# Shape measurement of radio galaxies using Equivariant CNNs

Priyamvad Tripathi, Sunrise Wang, Simon Prunet, André Ferrari

► To cite this version:

Priyamvad Tripathi, Sunrise Wang, Simon Prunet, André Ferrari. Shape measurement of radio galaxies using Equivariant CNNs. 2024 32nd European Signal Processing Conference (EUSIPCO 2024), Aug 2024, Lyon, France. pp.2377-2381, 10.23919/EUSIPCO63174.2024.10715370. hal-05318201

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Shape measurement of radio galaxies
using Equivariant CNNs

Priyamvad Tripathi, Sunrise Wang, Simon Prunet and André Ferrari

Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS
Laboratoire J.-L. Lagrange, Nice, France

*Abstract*—Images of distant galaxies are sheared due to gravitational lensing from intervening astrophysical structures. Shear can be estimated by measuring the shape of galaxies, however, such estimations are biased due to various non-cosmological artifacts introduced in the observation process. Traditional methods to mitigate such biases rely on using simulated images to estimate bias and then correcting the real observations with this bias. Such methods rely on the accuracy of the simulations, and are computationally expensive, particularly in the radio domain. As a first step towards a properly calibrated shear field estimator, we present a supervised deep-learning model based on equivariant convolutional neural networks (CNNs) that can recover ellipticities from reconstructed images of isolated radio galaxies. Our fiducial model recovers ellipticities from simulated galaxies with a measurement bias level of $10^{-3}$ or better which is competitive with the ellipticity measurements from the state-of-the-art studies in the radio domain.

I. INTRODUCTION

Images of distant galaxies are distorted due to the weak gravitational lensing from the large-scale structure in the Universe along the line of sight. This phenomenon is known as cosmic shear and it serves as a probe to study the properties and evolution of structures on large scales as well as, potentially, the geometry of the Universe [1]. Radio surveys typically have lower sub-arcsec resolution, sub-$\mu$Jy depth, and wide-area coverage when compared to optical surveys. This resulted in weak lensing studies mainly being limited to the optical waveband except for a handful of studies where a shear signal was detected in the radio domain [2], [3], [4] although with a lower detection significance. Upcoming radio surveys like the Low-Frequency Array for Radio Astronomy (LOFAR) [5] and the Square Kilometre Array (SKA) [6] are expected to reach sufficient sensitivity to resolve radio emission of ordinary galaxies and therefore will provide the necessary datasets for weak lensing studies. However, such studies would also require the development of highly accurate shape measurement techniques suitable for a radio interferometric dataset.

Radio-interferometers measure galaxies as discrete samples in the Fourier domain of the image space known as the visibility space. The general radio-interferometric approach to reconstructing an image from the visibilities is an iterative process. The visibilities are gridded on a regularly spaced grid and corrected for geometric and propagation effects. Then an inverse Fast Fourier Transform is performed on the gridded visibilities to get a so-called dirty image. The

Point-Spread function (PSF) is then deconvolved from the dirty image to get an image estimate. To account for the artifacts introduced due to gridding, degridded visibilities are computed from the estimated image and subtracted from visibility measurement. A residual image is then recomputed from these visibilities and added to the current image estimate to get an image estimate for the next iterative cycle. A proper radio-interferometric reconstruction process involves multiple such cycles. Although this process preserves the underlying cosmological information, it is typically much more expensive than the algorithms used to obtain galaxy images in the optical domain due to complex gridding/degridding operations. The image reconstructed after deconvolving the PSF from the dirty image does not fully preserve the original shape information [7]. This error mostly depends on the geometry of antenna arrays with respect to the galaxy position and the Fourier space sampling pattern, and this information can essentially be summarised using the PSF.

In this work, we propose a deep learning model based on equivariant CNNs that can recover the original galaxy shape information using a reconstructed image and the associated PSF. Our model eliminates the need for a full, computationally expensive, radio-interferometric image reconstruction process. In Sec. II, we present a mathematical formulation of the problem and a brief summary of general methods used for shape measurement and bias calibration. In Sec. III, we describe the details of our network. Finally in Sec. IV, we present the details and results of our simulations and show that our fiducial model can recover original galaxy ellipticities with an accuracy that is competitive with the state-of-the-art ellipticity measurement methods in the radio domain.

II. BACKGROUND

Cosmic shear is a phenomenon in which images from distant galaxies are distorted due to the gravitational lensing caused by the intervening matter between us and the galaxy. In the weak lensing regime, the effect of cosmic shear can be parameterized by the spin-2 shear vector $\gamma = (\gamma_1, \gamma_2)$. $\gamma$ is an observable that quantifies the anisotropic stretching of the galaxy image perpendicular to the line of sight [8] and it modifies the galaxy ellipticity as:

$$ \epsilon^{\text{true}} \approx \epsilon^I + \gamma \quad (1) $$
where $\epsilon^{\text{true}} \in \mathbb{R}^2$ is the apparent galaxy ellipticity and $\epsilon^I \in \mathbb{R}^2$ is the intrinsic ellipticity of the galaxy.

Shear only makes a small portion of the overall galaxy ellipticity i.e. $\|\gamma\| \ll \|\epsilon\|$ and to recover the shear signal, the ellipticity measurement needs to be very precise. However, estimating the galaxy ellipticity from a reconstructed image is a challenging task since galaxy images are typically noisy, faint, and small. They are also corrupted due to sparse sampling on the Fourier domain, discretization errors due to gridding/de-gridding operations, calibration errors, etc. This problem is illustrated in the example plotted in Fig. 1, where the ellipticity recovered from the reconstructed image is not the same as the true ellipticity.

This measurement bias in the ellipticity can be summarized using a linear model i.e. a multiplicative bias $m$ and an additive bias $c$ as:

$$ \Delta\epsilon_i \equiv \epsilon_i^{\text{meas}} - \epsilon_i^{\text{true}} = m_i\epsilon_i^{\text{true}} + c_i \quad (2) $$

where $\epsilon_i^{\text{meas}}$ and $\epsilon_i^{\text{true}} \equiv \epsilon_i^I + \gamma_i$ are the measured and true ellipticity respectively, and $i = 1, 2$ corresponds to the two ellipticity components $\epsilon = (\epsilon_1, \epsilon_2)$. Bias parameters $m$ and $c$ have non-trivial dependence on galaxy properties (e.g. size, magnitude, intrinsic ellipticity), redshift, PSF, etc. In the absence of intrinsic alignment of the background galaxies, i.e. $\mathbb{E}[\epsilon^I] = (0, 0)$, we can recover the shear signal by taking the average of measured ellipticity over a set of galaxies:

$$ \gamma_i^{\text{reco}} = \mathbb{E}[\epsilon_i^{\text{meas}}] = (1 + m_i)\gamma_i^{\text{true}} + c_i \quad (3) $$

where $\gamma_i^{\text{reco}}$ is the recovered and $\gamma_i^{\text{true}}$ the true shear signal. It should be noted that Eq. 3 is only a crude approximation since it ignores possible correlations between ellipticity measurement errors of different galaxies. Such correlations would potentially increase the shear bias, however, we do not investigate these effects in this work. The bias in individual galaxy ellipticity measurements that we calculate using Eq. 2 is only an estimate of the overall shear bias.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Example of a simulated galaxy with ellipticity $\epsilon = (0.236, 0.001)$. The images plotted from left to right are the true image $x$, dirty image $x^D$, PSF $h$, and reconstructed image $\hat{x}$. All the images are simulated on 128 × 128 pixel scale with a resolution 0.0717 arcsec per pixel. Only the central 64 × 64 pixels part of the images is shown for clarity. The red and yellow curves show the ellipticity measured from the true and reconstructed images respectively.</figcaption></figure>

In the optical domain, shear bias is calculated by applying known shear fields to real observations and measuring the

observed shear variations with respect to the applied input shear [9], [10]. This allows them to estimate the shear bias for each observed galaxy and the real observations are then corrected for this bias. Such techniques rely on simulating multiple sheared versions of real galaxies and hence would be very costly in the radio domain as the radio-interferometric reconstruction process is computationally much more expensive. There also have been studies to estimate shear from galaxy images using neural networks [11], [12] but in the radio domain, these methods would require a proper image reconstruction. There are other reconstruction algorithms that preserve the underlying shape information [13], [14], however, we do not use these methods since our tests showed that our fiducial network was able to recover shape information from basic reconstructed images. The current state-of-the-art methods provide ellipticity estimates in the radio domain with a measurement bias at the level of $10^{-3}$ or worse [15], [16], [13], [14].

### III. METHODOLOGY

Our main contribution is a model to predict the true ellipticities from reconstructed galaxy images and associated PSFs. Our model can be broken into two parts: a feature extraction layer that extracts features from the reconstructed image $\hat{x}$ and an encoder layer that encodes the PSF $h$ into a latent vector with reduced dimensions. The overall structure of the network is summarized in Fig. 2.

**Image features extraction** - We used an $E(2)$ equivariant convolutional neural network (CNN) for our feature extraction layer. The motivation behind using an equivariant network is that the galaxy ellipticity is equivariant to the actions of the $E(2)$ group on galaxy images, for example, if the galaxy image is rotated by a certain angle, the measured ellipticity should also rotate by the same angle. A similar argument can be given for the other actions of the $E(2)$ group which include translations, rotations, and reflections of $\mathbb{E}^2$. We choose equivariant CNNs since they preserve this equivariance relation between the input image and output features and have been shown to outperform traditional CNNs trained using data-augmentation techniques [17], [18].

We build the equivariant layer by replacing traditional convolutional kernels with kernels expressed in a steerable basis of the $E(2)$ group:

$$ k(\boldsymbol{x}|\boldsymbol{w}) = \sum_{\ell=1}^{8} w_{\ell}(r)Y_{\ell}(\alpha) \quad (4) $$

where $\boldsymbol{x} = (r, \alpha)$, $Y_{\ell}(\alpha) = e^{i\ell\alpha}$ are the basis vectors and the kernel weights $w_{\ell}(r)$ have a radial symmetry. We refer to [17] for a mathematical description of equivariant convolutions. A traditional convolution produces a scalar feature map that is equivariant to translations while a convolution with a steerable kernel produces a vector feature map that is equivariant to the actions of the $E(2)$ group:

$$ C_{E(2)}[G(\hat{x})] = G'[C_{E(2)}(\hat{x})] \quad (5) $$
<figure><img src="image_3.png" /><figcaption>Fig. 2: The architecture of the deep learning network used for the fiducial model. The reconstructed image $\hat{x}$ is passed through three $E(2)$ equivariant convolutional layers and two average pool layers. The PSF $h$ is passed through an encoder/decoder architecture (bottom row of the diagram) with three 2D convolutional/transposed convolutional layers followed by a dense layer. The latent vector $z(h)$ is converted into the correct dimensions using an unflattening layer. All the dense layers have a linear activation while the convolutional layers have a ReLU activation.</figcaption></figure>

where $\hat{x}$ is an image, $C_{E(2)}$ is an equivariant convolution layer, and $G$ and $G'$ are the representation of the $E(2)$ group acting on 2D images and the vector feature map respectively. We use multiple equivariant convolutions for our feature extraction layer $R_{E(2)}$ and the architecture parameter $\theta_{R_{E(2)}}$ are the set of all the kernel weights $w_l(r)$ from these convolutional layers.

**PSF encoding** - We also train our model to remove the bias caused by different PSFs. This is an important part of our analysis since we use a basic reconstruction process, the estimated image still contains artifacts from the PSF as shown in the example in Fig. 1. To do this we first train an autoencoder to encode the PSFs $h$ to a latent vector $z_{\theta_E}(h)$. The autoencoder is based on a standard convolutional encoder-decoder architecture [19] and is trained to minimize the MSE error:

$$ \{\hat{\theta}_E, \hat{\theta}_D\} = \underset{\{\theta_E, \theta_D\}}{\operatorname{argmin}} \mathbb{E}_h [\|h - \hat{h}_{\theta_D}[z_{\theta_E}(h)]\|^2] \quad (6) $$

where $\hat{h}_{\theta_D}$ is the output from the decoder, $z_{\theta_E}$ is the output from the encoder and $\{\theta_E, \theta_D\}$ the encoder-decoder architecture parameters.

**Fiducial network** - Our fiducial network $N_{\theta_E, \hat{\theta}_E}$ concatenates the outputs from the feature extraction layer $R_{E(2)}$ and the pre-trained encoder layer $z_{\hat{\theta}_E}$ and passes it to a fully connected dense layer to predict the ellipticities: $\epsilon^{\text{meas}} \equiv N_{\theta_E, \hat{\theta}_E}(\hat{x}, h)$. The network is trained to minimize the following loss function:

$$ \hat{\theta} = \underset{\theta}{\operatorname{argmin}} \mathbb{E}_{(\hat{x}, h, \epsilon^{\text{true}})} [\| N_{\theta, \hat{\theta}_E}(\hat{x}, h) - \epsilon^{\text{true}} \|^2] \quad (7) $$

where $\epsilon^{\text{true}}$ is the simulated ellipticity of the galaxy and $\theta$ are the model parameters that need to be optimized i.e. $\theta_{R_{E(2)}}$ and

the parameters of the dense layer. We use the python library `escnn` [20] to build the equivariant layer and our model is compiled using `PyTorch` [21].

## IV. SIMULATION RESULTS

### A. Generating Galaxy Images

We use the Star-Forming Galaxies (SFGs) catalogue from the Tiered-Radio Extragalactic Continuum Simulation (T-RECS) [22] for our population model. To simulate galaxy images, we use the GalSim galaxy image simulation toolkit [23] and model the galaxy shape using a Sérsic exponential brightness profile. The galaxy is then provided with an intrinsic ellipticity as specified by the T-RECS catalogue. This is equivalent to applying a shear equal to the intrinsic ellipticity to a circular galaxy. Every simulated galaxy varies in the following properties: flux, size, ellipticity, and position in the sky.

We then generate visibility positions based on the SKA-MID telescope configuration and assuming that the observations are made at a central frequency of 1.4GHz and last 4 hours with visibilities sampled every 300 s. We model the noise received by antennas by perturbing the visibilities with noise sampled from $N(0, (\sigma/20)^2)$, where $\sigma$ is the standard deviation of the visibilities. The noisy visibilities are then used to construct a dirty image. The PSFs are then deconvolved from the dirty images using the Multi-Scale CLEAN algorithm [24] to get our reconstructed image. We use the Radio Astronomy Simulation, Calibration, and Imaging Library, RASCIL<fnref n="1" /> to calculate the visibilities, dirty images, and PSFs. We plot an

<footnote n="1">¹RASCIL Documentation</footnote>
example of a simulated galaxy in Fig. 1: the true image, the
dirty image, the PSF, and the reconstructed image.

B. Traditional CNN vs Equivariant CNN for a fixed PSF

For the first part of the analysis, we compare the perfor-
mance of a traditional CNN layer versus an equivariant CNN
layer in extracting features from reconstructed images for a
fixed PSF. To do so we use a simpler architecture that contains
a feature extraction layer (either traditional or equivariant
CNN) followed by a dense layer. We simulated a dataset of
20,000 reconstructed galaxy images to train/test both models,
which is further split into 70%-10%-20% subsets for train-
validation-test purposes. To ensure that all the galaxies have
the same PSF, we simulated the galaxies at the same position
in the sky. In Fig. 3, we plot the measured vs true ellipticities
for the galaxies in the test subset. We estimate the ellipticity
measurement bias (defined in Eq. 2) by calculating the slope
$\hat{m}_i$ and intercept $\hat{c}_i$ of the best-fit linear relation between
$\Delta\epsilon_i = \epsilon_i^{\text{meas}} - \epsilon_i^{\text{true}}$ and $\epsilon_i^{\text{true}}$ where $\epsilon_i^{\text{meas}}$ is the output ellipticity
from the trained models and $\epsilon_i^{\text{true}}$ is the true ellipticity ($i = 1,2$
corresponds to the two ellipticity components).

We further tested the performance of the networks on sim-
ulated datasets of 1000 reconstructed galaxy images with
constant ellipticities. We passed the images to the two net-
works and estimated the squared bias $\mu^2 \equiv \|\mathbb{E}[\hat{\epsilon}] - \epsilon\|^2$
and variance $\sigma^2 \equiv \mathbb{E}[\|\hat{\epsilon} - \mathbb{E}[\hat{\epsilon}]\|^2]$ where $\hat{\epsilon}$ is the recovered
ellipticities and $\epsilon$ is the constant ellipticity. We calculated
$\mu^2$ and $\sigma^2$ on five such datasets with constant ellipticity
$\epsilon \in \{(0,0), (-0.2,-0.2), (-0.2,0.2), (0.2,-0.2), (0.2,0.2)\}$
and the results are reported in Tab. I. We find from these simu-
lations that both architectures recover the original ellipticities
with a measurement bias level of $10^{-2}$ or better. However,
the equivariant architecture has significantly lower biases. It
also beats the traditional architecture in terms of estimation
variance and bias for all input ellipticity values that we tested.

C. Fiducial Model

In this second part of the simulations, we evaluate the
performance of the model described in Fig. 2 that aims to
remove the bias caused by different PSFs. We first train
the autoencoder layer using 20,000 simulated PSFs. We then
simulated a dataset of 15,000 reconstructed galaxy images and
the associated PSFs and pass it to our network as explained
in Sec. III. The pre-trained autoencoder layer encodes PSF
$h$ with $\dim(h) = 128 \times 128$ to latent vector $z_{\theta_D}(h)$ with
$\dim(z) = 288$. We tested two scenarios for the weight ini-
tialization of our equivariant feature extraction layer $R_{E(2)}$.
In the first case $R_{E(2)}$ was initialized with weights $\hat{\theta}_{R_{E(2)}}$
learned using a single PSF (this is our fiducial model) and
second where it was initialized with random weights. In both
cases, we retrained the equivariant and dense layer, so that
the model can learn to remove the biases caused by arbitrary
PSFs. We plot the ellipticities measured by the two models in
Fig. 4 along with the estimated ellipticity measurement bias.
We also report the estimation bias and variance for the two
models in Tab. I (calculated using the procedure in Sec. IV-B).

<figure><img src="image_4.png" /><figcaption>(a) Traditional CNN architecture</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(b) Equivariant architecture $R_{E(2)}$</figcaption></figure>

Fig. 3: Comparison of $\Delta\epsilon = \epsilon^{\text{meas}} - \epsilon^{\text{true}}$ vs $\epsilon^{\text{true}}$ where $\epsilon^{\text{meas}}$ is the output from the trained models and $\epsilon^{\text{true}}$ the simulated ellipticity. The two components $\Delta\epsilon_1$ and $\Delta\epsilon_2$ are plotted in blue and orange respectively. The legend indicates the ellipticity measurement bias $\hat{m}_i$ and $\hat{c}_i$ in the order: $\{\hat{m}_i, \hat{c}_i\}$ for the two components. These models have been trained/tested on images with the same PSF.

We find that the model initialized with pre-learned weights has better estimation bias and variance for all input ellipticities that we tested except one (the MSE $\mu^2 + \sigma^2$ is lower for all test cases). This is expected because the original equivariant model had already learned weights that work well for a single PSF. If we transfer these pre-learned weights to the equivariant layer of the full model, the optimizer only needs to fine-tune these weights. This results in faster convergence as compared to the model initialized with random weights. In terms of ellipticity measurement bias, the fiducial model performs only slightly better than the other model. Both the models recover ellipticity with a bias at the order $10^{-3}$ or smaller, which is competitive with the ellipticity measurements from the state-of-the-art studies in radio domain [15], [13], [14], [16].

V. CONCLUSION

In conclusion, we proposed a neural network architecture
based on equivariant convolutions that predicts radio galaxies
ellipticity from deconvolved images and their associated PSF.
We showed that the concepts of equivariant convolutional net-
works and weight transfer can be used for faster convergences
in the case of a limited training dataset. Our tests showed
that our model recovers ellipticities at an accuracy that is
<table><caption>TABLE I: Bias $\mu$ and Variance $\sigma^2$ of the estimated ellipticity for the different datasets from different models. The best statistic for each case is highlighted in bold. $\epsilon$ corresponds to the input ellipticity and the subscripts 0, A, B, C and D corresponds to an input ellipticity of (0, 0), (-0.2, -0.2), (-0.2, 0.2), (0.2, -0.2) and (0.2, 0.2) respectively.</caption><thead><tr><th></th><th></th><th>μ<sup>2</sup><br>[10<sup>-6</sup>]</th><th>σ<sup>2</sup><br>[10<sup>-6</sup>]</th></tr></thead><tbody><tr><td rowspan="5">Fixed PSF</td><td rowspan="4">Traditional CNN<br>Architecture</td><td>ε<sub>0</sub></td><td>5.12</td><td>80.66</td></tr><tr><td>ε<sub>A</sub></td><td>7.51</td><td>149.12</td></tr><tr><td>ε<sub>B</sub></td><td>80.96</td><td>140.31</td></tr><tr><td>ε<sub>C</sub></td><td>5.29</td><td>163.71</td></tr><tr><td>ε<sub>D</sub></td><td>14.97</td><td>164.05</td></tr><tr><td rowspan="5">Varying PSF</td><td rowspan="4">Equivariant CNN<br>Architecture</td><td>ε<sub>0</sub></td><td><b>1.99</b></td><td><b>25.88</b></td></tr><tr><td>ε<sub>A</sub></td><td><b>0.08</b></td><td><b>59.83</b></td></tr><tr><td>ε<sub>B</sub></td><td><b>1.29</b></td><td><b>47.38</b></td></tr><tr><td>ε<sub>C</sub></td><td><b>3.64</b></td><td><b>42.85</b></td></tr><tr><td>ε<sub>D</sub></td><td><b>0.13</b></td><td><b>52.54</b></td></tr><tr><td rowspan="5"></td><td rowspan="4">Full model with<br>equivariant layer initialized<br>at random weights</td><td>ε<sub>0</sub></td><td>1.60</td><td>64.56</td></tr><tr><td>ε<sub>A</sub></td><td>13.30</td><td>113.69</td></tr><tr><td>ε<sub>B</sub></td><td>6.09</td><td>68.14</td></tr><tr><td>ε<sub>C</sub></td><td><b>4.24</b></td><td>120.98</td></tr><tr><td>ε<sub>D</sub></td><td>16.31</td><td><b>77.08</b></td></tr><tr><td rowspan="5"></td><td rowspan="4">Full model with<br>equivariant layer initialized<br>at pre-learned weights θ̂<sub>R<sub>E</sub>(2)</sub></td><td>ε<sub>0</sub></td><td><b>0.18</b></td><td><b>62.70</b></td></tr><tr><td>ε<sub>A</sub></td><td><b>2.26</b></td><td><b>89.03</b></td></tr><tr><td>ε<sub>B</sub></td><td><b>5.90</b></td><td><b>66.10</b></td></tr><tr><td>ε<sub>C</sub></td><td>5.69</td><td><b>109.37</b></td></tr><tr><td>ε<sub>D</sub></td><td><b>8.05</b></td><td>80.33</td></tr></tbody></table>

competitive with ellipticity measurements from state-of-the-art methods.

## ACKNOWLEDGMENT

This research was funded by the National Research Agency (ANR) under the project ANR-22-CE31-0014-01 TOSCA.

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Fig. 4: Comparison of $\Delta\epsilon = \epsilon^{\text{meas}} - \epsilon^{\text{true}}$ vs $\epsilon^{\text{true}}$ where $\epsilon^{\text{meas}}$ is the output from the trained models and $\epsilon^{\text{true}}$ the simulated ellipticity. The legend is the same as in Fig. 3. These models have been trained/tested on images with different PSFs.

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