Flexible Image modelling for deblending

Rémy Joseph

MLxCosmo December 17th 2020

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).

The problem of blending

Blending: The apparent ovelap of objects on the plane of the sky
Expected blending in Rubin: 67% of galaxies
Euclid: 43%
Sanchez et al. (in prep)

Affects galaxy shapes, counts and photometric redshift measurements

Modelling astro images for

Deblending

Galaxy light profile

Telescope refraction (convolution)

Instrument acquisition (pixelation)

Instrumental noise

(HM)_{[x,y]}+N_{[x,y]}

(R*P*M)_{[x,y]}

P*M

In practice

Model choices impact separation of blended galaxies
Real galaxies have complex shapes and colours

I_{[x,y]} = (H\sum_i M_i)_{[x,y]}+N_{[x,y]}

Pixelated model

M^{\star} = \underset{M}{argmin}||I-HM||_2^2 + C(M)

Get creative about what constraints to use

Example: The SDSS deblender: symmetry & Monotonicity

Robert Lupton

SCARLET

$$I_j = R*P_j * \sum_{i,n} a_{j,i,n}m_{i,n} +N_j$$

Melchior et al. 2016 ( arXiv:1802.10157)

GitHub: https://github.com/pmelchior/scarlet

morphological assumptions as constraints:
- Positivity: All non-zero pixels must have positive values
- Monotonicity: Profiles smoothly decrease for the centre out.
- ~~Symmetry~~: Pixels about the central pixel take the value of the minimum of the two (Obsolete since Melchior, Joseph, Moolekamp 2019)
- Bounding: Each galaxy profile is contained in a finite bounding box

SCARLET:

Modelling multi-band images

F435w

F606w

F814w

NASA/ESA: Hubble Frontier Fields, MACSJ 1149, Lotz et al. (2016)

RGB images are collections of band in different filters

SCARLET

Colour-based: each band is a linear combination of monochromatic components

F435w: $I_2$

F606w: $I_1$

F814w: $I_0$

$$I_j = H_j \sum_i a_{j,i}m_i + N_j$$

$$m_0$$

$$m_1$$

$$I$$

SCARLET

Melchior et al. 2016 ( arXiv:1802.10157)

GitHub: https://github.com/pmelchior/scarlet

Linear Optimisation

Constraints: Positivity, Monotonicity, Bounding.

Functional decompositions:

The Starlet transfrom

Starlet coefficients

Multiscale transformation
Decomposition in B-splines at different spatial scales

Starlet basis set

Constraints on starlet coefficients

Is achieved by reconstructing sparse fields in starlets:

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

Likelihood Sparsity Positivity

(smoothness constraint)

MuSCADeT: Joseph et al. 2016 (arxiv:1603.00473)

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

$$I_j = R*P_j * \sum_i a_{j,i}\Phi s_i + N_j, \qquad m_i = \Phi s_i$$

NASA/ESA: Hubble Frontier Fields, MACSJ 0416, Lotz et al. (2016)

Complex galaxies

F814w

F435w

RGB

MuSCADeT Red

MuSCADeT Blue

Low Surface Brightness Galaxies

On going work with Johnny Greco, Jiaxuan Li & Jenny Greene

HSC image

image-model

LSB model

Residuals

Reconstruction of strongly lensed source

I_{[x,y]} = H(FM)_{[x,y]}+N_{[x,y]}

Reconstruction of strongly lensed source

Credit: Aymeric Galan

SLITronomy

Joseph et al. 2018

Galan et al. 2020

$$I_j = R*P_j * \sum_{i,n} a_{j,i,n}m_{i,n}$$

PixelCNN as a prox

In scarlet

Scarlet is flexible to the kind of constraints we can impose on morphology. We are now implementing priors PixelCNN Lanusse et al. 2019:

$$p(m) = \prod_k p(m_k|m_{k-1}, ..., s_0) $$

$ \tilde{M} = \underset{M}{argmin}$ $ \frac{1}{2}||I-HAM||^2_2 $ $+$ $\sum_i p(m_i)$

Last thoughts

Modelling images:
- Understanding the formation of images
- Great data require great models (flexibility & scalability)
- Flexible models need flexible priors:
  - Knowledge of galaxy morphology: Monotonicity, Smoothness, Positivity, ML
Data can get even more complicated:
- Integral Field Units
- spectral information
- How to incorporate dust?
- varying PSF
- Multi-resolution processing