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).
F435w
F606w
F814w
NASA/ESA: Hubble Frontier Fields, Lotz et al. (2016)
F435w
F606w
F814w
A linear model for colour images:
$$y_i = \sum_j a_{i,j}s_j$$
$$s_0$$
$$s_1$$
$$y$$
A linear model for colour images:
(including PSF)
$$Y = (H)AS +N$$
$$A$$
$$S$$
$$Y$$
$$=$$
Red filter
Green filter
Blue filter
Joseph et al. 2016 (arxiv:1603.00473)
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
The algorithm
$$U \gets S+\mu A^T H^T (Y-HAS)$$
$$S \gets \Phi \underset{\alpha_S}{argmin}\frac{1}{2}||U-\Phi\alpha_S||_2^2+\lambda|| \alpha_S||_0$$
$$S[S<0]\gets 0$$
The algorithm
Colours are extracted from the scene using Principal Component Analysis (PCA) of the multi-band pixels
The algorithm
$$U \gets S+\mu A^T H^T (Y-HAS)$$
$$S \gets \Phi \underset{\alpha_S}{argmin}\frac{1}{2}||U-\Phi\alpha_S||_2^2+\lambda|| \alpha_S||_0$$
$$S[S<0]\gets 0$$
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
SLIT, Joseph et al. 2018
Lenstronomy, Birrer et Amara 2018
SLITronomy, Galan et al. 2021
Source inversion
Source inversion
\( \tilde{S} = \underset{S}{argmin}\) \( \frac{1}{2}||Y-HFS||^2_2 \) \(+\) \(\lambda||\Phi^T S||_1\) \(+\) \(\mathcal{i}_+(S) \)
Data attachement Sparsity Positivity
Functional decompositions:
The Starlet transfrom
Starlet coefficients
Starlet basis set
Model for a strong lens system
$$Y = HG+HFS+N$$
Constrained problem
$$HG, S = \Phi \underset{\alpha_S, \alpha_{HG}}{argmin}||Y-\Phi\alpha_{HG}-HF\Phi\alpha_S||_2^2+\lambda_S||\alpha_S||_1 + \lambda_{HG}||\alpha_{HG}||_1 \\ + \mathcal{i}_+(HG) + \mathcal{i}_+(S) $$
Alternate between optimisation over \(\alpha_S\)
The algorithm
And optimisation over \(\alpha_{HG}\)
SLIT
HST F814W
Reconstruction of strongly lensed source
LENSRTONOMY
Default reconstructions use lensed shapelets to model sources. Coefficients of shapelet components are the optimised parameters.
Reconstruction of strongly lensed source
Starlets => \(LN^2\) parameters to optimise
email: remyj@princeton.edu
Minimising \(\lambda|| \alpha_S||_0\) with Hard Thresholding
$$HT_{th}(x) = \begin{cases} 0 \quad if \quad x<th \\ x \quad otherwise\end{cases}$$
th
Minimising \(\lambda|| \alpha_S||_0\) with Hard Thresholding
Threshold is set according to noise level \(\sigma\) and decreases with iterations $$\lambda_{it} = k_{it} \sigma$$
FISTA algorithm
$$\alpha_S^{n+1} \gets \alpha_S^n+\mu \Phi^TA^T H^T (Y-HA\Phi\alpha_S^n)$$