I The authors would like to acknowledge the support provided by the European Research Councils,
through grant SparseAstro (ERC-228261). I This work was undertaken jointly as part of the SEDI/LCS and SAP/LCEG working groups at CEA
Saclay.
corresponding:
[email protected]
