Multi-resolution deblending with Scarlet:
Resampling, Detection and Astrometry

Rémy Joseph, Peter Melchior, Fred Moolekamp

SCMA VII: June 10 2021

Credit: Ranga Ram Chary

Scarlet default:

 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: \(I_0\)

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

$$m_0$$

$$m_1$$

$$I$$

SCARLET

Melchior et al. 2016 ( arXiv:1802.10157)

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

Constrained optimisation: Positivity, Monotonicity, Bounding.

pixels

wavelength

pixels

pixels

HST cosmos, F814w

HSC DR2, grizy

\(Y_1\):

\(Y_2\):

pixels

Interpolation at different resolution

Astronomical images have different samplings & different PSFs!!

We construct a common  model m and the adequate operators to fit both datasets

$$I^{hr}_j = H_j * \sum_i a^{hr}_{j,i}m_i + N^{hr}_j$$

$$I^{lr}_j = R_j * \sum_i a^{lr}_{j,i}m_i + N^{lr}_j$$

Interpolation at different resolution

Resampling and difference convolution can be done as one operation with a sinc kernel (Shannon-Whitaker interpolation):

Interpolation with difference kernel as the interpolation kernel.

\(R_j(x,y) = \mathcal{F}^{-1}(\frac{\tilde{h_j}^{lr}}{\tilde{h}^{hr}})(x,y)\)

$$I^{lr}_j = R_j . \sum_i a_{j,i}m_i + N_j$$

The problem of speed

\(I^{lr}(X, Y) = \sum_{x_k, y_j} m^{hr}(x_k, y_j) r(X-x_k, Y-y_j)\)

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

The problem of speed

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

As many operations as there are distances between red and blue points, i.e. \(M^2N^2\)

M

N

Comparison with galsim:

Bernstein & Gruen 2014

Rowe et al. 2015

Galsim quintic interpolation:

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

Complexities:

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

Comparison with galsim:

Reconstructions

Comparison of source distortion ratios between Galsim's default and Scarlet interpolations

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

Multi-resolution vs single resolution

Experimental test results on simulations (EUCLID+RUBIN):

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

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

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

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