Rémy Joseph
Princeton, September 10th 2020
Collaborators: Peter Melchior, Fred Moolekamp, Frederic Courbin (EPFL, SW), Jean-Luc Starck (CEA, FR), Aymeric Galan (EPFL, SW), Martin Millon (EPFL, SW), François Lanusse (CNRS, FR),
Modelling astro images
Galaxy light profile
Telescope refraction (convolution)
Instrument acquisition (pixelation)
Instrumental noise
Possible model:
Sersic profile
Modelling astro images
Galaxy light profile
Telescope convolution
Instrument acquisition (pixelation)
Instrumental noise
Analytic Sersic profile
Modelling astro images
Galaxy light profile
Telescope convolution
Instrument acquisition (pixelation)
Instrumental noise
Analytic Sersic profile
Residuals
In practice
Alternatives
Pixelated model
Solution for M is not unique
Solution space:
Pixelated model
Requires constraints
Example: limiting the 2-norm of the solution
Pixelated model
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)
SCARLET:
Modelling multi-band images
F435w
F606w
F814w
NASA/ESA: Hubble Frontier Fields, MACSJ 1149, Lotz et al. (2016)
SCARLET
F435w: \(I_2\)
F606w: \(I_1\)
F814w: \(I_0\)
$$I_j = R*P_j * \sum_i a_{j,i}m_i + N_j$$
$$m_0$$
$$m_1$$
$$I$$
SCARLET
Melchior et al. 2016 ( arXiv:1802.10157)
Linear Optimisation
Constraints: Positivity, Monotonicity, Bounding.
SCARLET
Melchior et al. 2016 ( arXiv:1802.10157)
Weakness (of the model): Monotonicity can causes trails and shadows.
Functional decompositions:
The Starlet transfrom
Starlet coefficients
Starlet basis set
Functional decompositions:
The Starlet transfrom
Illustration: Detection in crowded fields
Credit: Fred Moolekamp
NGC 6569
Sep detection
NGC 6569
Starlet+Sep detection
NGC 6569
Starlet level 1
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)
$$I_j = R*P_j * \sum_i a_{j,i}\Phi s_i + N_j, \qquad m_i = \Phi s_i$$
NASA/ESA: Hubble Frontier Fields, MACSJ 0416, Lotz et al. (2016)
Complex galaxies
F814w
F435w
RGB
MuSCADeT Red
MuSCADeT Blue
Low Surface Brightness Galaxies
Sep detection
Starlet + Sep detection
On going work with Johnny Greco
Low Surface Brightness Galaxies
On going work with Johnny Greco
Data
Scarlet model
Residuals
Monotonicity + Positivity:
Starlet sparsity + Positivity:
Low Surface Brightness Galaxies
On going work with Johnny Greco
Starlet sparsity + Positivity:
Model
Spectrum
g r i z y
Reconstruction of strongly lensed source
Reconstruction of strongly lensed source
Disclaimer: it can be improved
$$I_{j\in [0,k]} = H P_j * \sum_{i,n} a_{j,i,n}s_{i,n} + N_j$$
HST cosmos, F814w
HSC DR2, grizy
\(I_1\)
\(I_2\)
$$I_{j\in [k+1,n]} = R * P_j * \sum_{i,n} a_{j,i,n}s_{i,n} + N_j$$
pixels
wavelength
pixels
pixels
Combining surveys at different resolutions
HST cosmos, F814w
HSC DR2, grizy
\(I_1\):
\(I_2\):
pixels
Multi-resolution fitting: Finding a model at high resolution that fits datasets at different resolutions:
\(I_1\)
\(I_2\)
\( \tilde{M} = \underset{M}{argmin}\) \( \frac{1}{2}||I_1-HPAM||^2_2 \) \(+\) \( \frac{1}{2}||I_2-RP'AM||^2_2 \)\(+\)\(\sum_i p(m_i)\)
High resolution
Low resolution
HST cosmos, F814w
HSC DR2, grizy
Scarlet model
1
2
4
3
5
6
1
2
4
3
5
6
1
2
4
3
5
6
$$I_j = R*P_j * \sum_{i,n} a_{j,i,n}m_{i,n}$$
$$p(m) = \prod_k p(m_k|m_{k-1}, ..., s_0) $$
\( \tilde{M} = \underset{M}{argmin}\) \( \frac{1}{2}||I-RPAM||^2_2 \) \(+\) \(\sum_i p(m_i)\)
Last thoughts
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 problem of speed
\(f_2(X, Y) = \sum_{x_k, y_j} f_1(x_k, y_j) P(X-x_k, Y-y_j)\)
As many operations as there are distances between red and blue points, i.e. \(M^2N^2\)
M
N
Interpolation at different resolution
Euclid-like
HST-like
Resampling and difference convolution can be done as one operation:
\(M(X, Y) = \sum_{x_k, y_j} m(x_k, y_j) H(X-x_k, Y-y_j)\)
Interpolation with difference kernel as the interpolation kernel. (Demo-ish in back up slide)
\(H(x,y) = \mathcal{F}^{-1}(\frac{\tilde{p_2}}{\tilde{p_1}})(-x,-y)\)
The problem of speed
\(f_2(X, Y) = \sum_{x_k, y_j} f_1(x_k, y_j) P(\frac{X}{h}-x_k, \frac{Y}{h}-y_j)\)
Reformulate to share the distance calculations along directions of constant coordinates:
\(f_2(X, Y) = \sum_{x_k, y_j} f_1(x_k-X_1cos\theta, y_j+X_1sin\theta) \times P(-Y_1sin\theta-x_k, -Y_1cos\theta-y_j)\)
\(X = hX_1cos\theta + hY_1sin\theta\)
\(Y = -hX_1sin\theta + hY_1cos\theta\)
Shifts along two directions
The problem of speed
\(f_2(X, Y) = \sum_{x_k, y_j} f_1(x_k-X_1cos\theta, y_j+X_1sin\theta) \times P(-Y_1sin\theta-x_k, -Y_1cos\theta-y_j)\)
\(f_2(X) = (f_1*P)(X) = (\sum_{x_k} f_1(x_k)sinc(\frac{X-x_k}{h}))*(\sum_{x_p}P(x_p)sinc(\frac{X-x_p}{h}))\)
\(f_2(X) = (f_1*P)(X) = \sum_{x_k} f_1(x_k)\sum_{x_l}P(x_l)sinc(\frac{X-x_k - x_l}{h})\)
\(f_2(X) = (f_1*P)(X) = \sum_{x_k} f_1(x_k)\sum_{x_l}P(x_l- X)sinc(\frac{x_k + x_l}{h})\)
\(I_2(x_{i2},y_{j2}).\) = \((rect_{h_1}*(f*p_1) *{F}^{-1}(\frac{\hat{rect_{h_2}}}{\hat{rect_{h_1}}}\frac{\hat{p_2}}{\hat{p_1}}))(x_{i2},y_{j2})\)
\(I_2(x_{i2},y_{j2}) = (rect_{h_2}*p_2*f)(x_{i2},y_{j2})\)