Multi-resolution Scarlet

Rémy Joseph, Peter Melchior, Fred Moolekamp, JiaXuan Li

 

Modelling astro images for

Deblending

 

Galaxy light profile

Telescope refraction (convolution)

Instrument acquisition (pixelation)

Instrumental noise

(PM)_{[x,y]}+N_{[x,y]}
(R*H*M)_{[x,y]}
H*M
M

PSF:

R*H

In practice

 

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

Scarlet:

 Multi-band deblending

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: \(I_0\)

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

$$m_0$$

$$m_1$$

$$I$$

pixels

wavelength

pixels

pixels

Multi-resolution deblending: Same idea, different resolutions

HST cosmos, F814w

HSC DR2, grizy

\(Y_1\):

\(Y_2\):

pixels

Scarlet model

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Interpolation at different resolution

General formula for interpolation

\(f(x, y) = \sum_{x_k, y_j} f(x_k, y_j) K(x-x_k, y-y_j)\)

Interpolation kernel

Known samples

Samples at desired position (x, y)

Euclid-like

HST-like

?

Euclid-like

HST-like

Interpolation at different resolution

Astronomical images have different samplings & different PSFs!!

\(f_1(x, y) = (f*h_1)(x_{}, y)\)

\(f_2(X, Y) = (f*h_2)(X, Y)\)

Shannon-Whittaker interpolation

\((f*p)(t_m) = (F*S)*(P*S)\) = F*P*S

\(f(t_m) = h\sum_{t_k}f(t_k)sinc(\frac{t_k-t_m}{h}) = F*S\)

Interpolation and convolution

\( (f*p)(x_{mx},y_{my}) = h^2 \sum_{x_{kx},x_{ky}} f(x_{kx}, y_{ky})\sum_{x_{lx}, y_{ly}}p(x_{lx},y_{ly})sinc(\frac{x_{mx}-x_{kx}-x_{lx}}{h})sinc(\frac{y_{my}-y_{ky}-x_{ly}}{h})\)

\(I_2(x_{i2},y_{j2}) = (f*p_2) (x_{i2},y_{j2}) = \\ h^2 \sum_{y_{j1}} \sum_{x_{k1}}\sum_{x_{i1}} F(x_{i1}, y_{j1}) sinc(\frac{x_{i2}-x_{i1}-x_{k1}}{h}  )\sum_{y_{l1}} P_d(x_{k1}, y_{l1})sinc(\frac{y_{j2}-y_{j1}-y_{l1}}{h}) \)

The problem of speed

\(f(X, Y) = \sum_{x_k, y_j} f(x_k, y_j) p(X-x_k, Y-y_j)\)

\((X,Y)\) and \((x_k, y_j)\) are not on the same grid.

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

The problem of speed

\(f_2(X, Y) = \sum_{x_k, y_j} f_1(x_k, y_j) P(\frac{X}{h}-x_k, \frac{Y}{h}-y_j)\)

\(X = hX_1cos\theta + hY_1sin\theta\)

\(Y = -hX_1sin\theta + hY_1cos\theta\)

Shifts along two directions

Reformulate to share the distance calculations along directions of constant coordinates:

\(f_2(X, Y) = \sum_{x_k, y_j} f_1(x_k-X_1cos\theta, y_j+X_1sin\theta) \times P(-Y_1sin\theta-x_k, -Y_1cos\theta-y_j)\)

The problem of speed

\(f_2(X, Y) = \sum_{x_k, y_j} f_1(x_k-X_1cos\theta, y_j+X_1sin\theta) \times P(-Y_1sin\theta-x_k, -Y_1cos\theta-y_j)\)

  • Shifts along \(2\times M\) directions instead of \(M^2\)
  • Multiplcation of two matrices :                    \(N^2M \times MN^2\)  instead of \(N^2\times N^2M^2\)
  • Orders of magnitude faster than computing and multiplying by the \(P(X-x_k, Y-y_j)\) kernel

Interpolation at different resolution

Euclid-like

HST-like

Resampling and difference convolution can be done as one operation:

\(f_2(X, Y) = \sum_{x_k, y_j} f_1(x_k, y_j) H(X-x_k, Y-y_j)\)

Interpolation with difference kernel as the interpolation kernel. (Demo-ish in back up slide)

\(P(x,y) = \mathcal{F}^{-1}(\frac{\tilde{h_2}}{\tilde{h_1}})(-x,-y)\)

Comparison with galsim:

t = 0.046M - 0.2

Galsim interpolation scheme:

Interpolation kernel is reduced to a quintic kernel of size kxk = 6x6. requires s=4-fold padding.

Complexities:

  • Galsim: $$O(N^2(k^2+s^2log(sN))+M^2(3+log(M)))$$
  • Scarlet: $$O(N^2(M+1)(log(N)+M))$$

Comparison with galsim:

Reconstructions

Comparison of source distortion ratios between Galsim and Scarlet interpolations

$$SDR(\tilde{X}) = 10\log_{10}(\frac{||X_{true}||}{||\tilde{X} - X_{true}||} )$$

Gains from Multi-resolution

Comparison of SDR in deblending of Euclid, Rubin and Euclid+Rubin images.

 

Gains from Multi-resolution

Comparison of SDR in deblending of Euclid, Rubin and Euclid+Rubin images.

 

The issue: initialisation

The problem of detection

Credit: Ranga Ram Chary

The problem of detection

Multi-band detection with SEP:

  • Build a coadd of all bands
  • Apply Starlet filter to select High frequencies
  • Run sep on the filtered Coadds

Multi-resolution detection

  • Interpolation of low resolution image to high resolution
  • Coadd interpolated images and High resolution bands
  • Idem as before

Side note:

SEP detection

SEP + Starlet detection

Credit: Fred Moolekamp

Astrometry

COSMOS Fields

Matching observations, requires matching object positions

Astrometry

HST - HSC positions

HSC s18 dud catalog

Astrometry

HSC s18 dud catalog

Gaia dr2

HSC - Gaia

HST - Gaia

HST cosmos, F814w

HSC DR2, grizy

Scarlet model

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Multi-resolution in Scarlet

Takes advantage of both colour and resolution 

To do:

  • Improve simulations: realistic profiles, magnitudes
  • Quantify the gain from multi-resolution detection
  • Test multi-detection vs multi-resolution
  • Quantify the allowed astrometric error
  • Tests with BTK: New multi-resolution capabilities
  • Allow PSFs to shift to alleviate astrometric errors

Interpolation kernel is actually a sinc

\(f_2(X) = (f_1*P)(X) = (\sum_{x_k} f_1(x_k)sinc(\frac{X-x_k}{h}))*(\sum_{x_p}P(x_p)sinc(\frac{X-x_p}{h}))\)

\(f_2(X) = (f_1*P)(X) = \sum_{x_k} f_1(x_k)\sum_{x_l}P(x_l)sinc(\frac{X-x_k - x_l}{h})\)

\(f_2(X) = (f_1*P)(X) = \sum_{x_k} f_1(x_k)\sum_{x_l}P(x_l- X)sinc(\frac{x_k + x_l}{h})\)

Making sure we get the pixel response right

\(I_2(x_{i2},y_{j2}).\) = \((rect_{h_1}*(f*p_1) *{F}^{-1}(\frac{\hat{rect_{h_2}}}{\hat{rect_{h_1}}}\frac{\hat{p_2}}{\hat{p_1}}))(x_{i2},y_{j2})\)

\(I_2(x_{i2},y_{j2}) = (rect_{h_2}*p_2*f)(x_{i2},y_{j2})\)

Multi-resolution Scarlet and LSBG deblending

By herjy

Multi-resolution Scarlet and LSBG deblending

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