S-R2D2: a spherical extension of the R2D2 deep neural network series paradigm for wide-field radio-interferometric imaging

TL;DR

S-R2D2 extends the R2D2 deep learning framework to spherical wide-field radio interferometric imaging by embedding a fast Fourier-based sphere-to-plane interpolator $\Gamma$ and its adjoint $\Gamma^{\dagger}$ into a plane-based DNN residual series. The method preserves spherical topology throughout reconstruction, training 2D U-Nets on the plane while enforcing consistency with spherical ground truth via back-projection through $\Gamma^{\dagger}$. A critical balance between interpolation accuracy and computational efficiency is achieved by operating $\Gamma$ and $\Gamma^{\dagger}$ at a lower-than-optimal plane resolution and letting the DNNs learn to correct interpolation errors. Across simulations, S-R2D2 yields significantly higher image-domain fidelity (higher $\mathrm{SNR}$ and $\mathrm{logSNR}$) and robust data fidelity (lower $\mathrm{RDR}$) than R2D2, with the best results around $\mathrm{N}_{\mathrm{p}}=600^2$ and $\mathrm{SR}=2.25$, while maintaining fast runtimes on GPUs. These findings demonstrate a scalable path to high-resolution spherical RI reconstructions and motivate validation on real wide-field RI data and extension to include the $w$-component in future work.

Abstract

Recently, the R2D2 paradigm, standing for ''Residual-to-Residual DNN series for high-Dynamic-range imaging'', was introduced for image formation in Radio Interferometry (RI) as a learned version of the traditional algorithm CLEAN. The first incarnations of R2D2 are limited to planar imaging on small fields of view, failing to meet the spherical-imaging requirement of modern telescopes observing wide fields. To address this limitation, we propose the spherical-imaging extension S-R2D2. Firstly, as R2D2, S-R2D2 encapsulates its minor cycles in existing 2D-Euclidean deep neural network (DNN) architectures, but adapts its iterative scheme to incorporate the wide-field measurement model mapping a spherical image to visibility data. We implemented this model as the composition of an efficient Fourier-based interpolator mapping the spherical image onto the equatorial plane, with the standard RI operator mapping the equatorial-plane image to visibility data. Importantly, the interpolation step must inevitably be performed at a lower-than-optimal resolution on the plane, to meet the high-resolution requirement on the sphere of wide-field imaging while preserving scalability. Therefore, secondly, we design S-R2D2's DNN training loss to jointly learn to correct the interpolation approximations and identify residual image structures on the sphere, ensuring consistency with the spherical ground truth using the adjoint plane-to-sphere interpolator. Finally, we demonstrate through simulations S-R2D2's capability to perform fast and accurate reconstructions of spherical monochromatic intensity images, across high-resolution, high-dynamic-range settings.

Figures (13)

• Figure 1: Illustration of a wide-field RI inverse problem, imaged on the sphere. Panel (a) shows a VLA-type sampling pattern. Panel (b) shows, in a linear scale, the back-projection of RI measurements onto the sphere, generated from the measurement equation \ref{['eqn:vis_discrete_2D_sphere']}, using the sampling pattern (a) and the ground truth (c). This spherical ground truth (c) is generated from the Messier 106 radio galaxy shimwell2022 with the procedure depicted in Section \ref{['subsec:sec_4_subsec_1']}, and displayed in logarithmic scale with the logarithmic exponent equals to its dynamic range ($\textrm{DR}=1.4\!\times\!10^5$). Solving this inverse problem consists of reconstructing the target (c) from the back-projected measurements (b). We visualise the Northern hemisphere in the orthographic projection perspective in panels (b) and (c).
• Figure 2: Panel (a) illustrates the sample arrangement on the sphere using the uniform equal-area HEALPix scheme. The invertible projection, depicted with dashed lines in panel (b), maps a point on the sphere with coordinates $(l,m,n)$ to the equatorial plane by setting $n=0$. Thus, without loss of information, spherical samples, represented on the HEALPix grid, are equivalently nonuniform samples on the equatorial plane, as shown in panel (c). Moreover, panel (b) showcases that at the north pole, the frequency content is preserved, because the distances after projection are preserved. The maximum projected frequency content arises at the extents of the FOV, where the frequency content is projected onto much higher frequency content on the $(l, m)$ plane.
• Figure 3: Illustration of the effect of the application of the super-resolution equation \ref{['eqn:SR_rule']} on the visibility domain using three different $\textrm{SR}$ levels: $\textrm{SR}=1.5$ (panel (a)), $\textrm{SR}=2.25$ (panel (b)) and $\textrm{SR}=3$ (panel (c)), which respectively represent visibility domains with $\mathrm{N}_{\mathrm{p}}=400^2$, $\mathrm{N}_{\mathrm{p}}=600^2$ and $\mathrm{N}_{\mathrm{p}}=800^2$ pixels. The sampling pattern is framed and preserved across all $\textrm{SR}$ levels.
• Figure 4: Illustration of the effect of the operator $\mathbf{\Gamma}^{\dagger}\mathbf{\Gamma}$ on ground-truth signals $\mathbf{x}_{\mathrm{s}}^{\star}$. The two metric plots, on the left, show respectively the evolution of $\textrm{SNR}$ (top) and $\textrm{logSNR}$ (bottom) metrics (defined in Section \ref{['subsec:sec_5_subsec_2']}) between $\mathbf{x}_{\mathrm{s}}^{\star}$ and $\mathbf{\Gamma}^{\dagger}\mathbf{\Gamma}\mathbf{x}_{\mathrm{s}}^{\star}$, as a function of the resolution on the plane $\mathrm{N}_{\mathrm{p}}$. The values represent the averages computed over the 60 signals part of the test dataset, defined in Section \ref{['subsec:sec_4_subsec_3']}. $\mathrm{N}_{\mathrm{s}}$ is fixed for all images and its choice is detailed in Section \ref{['subsec:sec_5_subsec_1']}. Lower values of $\mathrm{N}_{\mathrm{p}}$ result in reduced interpolation accuracy with critical aliasing artifacts. The signals $\mathbf{x}_{\mathrm{s}}^{\star}$ do not satisfy any band-limit assumption. Therefore, we do not observe a saturation in metric values beyond the resolution-rule value, denoted $\mathrm{N}_{\mathrm{p}}^{\mathrm{r}}$. On the right, we visualise a specific ground truth $\mathbf{x}_{\mathrm{s}}^{\star}$ alongside the corresponding $\mathbf{\Gamma}^{\dagger}\mathbf{\Gamma}\mathbf{x}_{\mathrm{s}}^{\star}$ for different $\mathrm{N}_{\mathrm{p}}$. The ground truth was generated with a dynamic range $\textrm{DR} = 1.1\!\times\!10^5$ from the radio galaxy 3c354 (NRAO Archives), following the procedure described in Section \ref{['subsec:sec_4_subsec_1']}. Values of ($\textrm{SNR}$, $\textrm{logSNR}$) are reported below each panel. We visualise the Northern hemisphere in the orthographic projection perspective and in logarithmic scale (with the logarithmic exponent being equal to $\textrm{DR}$), clipping the negative values for visualisation purposes.
• Figure 5: Illustration of the generation of a high-$\textrm{DR}$ spherical ground truth (panel (iv), $\textrm{DR}=1.7\!\times\!10^5$), starting from the initial low-$\textrm{DR}$ planar image (panel (i), $\textrm{DR}=45$). The domain (plane or sphere) and the $\textrm{DR}$ are noted below each image. Firstly, we apply the disk selection step from (panel (i)) to (panel (ii)). Then, from (panel (ii)) to (panel (iii)), we back-project onto the sphere the disk with the ray-tracing operator which generates a low-$\textrm{DR}$ spherical signal. Finally, we apply the $\textrm{DR}$ exponentiation and tapering operations to produce the final high-$\textrm{DR}$ spherical signal (panel (iv)). All images are displayed in logarithmic scale, with the logarithmic exponent equals to their $\textrm{DR}$. For the spherical signals, we visualise the Northern hemisphere in the orthographic projection perspective.