Bayesian imaging inverse problem with scattering transform

TL;DR

The paper tackles Bayesian imaging inverse problems in astrophysics under extreme data scarcity and non-Gaussian signals. It shifts inference from high-dimensional pixel space to a low-dimensional space of Scattering Transform statistics $\mu_S$, using a Taylor-expanded Gaussian likelihood $p(\phi(d)\mid \mu_S) \approx \mathcal{N}(A\mu_S + b, \Sigma)$ and an adaptive sequential estimator to approximate $p(\mu_S \mid \phi(d_0))$. The main contributions are (i) a tractable SBI-inspired algorithm that refines Gaussian proposals over $\mu_S$, (ii) demonstration on Quijote large-scale structure maps with stochastic forward contamination showing that samples from $p(\mu_S \mid \phi(d_0))$ reproduce the true field’s statistics and are indistinguishable from data after forward processing, and (iii) a pixel-level reconstruction pathway using a neural denoiser trained on ST-posterior samples. This approach enables robust non-Gaussian signal inference in regimes with little or no external priors, with potential applications to Galactic dust and multi-frequency cosmological data.

Abstract

Bayesian imaging inverse problems in astrophysics and cosmology remain challenging, particularly in low-data regimes, due to complex forward operators and the frequent lack of well-motivated priors for non-Gaussian signals. In this paper, we introduce a Bayesian approach that addresses these difficulties by relying on a low-dimensional representation of physical fields built from Scattering Transform statistics. This representation enables inference to be performed in a compact model space, where we recover a posterior distribution over signal models that are consistent with the observed data. We propose an iterative adaptive algorithm to efficiently approximate this posterior distribution. We apply our method to a large-scale structure column density field from the Quijote simulations, using a realistic instrumental forward operator. We demonstrate both accurate statistical inference and deterministic signal reconstruction from a single contaminated image, without relying on any external prior distribution for the field of interest. These results demonstrate that Scattering Transform statistics provide an effective representation for solving complex imaging inverse problems in challenging low-data regimes. Our approach opens the way to new applications for non-Gaussian astrophysical and cosmological signals for which little or no prior modeling is available.

Figures (5)

• Figure 1: Top: True field $s_{0}$ and three fields generated from ST statistics sampled from the posterior $p(\mu_{S} \mid \phi(d_{0}))$. Bottom: Observation $d_{0}$ and the corresponding posterior predictive samples obtained by applying the pixel-space forward operator $F$ to the generated fields. The predictive samples are visually indistinguishable from the observation, while the generated fields seem consistent with LSS maps, with variability primarily at the smallest, noise-dominated scales.
• Figure 2: Schematic of the iterative algorithm used in this paper. See Sec. \ref{['subsec:algorithm']} for more details.
• Figure 3: Top: Comparison of summary statistics for the true field $s_{0}$ (red), the observation $d_{0}$ (blue), samples generated from the posterior distribution $p(\mu_{S} \mid \phi(d_{0}))$ (green), and the posterior predictive (PP) distribution (purple). The statistics are power spectrum, and one-point probability distribution function (PDF) shown in both linear and logarithmic scales. Bottom: Residual of the statistics of $s_{0}$ normalized by the standard deviation of the posterior distribution. We see that the posterior samples accurately reproduce the statistical properties of the true field, while the posterior predictive samples match those of the observation, up to sample variance.
• Figure 4: True signal $s_{0}$, observation $d_{0}$, and posterior mean and standard deviation maps, both estimated pixel-wise using a moment network. The posterior mean recovers the large-scale features of the true signal while filtering the noise-dominated small scales. In the masked regions, where the data provide no information, the posterior uncertainty is large, as expected in a Bayesian framework.
• Figure 5: Comparison of Minkowski functionals for the true field $s_{0}$ (red), the observation $d_{0}$ (blue), samples generated from the posterior distribution $p(\mu_{S} \mid \phi(d_{0}))$ (green), and the posterior predictive distribution (purple). We see that the posterior samples accurately reproduce the statistical properties of the true field with a slight bias for $M1$, while the posterior predictive samples match those of the observation, demonstrating the consistency of the inferred posterior with the data under the forward model.