Galaxy Image Restoration with Shape Constraint

Abstract

Images acquired with a telescope are blurred and corrupted by noise. The blurring is usually modelled by a convolution with the Point Spread Function and the noise by Additive Gaussian Noise. Recovering the observed image is an ill-posed inverse problem. Sparse deconvolution is well known to be an efficient deconvolution technique, leading to optimized pixel Mean Square Errors, but without any guarantee that the shapes of objects (e.g. galaxy images) contained in the data will be preserved. In this paper, we introduce a new shape constraint and exhibit its properties. By combining it with a standard sparse regularization in the wavelet domain, we introduce the Shape COnstraint REstoration algorithm (SCORE), which performs a standard sparse deconvolution, while preserving galaxy shapes. We show through numerical experiments that this new approach leads to a reduction of galaxy ellipticitiy measurement errors by at least 44%.

Appendices

Appendices

Expressing Galaxy Ellipticity with Inner Products

Assume that we have $x \in {\mathbb {R}}^{n\times n}$ such that

$$\begin{aligned} \mathrm {e}(x) = \frac{\mu _{2,0}(x)-\mu _{0,2}(x)+2i\mu _{1,1}}{\mu _{2,0}(x)+\mu _{0,2}(x)}\quad , \end{aligned}$$

(39)

where $\mu _{s,t}$ is a centered moment of order $(s+t)$, defined as follows:

$$\begin{aligned} \mu _{s,t}(x)=\sum _{i=1}^n\sum _{j=1}^n x\left[ (i-1) n+j\right] (i-i_c)^s (j-j_c)^t , \end{aligned}$$

(40)

and $(i_c,j_c)$ are the coordinates of the centroid of the two dimensional image encoded by x:

$$\begin{aligned} i_c = \frac{\sum _{i=1}^n\sum _{j=1}^n i\cdot x[(i-1) n+j]}{\sum _{i=1}^n\sum _{j=1}^n x[(i-1) n+j]} \text { and } j_c = \frac{\sum _{i=1}^n\sum _{j=1}^n j\cdot x[(i-1) n+j]}{\sum _{i=1}^n\sum _{j=1}^n x[(i-1) n+j]}.\nonumber \\ \end{aligned}$$

(41)

We want to show that

$$\begin{aligned} \mathrm {e}(x) = \frac{\left<x,u_3\right>\left<x,u_5\right>-\left<x,u_1\right>^2+\left<x,u_2\right>^2+2i\left( \left<x,u_3\right>\left<x,u_6\right>-\left<x,u_1\right>\left<x,u_2\right>\right) }{\left<x,u_3\right>\left<x,u_4\right>-\left<x,u_1\right>^2-\left<x,u_2\right>^2},\nonumber \\ \end{aligned}$$

(42)

where $\forall i,j \in \{1,\ldots ,n\}$,

$$\begin{aligned} \begin{array}{lll} u_1[(i-1)n+j] = (i), &{}&{}u_2[(i-1)n+j] = (j),\\ u_3[(i-1)n+j] = (1), &{}&{}u_4[(i-1)n+j] = (i^2+j^2),\\ u_5[(i-1)n+j] = (i^2-j^2), &{}&{}u_6[(i-1)n+j] = (ij). \end{array} \end{aligned}$$

(43)

To do so, we only have to express $\mu _{0,2}(x)$; $\mu _{1,1}(x)$ and $\mu _{2,0}(x)$ using $\left( \left<x,u_i\right>\right) _{1\le i\le 6}$. We start by introducing $m_{s,t}(x)$, the non-centered moment of order $(s+t)$,

$$\begin{aligned} m_{s,t} = \sum _{i=1}^n\sum _{j=1}^n x\left[ (i-1) n+j\right] i^s j^t . \end{aligned}$$

(44)

We then express $\mu _{0,2}(x)$; $\mu _{1,1}(x)$ and $\mu _{2,0}(x)$ using the non-centered moments (of order equal or less than 2), as follows:

$$\begin{aligned} \begin{aligned} \mu _{0,2}(x)&=m_{0,2}(x)-\frac{m^2_{0,1}(x)}{m_{0,0}(x)} ,\\ \mu _{1,1}(x)&=m_{1,1}(x)-\frac{m_{0,1}(x)\cdot m_{1,0}(x)}{m_{0,0}(x)},\\ \mu _{2,0}(x)&=m_{2,0}(x)-\frac{m^2_{1,0}(x)}{m_{0,0}(x)}. \end{aligned} \end{aligned}$$

(45)

Expressing these non-centered moments using $\left( \left<x,u_i\right>\right) _{1\le i\le 6}$, we obtain

$$\begin{aligned} \begin{array}{lll} m_{0,0}(x) = \left<x,u_3\right>, &{}&{}m_{1,0}(x) = \left<x,u_1\right>,\\ m_{0,1}(x) = \left<x,u_2\right>, &{}&{}m_{1,1}(x) = \left<x,u_6\right>,\\ m_{0,2}(x) =\frac{1}{2}\left( \left<x,u_4\right>-\left<x,u_5\right>\right) , &{}&{}m_{2,0}(x) = \frac{1}{2}\left( \left<x,u_4\right>+\left<x,u_5\right>\right) . \end{array} \end{aligned}$$

(46)

Using Eq. 45 and  46, we can express $\mu _{0,2}(x)$, $\mu _{1,1}(x)$, and $\mu _{2,0}(x)$ using $\left( \left<x,u_i\right>\right) _{1\le i\le 6}$:

$$\begin{aligned} \begin{aligned} \mu _{0,2}(x)&=\frac{1}{2}\left( \left<x,u_4\right>-\left<x,u_5\right>\right) -\frac{\left<x,u_2\right>^2}{\left<x,u_3\right>},\\ \mu _{1,1}(x)&=\left<x,u_6\right>-\frac{\left<x,u_1\right>\left<x,u_2\right>}{\left<x,u_3\right>},\\ \mu _{2,0}(x)&=\frac{1}{2}\left( \left<x,u_4\right>+\left<x,u_5\right>\right) -\frac{\left<x,u_1\right>^2}{\left<x,u_3\right>}. \end{aligned} \end{aligned}$$

(47)

We finish the proof by inserting Eq. 47 in Eq. 39 to obtain Eq. 42.

Dataset Generation

To generate the galaxy and the PSF images, we chose simple and commonly used profiles in astrophysics which are repectively Sersic and Moffat profiles. First, the light intensity $I_G$ of a galaxy is modeled with a Sersic using the following formula:

$$\begin{aligned} I_G(R)=I_e \cdot \text {exp}\left( b_n\left[ \left( \frac{R}{R_e}\right) ^{\frac{1}{n}}-1\right] \right) \quad , \end{aligned}$$

(48)

where $n \in {\mathbb {R}}_+$ is the Sersic index, $R_e$ is the half-light radius, $I_e$ is the light intensity at $R_e$ and $b_n$ satisfies $\gamma (2n;b_n)=\frac{1}{2}\varGamma (2n)$ with $\varGamma$ and $\gamma$ respectively the Gamma function and the lower incomplete gamma function. We draw the values of the parameters n, $I_e$ and $R_e$ from the catalog COSMOS [20] to generate isotropic galaxy images to which we will later give a non-zero ellipticity.

Second, the light intensity $I_P$ of a PSF is modeled with a Moffat profile using the following formula:

$$\begin{aligned} I_P(R) = 2\frac{\beta -1}{\sigma ^2}\left( 1+\left[ \frac{R}{\sigma }\right] ^2\right) ^{-\beta } , \end{aligned}$$

(49)

where $\beta$ is set to 4.765 (cf. Ref. [34]) and $\sigma$ is calculated using the following relation:

$$\begin{aligned} \text {FWHM} = 2\sigma \sqrt{2^{\frac{1}{\beta }}-1}, \end{aligned}$$

(50)

where FWHM is the Full Width Half Maximum of the Moffat profile. Its value is drawn from a uniform distribution between 0.1 and 0.2 arcsec, which correspond respectively to Hubble Space Telescope³ and Euclid space telescope⁴ observations. This gives us preliminary, isotropic PSF images.

We finally give an ellipticity to the both these galaxy and PSF images. To do so, we draw the values of the ellipticity components from a centered normal distribution truncated between $-1$ and 1. The standard deviations are chosen as 0.3 for the galaxies and 0.03 for the PSF [2].