Estimation of instrument and noise parameters for inverse problem based on prior diffusion model

TL;DR

The paper tackles joint estimation of instrument and noise parameters in Bayesian inverse problems with a diffusion-prior for the image, framed by a linear observation model $\mathbf{y}=\mathbf{H}_{\iota}\mathbf{x}_0+\mathbf{e}$. It introduces a Gibbs sampling approach built on a diffusion prior (with latent ${x}_{1:T}$ and forward/backward processes) to sample the full posterior $\pi_{0:T}(\mathbf{x}_{0:T},\boldsymbol{\theta}|\mathbf{y})$, including conjugate updates for $m_e$ and $\gamma_e$ and a Metropolis-Hastings step for $\boldsymbol{\iota}$. The method yields optimal estimators (e.g., posterior mean) and uncertainty quantification, demonstrated on a MNIST-based diffusion-prior problem with a Lorentz PSF, achieving accurate parameter recovery and high-quality image restoration with favorable computational efficiency. This work enables robust auto-calibration of observation parameters within diffusion-prior inverse problems, offering scalable sampling and practical impact for applications requiring joint image reconstruction and instrument/noise calibration.

Abstract

This article addresses the issue of estimating observation parameters (response and error parameters) in inverse problems. The focus is on cases where regularization is introduced in a Bayesian framework and the prior is modeled by a diffusion process. In this context, the issue of posterior sampling is well known to be thorny, and a recent paper proposes a notably simple and effective solution. Consequently, it offers an remarkable additional flexibility when it comes to estimating observation parameters. The proposed strategy enables us to define an optimal estimator for both the observation parameters and the image of interest. Furthermore, the strategy provides a means of quantifying uncertainty. In addition, MCMC algorithms allow for the efficient computation of estimates and properties of posteriors, while offering some guarantees. The paper presents several numerical experiments that clearly confirm the computational efficiency and the quality of both estimates and uncertainties quantification.

Figures (5)

• Figure 1: Hierarchy --- ${\mathb{x}}\xspace_{ 0}$ is the image of interest, ${\mathb{x}}\xspace_{ 1:T}$ are the latent images and ${\mathb{y}}\xspace$ is the measured image (blurred and noisy version of the true ${\mathb{x}}\xspace_{ 0}$). ${\boldsymbol{\theta}}\xspace$ contains the parameters of the observation (response and error), and its estimation is the core of the article. This graph already shows that if we know how to sample the ${\mathb{x}}\xspace_{ t}$ properly including the conditional independences encoded by this hierarchy, the difficulty of sampling ${\boldsymbol{\theta}}\xspace$ is greatly alleviated.
• Figure 2: Samples provided by the Gibbs algorithm for three unknown parameters, from top to bottom: $\iota$, $m_e$ and $v_e$. They are shown as a function of iteration index (left) and as histograms (right). They are samples of one dimensional marginal pdfs.. The green lines / dots give the true value. Quantitative results are given in Tab. \ref{['Tab:ParamEstimate']}.
• Figure 3: Point clouds for two dimensional marginals pdfs for three unknown parameters: $\iota$, $m_e$ and $v_e$. From left to right: $(m_e,v_e)$, $(v_e,\iota)$, and $(m_e,\iota)$. The sample are given in blue and the true value is given in green. See also Fig. \ref{['Fig:ParamChains']} for one dimensional plots and Tab. \ref{['Tab:ParamEstimate']} for quantitative assessment.
• Figure 4: Left to right: true image ${\mathb{x}}\xspace^\star$, measurements ${\mathb{y}}\xspace$ and estimated image $\widehat{{\mathb{x}}\xspace}$. The figure shows the images themselves (top) and cross-sections (bottom).
• Figure 5: Cross-sections of ${\mathb{x}}\xspace^\star$ (plain green) and the "uncertainty" intervals (dashed blue) centered on the estimate and of width two standard deviations.