Event-horizon-scale Imaging of M87* under Different Assumptions via Deep Generative Image Priors

Abstract

Reconstructing images from the Event Horizon Telescope (EHT) observations of M87*, the supermassive black hole at the center of the galaxy M87, depends on a prior to impose desired image statistics. However, given the impossibility of directly observing black holes, there is no clear choice for a prior. We present a framework for flexibly designing a range of priors, each bringing different biases to the image reconstruction. These priors can be weak (e.g., impose only basic natural-image statistics) or strong (e.g., impose assumptions of black-hole structure). Our framework uses Bayesian inference with score-based priors, which are data-driven priors arising from a deep generative model that can learn complicated image distributions. Using our Bayesian imaging approach with sophisticated data-driven priors, we can assess how visual features and uncertainty of reconstructed images change depending on the prior. In addition to simulated data, we image the real EHT M87* data and discuss how recovered features are influenced by the choice of prior.

Figures (3)

• Figure 1: Method illustration. The CIFAR-10 prior was used for these examples; images are shown as $32\times 32$ pixels on a $[0,1]$ scale. At a high level, we optimize a variational distribution $q_\phi$ to approximate the image posterior $p_\theta(\cdot\mid\mathbf{y})$ given a score-based prior $p_\theta$ and log likelihood based on EHT measurements. Panel (a) illustrates our particular variational distribution: a RealNVP with parameters $\phi$. At each optimization iteration $i$, the measurement log likelihood (Equation \ref{['eq:llh']}) and the log density under the score-based prior of each sample $\mathbf{x}$ from $q_\phi= q_{\phi^{(i)}}$ are evaluated. The average gradient is computed with respect to $\phi$ to update $\phi^{(i)}$. In other words, $q_\phi$ is optimized to produce samples that have high probability under both measurement likelihood and prior. Panel (b) zooms in to the score-based prior. A score-based prior is based on a score-based diffusion model, a deep generative model with parameters $\theta$, that is trained on images from a target prior. Once trained, the diffusion model generates samples from a generative image distribution $p_\theta$. There is an analytical formula for computing the ELBO $b_\theta(\mathbf{x})$ of the log probability $\log p_\theta(\mathbf{x})$ for any image $\mathbf{x}$, even for out-of-distribution images and images of pure noise.
• Figure 2: Score-based priors used in this work. Nine samples from each learned prior are shown.
• Figure 3: Image reconstructions from simulated data. A sample (one sample from each mode if the posterior is bimodal) is shown from each estimated posterior. Qualitatively, the CIFAR-10 prior adds the least amount of bias, producing reasonable reconstructions of each image in this dataset. The GRMHD prior strongly prefers a centered ring in the image. The RIAF prior prefers a centered ring- or disk-like structure in the image. The CelebA prior struggles to recover these source images, in some cases adding face features, and it leads to the most multimodal posteriors. However, it performs decently well on certain images like the Crescent and GRMHD images. When the source image is known to be well-approximated by a GRMHD or RIAF model, the more constrained GRMHD or RIAF prior may be the best choice.