Hierarchical Interferometric Bayesian Imaging

TL;DR

This work reframes VLBI imaging as hierarchical Bayesian inference (HIBI), introducing latent image variables and hyperparameters to jointly model the sky and instrumental effects while quantifying uncertainty. By adopting Gaussian Markov random field priors with a first-order structure, the approach yields a scalable, data-driven prior that enforces positivity and spatial correlation and remains efficient through sparse precision matrices. The authors demonstrate HIBI on synthetic EHT-like data and real 2017 EHT M87* observations, showing robust recovery of ring-like structures and enabling direct measurement of ring properties, such as width, with credible uncertainties. They further apply HIBI to VLBA data (OJ 287) to illustrate super-resolution capabilities and discuss extensions, including non-Gaussian and higher-order MRFs, as well as connections to RML methods and RESOLVE. The paper provides extensive validation, shows improved feature extraction via ring-based priors, and releases the Comrade.jl implementation for public use, promising more reliable, uncertainty-aware VLBI imaging and calibration.

Abstract

Very long baseline interferometry (VLBI) achieves the highest angular resolution in astronomy. VLBI measures corrupted Fourier components, known as visibilities. Reconstructing on-sky images from these visibilities is a challenging inverse problem, particularly for sparse arrays such as the Event Horizon Telescope (EHT) and the Very Long Baseline Array (VLBA), where incomplete sampling and severe calibration errors introduce significant uncertainty in the image. To help guide convergence and control the uncertainty in image reconstructions, regularization on the space of images is utilized, such as enforcing smoothness or similarity to a fiducial image. Coupled with this regularization is the introduction of a new set of parameters that modulate its strength. We present a hierarchical Bayesian imaging approach (Hierarchical Interferometric Bayesian Imaging, HIBI) that enables the quantification of uncertainty for al parameters. Incorporating instrumental effects within HIBI is straightforward, allowing for simultaneous imaging and calibration of data. To showcase HIBI's effectiveness and flexibility, we build a simple imaging model based on Markov random fields and demonstrate how different physical components can be included, e.g., black hole shadow size, and their uncertainties can be inferred. For example, while the original EHT publications were unable to constrain the ring width of M87*, HIBI measures a width of $9.3\pm 1.3\,μ{\rm as}$. We apply HIBI to image and calibrate EHT synthetic data, real EHT observations of M87*, and multifrequency observations of \oj287. Across these tests, HIBI accurately recovers a wide variety of image structures and quantifies their uncertainties. HIBI is publicly available in the Comrade.jl VLBI software repository.

Figures (12)

• Figure 1: Draws from the GMRF prior in the $\bm{r}$ space (left) and in the linear image space $\bm{\eta}$ (right) assuming that they are related by \ref{['eq:softplus']} The columns of the figures show different correlation lengths for the log-ratio image, while the rows are different values of the image dispersion.
• Figure 2: Example reconstructions of various geometric model morphologies using the GMRF prior. The prior is flexible enough to capture various image morphologies and features, including the ring brightness profile.
• Figure 3: The parameter estimation results from the synthetic data tests in \ref{['fig:synthetic_data_results']}. Overall, HIBI can measure all the parameters of the ground truth image except the disk, where the diameter definition gives a slightly too small result. Analyzing the profile of the disk reconstruction, shown between the disk diameter marginal and on-sky truth, we see that at smaller distances from the origin ($< 20~\mu{\rm as})$, the disk is slightly too dim, while at larger distances ($> 40~\mu{\rm as}$) it is too bright.
• Figure 4: HIBI image reconstructions of M87$^*$ for each observing day and frequency band. The top row shows the MAP image, which is often littered with artifacts, such as ghost rings. The second and third rows display the mean image and its relative uncertainty, respectively. Finally, the bottom row shows a random sample from the posterior, blurred to match the resolution of the mean image using the NXCORR metric.
• Figure 5: Reconstructions of M87* using the ring profile prior and first order GMRF for multiplicative noise.