Flexible Image modelling for deblending and strong gravitational lensing

Rémy Joseph

Collaborators: Peter Melchior, Fred Moolekamp, Frederic Courbin (EPFL, SW), Jean-Luc Starck (CEA, FR), Aymeric Galan (EPFL), Austin Peel, Martin Millon (EPFL), François Lanusse (CNRS, FR), Jiaxuan Li (PKU), Jenny Greene, Johnny Greco (OSU).

Deblending multi-band images

F435w

F606w

F814w

NASA/ESA: Hubble Frontier Fields, Lotz et al. (2016)

Deblending multi-band images

F435w

F606w

F814w

A linear model for colour images:

$$y_i = \sum_j a_{i,j}s_j$$

$$s_0$$

$$s_1$$

$$y$$

Deblending multi-band images

A linear model for colour images:

(including PSF)

$$Y = (H)AS +N$$

$$A$$

$$S$$

$$Y$$

$$=$$

Red filter

Green filter

Blue filter

MuSCADeT

Joseph et al. 2016 (arxiv:1603.00473)

GitHub: https://github.com/herjy/MuSCADeT

Sparsity as a sieve

MuSCADeT

Inverting

Y = HAS+N

Is achieved by reconstructing sparse fields in starlets:

$ \tilde{S} = \underset{S}{argmin}$ $ \frac{1}{2}||Y-HAS||^2_2 $ $+$ $\lambda||\Phi^T S||_1$ $+$ $\mathcal{i}_+(S) $

Data attachement Sparsity Positivity

MuSCADeT

The algorithm

Gradient step

Find S close from U with the smallest $\ell_0$ norm

Set negative entries to 0

$$U \gets S+\mu A^T H^T (Y-HAS)$$

$$S \gets \Phi \underset{\alpha_S}{argmin}\frac{1}{2}||U-\Phi\alpha_S||_2^2+\lambda|| \alpha_S||_0$$

$$S[S<0]\gets 0$$

Estimate the mixing matrix A

MuSCADeT

The algorithm

Estimate the mixing matrix A (default)

Colours are extracted from the scene using Principal Component Analysis (PCA) of the multi-band pixels

MuSCADeT

The algorithm

Gradient step

Find S close from U with the smallest $\ell_0$ norm

Set negative entries to 0

$$U \gets S+\mu A^T H^T (Y-HAS)$$

$$S \gets \Phi \underset{\alpha_S}{argmin}\frac{1}{2}||U-\Phi\alpha_S||_2^2+\lambda|| \alpha_S||_0$$

$$S[S<0]\gets 0$$

Estimate the mixing matrix A

Applications

The Hubble Frontier Fields

(HFF Lotz et al. 2017)

Zooming in

Zooming in

Detections: Jauzax et al. (2014)

Zooming in

Detections: Jauzax et al. (2014)

Undetected Blends

MuSCARLET

Undetected Blends

MuSCARLET

Undetected Blends

MuSCARLET

Undetected Blends

MuSCARLET

Undetected Blends

MuSCARLET

Undetected Blends

MuSCARLET

Deblending and linear inversion of lensed sources

SLIT, Joseph et al. 2018

Lenstronomy, Birrer et Amara 2018

SLITronomy, Galan et al. 2021

Strong Gravitational lensing

Strong Gravitational lens system

Lensed source

Lens galaxy

Strong Gravitational lensing

Source inversion

Background source galaxy $S$

Lensed Background galaxy $HFS$

Lensing $F$

Strong Gravitational lensing

Source inversion

Model for images of lensed galaxies

$$Y = HFS+N $$

Constrained minimisation problem

$ \tilde{S} = \underset{S}{argmin}$ $ \frac{1}{2}||Y-HFS||^2_2 $ $+$ $\lambda||\Phi^T S||_1$ $+$ $\mathcal{i}_+(S) $

Data attachement Sparsity Positivity

Functional decompositions:

The Starlet transfrom

Starlet coefficients

Multiscale transformation
Decomposition in B-splines at different spatial scales

Starlet basis set

Strong Gravitational lens system

Lensed source

Lens galaxy

Sparse in Starlet

Spars(er) in Starlet

SLIT MCA

Model for a strong lens system

$$Y = HG+HFS+N$$

Constrained problem

$$HG, S = \Phi \underset{\alpha_S, \alpha_{HG}}{argmin}||Y-\Phi\alpha_{HG}-HF\Phi\alpha_S||_2^2+\lambda_S||\alpha_S||_1 + \lambda_{HG}||\alpha_{HG}||_1 \\ + \mathcal{i}_+(HG) + \mathcal{i}_+(S) $$